Knot invariants and their applications to constructions of quasi ...

Knot invariants and their applicationsto constructions of quasi-morphismson groupsMichael Brandenbursky

Knot invariants and their applicationsto constructions of quasi-morphismson groupsResearch ThesisIn partial fulfillment of the requirements of theDegree of Doctor of PhilosophyMichael BrandenburskySubmitted to the Senate of theTechnion - Israel Institute of TechnologyIyar 5770 Haifa May 2010

The research was carried out under the supervision of ProfessorMichael Entov and Professor Michael Polyak in the Departmentof Mathematics.I wish to thank my advisors for suggesting the main problems,and for their constant help, both mathematical and nonmathematical,and encouragement during last four years.I also owe thanks to other people:To Professor Leonid Polterovich for many very helpful comments,and for giving me the opportunity to present a part of this workat the University of Chicago.To Professor Michah Sageev for suggesting the proof of Lemma1.3.5.To Sergei Lanzat for many valuable discussions.To my parents and my sister for their constant support.I wish to thank my wonderful wife Carolina, whom I love verymuch, for her endless support and for being there for me for allthese years.The generous financial support of the Technion—Israel Instituteof Technology is gratefully acknowledged.

ContentsAbstract 1Introduction. 31 Preliminaries in knot theory and symplectic geometry 71.1 Classical invariants of knots and links . . . . . . . . . . . . . . . 71.1.1 Knot signatures . . . . . . . . . . . . . . . . . . . . . . . 71.1.2 Concordance group, four ball genus and genus of a knot . 91.1.3 Rasmussen invariant s . . . . . . . . . . . . . . . . . . . 91.1.4 Ozsvath-Szabo invariant τ . . . . . . . . . . . . . . . . . 121.2 Polynomial and Vassiliev invariantsof classical and virtual links . . . . . . . . . . . . . . . . . . . . 131.2.1 Conway and HOMFLYPT polynomials . . . . . . . . . . 131.2.2 Long classical and virtual links . . . . . . . . . . . . . . 141.2.3 Finite type invariants of classical and virtual links . . . . 161.3 Braid groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.4 Quasi-morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . 201.5 Calabi homomorphism . . . . . . . . . . . . . . . . . . . . . . . 212 Quasi-morphisms on braid groups defined by knot invariants 232.1 Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.2 Examples and applications . . . . . . . . . . . . . . . . . . . . . 272.2.1 Rasmussen quasi-morphism . . . . . . . . . . . . . . . . 272.2.2 Connection between τ, s and braid number . . . . . . . . 302.2.3 Signature quasi-morphisms . . . . . . . . . . . . . . . . . 313 Generalized Gambaudo-Ghys construction 363.1 Gambaudo-Ghys construction . . . . . . . . . . . . . . . . . . . 36

3.2 Reeb graphs and computations forautonomous diffeomorphisms . . . . . . . . . . . . . . . . . . . . 403.2.1 Angle-action coordinates and the Reeb graph . . . . . . 413.2.2 Main Theorem . . . . . . . . . . . . . . . . . . . . . . . 433.3 Properties of induced quasi-morphisms on the group D . . . . . 493.4 Induced quasi-morphisms and Calabihomomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.4.1 Rasmussen quasi-morphisms on D . . . . . . . . . . . . . 543.4.2 Signature quasi-morphisms and Calabihomomorphism . . . . . . . . . . . . . . . . . . . . . . . 553.5 Open problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 584 Link invariants via counting surfaces 594.1 Gauss diagrams and arrow diagrams . . . . . . . . . . . . . . . 594.1.1 Gauss diagrams of classical and virtual links . . . . . . . 604.1.2 Arrow diagrams and Gauss diagram formulas . . . . . . 624.1.3 Surfaces corresponding to arrow diagrams . . . . . . . . 634.1.4 Ascending and descending arrow diagrams . . . . . . . . 644.1.5 Separating states . . . . . . . . . . . . . . . . . . . . . . 664.2 Counting surfaces with one boundarycomponent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.2.1 Invariants of long links . . . . . . . . . . . . . . . . . . . 694.2.2 Properties of A n,m (G) and D n,m (G) . . . . . . . . . . . . 714.2.3 Alexander-Conway polynomials of long virtual links . . . 744.3 Counting surfaces with two boundarycomponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794.3.1 Link invariants and diagrams with two boundary components. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794.3.2 Properties of I n . . . . . . . . . . . . . . . . . . . . . . . 864.3.3 The case of knots . . . . . . . . . . . . . . . . . . . . . . 90Bibliography 92

List of Figures1.1 Smoothing of a crossing . . . . . . . . . . . . . . . . . . . . . . 71.2 Seifert surface for the Hopf link obtained by the Seifert algorithm 81.3 Closure ̂α of a braid α . . . . . . . . . . . . . . . . . . . . . . . 121.4 Conway triple . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.5 Reidemeister moves . . . . . . . . . . . . . . . . . . . . . . . . . 151.6 Virtual Reidemeister moves . . . . . . . . . . . . . . . . . . . . 151.7 Connected sum of long trefoil knot and long Hopf link . . . . . . 161.8 Diagrams of classical, long, virtual and long virtual trefoil knots 161.9 Generator σ i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.1 The braid α = σ 1 · . . . · σ n−1 . . . . . . . . . . . . . . . . . . . . 282.2 Braid η i,n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.1 Reeb graph which corresponds to the function H with 3 maximumpoints and 2 saddle points . . . . . . . . . . . . . . . . . . 433.2 The braid η i,k,n . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.3 Curve c together with annuli A l and A l+1 . . . . . . . . . . . . . 474.1 Diagrams of based and long classical Hopf links together withthe associated Gauss diagrams . . . . . . . . . . . . . . . . . . . 604.2 Virtual trefoil and a virtual Hopf link with the correspondingGauss diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.3 Based and long virtual trefoil with the corresponding Gauss diagrams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.4 Reidemeister moves of Gauss diagrams . . . . . . . . . . . . . . 624.5 Connected arrow diagrams . . . . . . . . . . . . . . . . . . . . . 624.6 Constructing a surface from an arrow diagram . . . . . . . . . . 644.7 Ascending and descending labeling . . . . . . . . . . . . . . . . 674.8 Smoothing of an arrow . . . . . . . . . . . . . . . . . . . . . . . 67

4.9 A Conway triple of Gauss diagrams . . . . . . . . . . . . . . . . 714.10 Dependence on a basepoint . . . . . . . . . . . . . . . . . . . . . 724.11 The virtualization move . . . . . . . . . . . . . . . . . . . . . . 754.12 Diagrams of long virtual knots that differ by an application ofthe virtualization move . . . . . . . . . . . . . . . . . . . . . . . 764.13 A diagram of the Kishino knot . . . . . . . . . . . . . . . . . . . 764.14 Gauss diagrams of the Kishino knot . . . . . . . . . . . . . . . . 774.15 A mirror pair of Gauss diagrams . . . . . . . . . . . . . . . . . . 784.16 A virtual knot and two of its based Gauss diagrams . . . . . . . 794.17 Identifying ascending separating states containing α l . . . . . . 824.18 Identifying other ascending separating states containing α l . . . 824.19 Comparison of ascending separating states of G and ˜G . . . . . 834.20 Comparison of other ascending separating states of G and ˜G . . 844.21 Gauss diagrams G, G 1 and G 2 . . . . . . . . . . . . . . . . . . . 854.22 Correspondence of separating states of G 0 and G ± . . . . . . . . 884.23 Link H 2 and its Gauss diagram . . . . . . . . . . . . . . . . . . 904.24 Two versions of the Reidemeister move Ω 1 . . . . . . . . . . . . 91

AbstractIn this work we study knot and link invariants and their applications to constructionsof quasi-morphisms on braids groups and the group D of area-preservingcompactly supported diffeomorphisms of a two-dimensional open disc.In the first part of the thesis, which is independent from the subsequentchapters, we give a sufficient criterion for a knot/link invariant to define quasimorphismson braid groups, provided a certain way of closing braids into knotsor links. We then discuss a generalized Gambaudo-Ghys construction whichallows to build quasi-morphisms on D. In particular, we study quasi-morphismson braid groups and D defined in this way by knot and link invariants comingfrom the knot Floer homology and Khovanov-type link homology. We alsocompute the values of quasi-morphisms obtained by this construction on thetime-one flow of a generic time-independent Hamiltonian H in terms of theReeb graph of H.In the second part of the thesis we consider link invariants arising fromthe Conway and HOMFLYPT polynomials. It is known that any Vassilievinvariant may be obtained from a Gauss diagram formula that involves counting(with signs and multiplicities) subdiagrams of certain geometric-combinatorialtypes. These formulas generalize the calculation of a linking number by countingsigns of crossings in a link diagram. Until recently, explicit formulas of thistype were known only for few invariants of low degrees. We generalize theresult of Chmutov-CapKhoury-Rossi and present Gauss diagram formulas for allcoefficients of ∇(L), where L is a link with arbitrary number of components, and∇(L) is the Conway polynomial of L. We discuss an interesting interpretationof these formulas in terms of counting surfaces of a certain genus and with oneboundary component. We also present two different extensions of the Conwaypolynomial to long virtual links. We compare these extensions with the existingextensions of the Alexander and Conway polynomials and show that they arenew.1

In the remaining part of the thesis we modify the Chmutov-CapKhoury-Rossi construction and present Gauss diagram formulas for the coefficients ofthe first partial derivative of the HOMFLYPT polynomial, with respect to thevariable a, evaluated at a = 1. These formulas are related, in a similar way, toa certain count of orientable surfaces with two boundary components. At theend we present a modification of these formulas in case of knots.2

Introduction.Knot theory is a field of mathematics which studies knots and links in 3-manifolds. Being of its own interest, knot theory plays an important role inthe study of braid groups and low-dimensional topology and has many applicationsin symplectic geometry and other mathematical fields. Invariants of linksin the 3-sphere, S 3 , are basic tools of knot theory which have been extensivelystudied for many years.Among the most intriguing link invariants there are Alexander, Jones andHOMFLYPT polynomials. The Alexander polynomial is a classical invariantof links in S 3 (see e.g. [2], [40]). For many years, before the discovery of theJones polynomial in the 1980s (see e.g. [33], [34]), it was thought to be theunique polynomial invariant, i.e a polynomial whose coefficients are link invariants.The HOMFLYPT polynomial (see e.g. [24], [41], [54]), discoveredshortly after the Jones polynomial, is a 2-variable polynomial which reducesto Alexander and Jones 1-variable polynomials by certain changes of variables.Recently new homological knot invariants were discovered by Ozsvath-Szabo[49], Rasmussen [55], Khovanov [36] and Khovanov-Rozansky [37]. These homologicaltheories generalize Alexander, Jones and HOMFLYPT polynomialsin the following sense: their graded Euler characteristics are equal (up to somenormalization) to Alexander, Jones and HOMFLYPT polynomials respectively.In the first part of the thesis we discuss link invariants, arising from knotFloer homology and Khovanov homology. We investigate some of the applicationsof these invariants in the study of braid groups and in symplectic geometry.In the second part of the thesis we establish a connection between invariantscoming from the HOMFLYPT polynomial and orientable surfaces.The results in the first part are based on a connection between link invariantsand quasi-morphisms on braid groups. Recall that a quasi-morphism on a groupG is a function φ : G → R such that for some K φ ≥ 0 one has|φ(ab) − φ(a) − φ(b)| ≤ K φ3

for all a, b ∈ G. A quasi-morphism φ is called homogeneous if φ(a m ) = mφ(a)for all a ∈ G and m ∈ Z. Any quasi-morphism φ can be homogenized: setting˜φ(a) :=limk→+∞ φ(ak )/k (1)we get a homogeneous (possibly trivial) quasi-morphism ˜φ. Quasi-morphismsare known to be a helpful tool in the study of non-Abelian groups, especially theones that admit a few or no (linearly independent) real-valued homomorphisms.In this work we discuss quasi-morphisms on the groups P n of pure braids onn strings. These quasi-morphisms on P n are constructed from certain invariantsof n-component links in R 3 in the following way: close up a pure braid to alink in the standard way and take the value of the invariant on that link. Weformulate sufficient conditions for a knot invariant to yield a quasi-morphismon the full braid group B n on n strings and hence on P n . We also considertwo specific remarkable knot/link invariants: the Rasmussen link invariant s[56], which comes from a Khovanov-type theory, and the Ozsvath-Szabo knotinvariant τ [50], which comes from the knot Floer homology. In [4] Baader hasshown that the Rasmussen link invariant s defines a quasi-morphism on B n . Weshow that the homogenization of this quasi-morphism is equal to the classicallinking number homomorphism on B n . We also show that the situation withthe Ozsvath-Szabo invariant τ is similar: it defines a quasi-morphism on B nand its homogenization is again the linking number homomorphism multipliedby 2. We also present an inequality which connects s, τ and the braid numberof a knot.Later we discuss the group D of compactly supported area-preserving diffeomorphismsof a unit open two-dimensional disc in the Euclidean plane. Thegroup D admits a unique (continuous, in the proper sense) homomorphismto the reals – the famous Calabi homomorphism (see e.g. [5], [13], [25]). Atthe same time D is known to admit many (linearly independent) homogeneousquasi-morphisms (see e.g. [8], [26], [11]). In this thesis we consider a particularconstruction of such quasi-morphisms, due to Gambaudo and Ghys [26]. This isan explicit geometric construction which produces quasi-morphisms on D fromquasi-morphisms on the groups P n . To show that the homogenizations of theresulting quasi-morphism on D is non-trivial, we follow Gambaudo and Ghysand evaluate it on the diffeomorphism generated by a certain time-independentHamiltonian function on the disc.We discuss the applications of the Gambaudo-Ghys construction to thequasi-morphisms on P n defined by τ, s and ω-signature knot/link invariants.4

In the first two cases the resulting homogeneous quasi-morphisms on D are homomorphisms,equal, up to a non-zero multiplicative constant, to the Calabihomomorphism. In the third case many of the resulting homogeneous quasimorphismsare not homomorphisms.We also discuss the computation of the quasi-morphisms on D, obtainedby Gambaudo-Ghys construction, on diffeomorphisms generated by time-independent(compactly supported) Hamiltonians. For a generic Hamiltonian Hof this sort we present the result of the computation in terms of the Reebgraph of H and the integral of the push-forward of H to the graph against acertain signed measure on the graph. This result enables us to show that theCalabi homomorphism and the quasi-morphism on D induced by the signatureinvariant of n-component links are asymptotically equivalent, as n → ∞, onthe flows generated by time-independent (compactly supported) Hamiltonians.In the second part of the thesis, based on a joint work with M. Polyak, weconsider link invariants arising from the Conway and HOMFLYPT polynomials.The HOMFLYPT polynomial P (L) is an invariant of an oriented link L (seee.g. [24], [41], [54]). It is a Laurent polynomial in two variables a and z, whichsatisfies the following skein relation:( ) ( ) ( )aP − a −1 P = zP .+The HOMFLYPT polynomial is normalized(in)the following way. If O m is anm−1.m-component unlink, then P (O m ) =The Conway polynomial−a−a −1z∇ may be defined as ∇(L) := P (L)| a=1 . This polynomial is a renormalizedversion of the Alexander polynomial (see e.g. [20], [40]). All coefficients of ∇are so-called finite type or Vassiliev invariants.One of the mainstream and simplest techniques for producing Vassiliev invariantsare so-called Gauss diagram formulas (see [30], [53]). These formulasgeneralize the calculation of a linking number by counting subdiagrams ofspecial geometric-combinatorial types with signs and weights in a given linkdiagram. This technique is also very helpful in the rapidly developing field ofvirtual knot theory (see [35]), as well as in 3-manifold theory (see [44]).Until recently, explicit formulas of this type were known only for few invariantsof low degrees. The situation has changed with works of Chmutov-CapKhoury-Rossi [15] and Chmutov-Polyak [17]. In [15] Chmutov-CapKhoury-Rossi presented an infinite family of Gauss diagram formulas for all coefficientsof ∇(L), where L is a knot or a two-component oriented link. We explain how50

each formula for the coefficient c n of z n is related to a certain count of orientablesurfaces of a certain genus, and with one boundary component. The genus dependsonly on n and the number of the components of L. These formulas maybe viewed as a certain combinatorial analog of Gromov-Witten invariants.In this work we generalize the result of Chmutov-CapKhoury-Rossi to linkswith arbitrary number of components. We present a direct proof of this result,without any prior assumption on the existence of the Conway polynomial. Itenables us to present two different extensions of the Conway polynomial tolong virtual links. We compare these extensions with the existing versions ofthe Alexander and Conway polynomials for virtual links, and show that theyare new. In particular, we give a new proof of the fact that the famous Kishinoknot K T [38] is non-classical, by calculating these polynomials for K T .This leads to a natural question: how to produce link invariants by countingorientable surfaces with an arbitrary number of boundary components? In thisthesis we deal with a model case, when the number of boundary componentsis two. We modify Chmutov-CapKhoury-Rossi construction and present aninfinite family of Gauss diagram formulas for the coefficients of the first partialderivative of the HOMFLYPT polynomial, w.r.t. the variable a, evaluated ata = 1. This family is related, in a similar way, to the family of orientablesurfaces with two boundary components. We conjecture that a similar countof orientable surfaces with n boundary components (up to some normalization)will give an infinite family of Gauss diagram formulas for the coefficients of then−1 derivative, w.r.t. the variable a in the HOMFLYPT polynomial, evaluatedat a = 1. At the end we present a modification of these formulas in case ofknots.A challenging problem is to try to build a theory similar to knot Floer homology,which will categorify the Conway polynomial. While we get the Conwaypolynomial from the Alexander polynomial by the change z = t − 1 2 − t 1 2 of variables,it is not clear how to ”change variables” in knot Floer homology. It seemsplausible that the first step in this direction would be a better understanding ofthe connection between orientable surfaces and the coefficients of the Conwaypolynomial. We hope that the techniques described in this thesis will help todeal with this problem.6

Chapter 1Preliminaries in knot theory andsymplectic geometryIn this chapter we shall recall the basic notions and facts in knot theory andsymplectic geometry that will be repeatedly used in the following chapters.1.1 Classical invariants of knots and linksIn this section we recall some classical and new invariants of knots and links inS 3 .1.1.1 Knot signaturesIn this subsection we recall the definition of the signature link invariant.Let L be an oriented link in S 3 , then there exists an oriented surface S Lwith boundary L. Each such orientable surface S L is called a Seifert surface ofL. For the reader’s convenience we recall the Seifert algorithm for constructionof Seifert surfaces [40]. Let D be a diagram of an oriented link L. Let usFigure 1.1: Smoothing of a crossing7

smoothen each crossing in D as shown in Figure 1.1. The resulting smootheddiagram ̂D consists of oriented simple closed curves, which are called the Seifertcircles. Thus ̂D is the boundary of a union of disjoint discs. We join these discstogether with half-twisted strips corresponding to the crossings in the diagram.This yields an oriented surface bounded by L. If this surface is disconnected,then we connect its components by the connected sum operation.Example 1.1.1. In Figure 1.2 we present a diagram of the positive Hopf linkL together with a Seifert surface S L obtained by the Seifert algorithm.Figure 1.2: Seifert surface for the Hopf link obtained by the Seifert algorithmWe choose a basis {a 1 , . . . , a 2g+|L|−1 } in H 1 (S L , Z). There exists a symmetricbilinear form on H 1 (S L , Z), which is defined as follows:Ω(a i , a j ) := lk(a i , a + j ) + lk(a j, a + i ),where lk is the linking number and a + i is a push of the curve, which representsa i in S L , from S L along the positive normal direction to S L . Tensoring by Rwe get a symmetric bilinear form on H 1 (S L , R). The signature of this form isindependent of the choices of the Seifert surface S L and the basis of H 1 (S L , Z),see [40]. Thus it is an invariant of L and is denoted by sign(L).For any complex number ω ≠ 1 and link L there exists ω-signature linkinvariant sign ω (L), such that sign −1 (L) = sign(L). It is defined as follows. Wetensor the bilinear form Ω by C, and we get a bilinear form on H 1 (S L , C). Thesignature of the following hermitian form on H 1 (S L , C)Ω ω (a i , a j ) = (1 − ω)Ω(a i , a j ) + (1 − ω)Ω(a j , a i )is independent of the choice of the Seifert surface S L and the base for H 1 (S L , Z),see [40]. Thus sign ω (L) is a link invariant for each ω ≠ 1.8

1.1.2 Concordance group, four ball genus and genus ofa knotLet K be a knot in S 3 . The genus of a knot K is the minimal genus of anySeifert surface S K . It is denoted by g(K). The four-ball genus of a knot K in S 3is the minimal genus of an oriented surface with boundary, which is embeddedin D 4 such that its boundary is a knot K ⊂ S 3 = ∂D 4 . It is denoted by g 4 (K).Note that g 4 (K) ≤ g(K).For any knot K, the knot K ∗ denotes its mirror image, and the knot −Kdenotes the same knot with the reversed orientation. For any two knots K 0and K 1 the knot K 0 #K 1 represents their connected sum. We say that K 0 isequivalent to K 1 , if there exists an embedding Ψ : S 1 × [0, 1] → S 3 × [0, 1]such that Ψ(S 1 × {0}) = K 0 and Ψ(S 1 × {1}) = K 1 . We denote by Conc(S 3 )the Abelian group whose elements are equivalent classes of knots in S 3 , andthe multiplication is the connected sum operation. It is easy to see that themultiplication is well-defined. The following lemma is a well-known fact in knottheory, see e.g. [40, 57]. For the reader’s convenience we will prove this lemma.Lemma 1.1.2. For any knot K the knot K# − K ∗ bounds an embedded discin D 4 .Proof. Let Ψ : S 1 × [0, 1] → S 3 × [0, 1] be an embedding such thatΨ(S 1 × {t}) = K × {t}for every t ∈ [0, 1]. There exists an arc ξ in S 1 such that Ψ((S 1 \ ξ) × [0, 1]) ishomeomorphic to a disc D 2 and bounds K# − K ∗ in D 4 . We can smooth thecorners of (S 1 \ξ)×[0, 1] so that Im(Ψ) of the resulting manifold is diffeomorphicto D 2 and bounds K# − K ∗ in D 4 .It follows from Lemma 1.1.2 that for any knot K the equivalence class of theknot −K ∗ represents the inverse element in Conc(S 3 ). The identity element inConc(S 3 ) is the equivalence class of the unknot.A knot K in S 3 = ∂D 4 is called slice if there exists an embedding of the unitdisc D 2 ⊂ R 2 in D 4 such that the image of an embedding of ∂D 2 is the knot K.1.1.3 Rasmussen invariant sThe Rasmussen invariant s(K) of a knot K in S 3 was discovered by JacobRasmussen in 2004, see [56]. It comes from the Lee theory [39] which is closely9

elated to the Khovanov homology [36]. This is a very powerful knot invariantwhich, for example, gives a lower bound on the four-ball genus of a knot. Itwas also used by Rasmussen to give the first combinatorial proof of the Milnorconjecture, which states that g(K) = g 4 (K) for any torus knot K. Recently thisinvariant was extended to links by Beliakova and Wehrli [10]. For the reader’sconvenience we will recall the definition and some properties of s.Let D be a diagram of an oriented link L. In [36] Khovanov associated toD a bigraded chain complex ( ⊕ C i,j (D), d) over Q, such thati,jd : C i,j (D) → C i+1,j (D).It turns out that the homology groups of this complex are invariants of L andthe graded Euler characteristic is equal to the unnormalized Jones polynomial.For more information about Khovanov homology see e.g. [6, 62].In [39] Lee associated to D a chain complex CLee ∗ (D) := (⊕ C i (D), d Lee )iover Q, such that d Lee : C i (D) → C i+1 (D) and C i (D) = ⊕ C i,j (D). Shejproved that the homology of this complex is an invariant of L. In addition sheshowed that Rank(HLee ∗ (L)) = 2m , where m is the number of components of L.She also showed that any orientation θ of L defines a cycle s θ in CLee ∗ (D) andthere is an explicit isomorphismH ∗ Lee(L) ∼ = Q{[s θ ]| θ is an orientation of L}.In Lee theory the differential d Lee does not respect the j grading. Everyelement u ∈ C i (D) can be written as a sum of monomials u = u 1 + · · · + u k .We setj(u) := min{j(u i )| i = 1, . . . , k}.Thus j defines a filtration on CLee ∗ (D) by settingF r C ∗ Lee(D) := {u ∈ C ∗ Lee(D)| j(u) ≥ r}.It follows from the definition of d Lee that it is a filtered map for a filtration F ron H ∗ Lee (L) defined as follows: a class x ∈ H∗ Lee (L) is in F r , if and only if, it hasa representative which is an element of F r C ∗ Lee (D). Thus for each x ∈ H∗ Lee (L)we defines(x) := max{j(u)| [u] = x}and F r := {x ∈ H ∗ Lee(L)| s(x) ≥ r}.10

For an oriented knot K defines min (K) := min{s(x)| x ∈ H ∗ Lee(K), x ≠ 0},s max (K) := max{s(x)| x ∈ H ∗ Lee(K), x ≠ 0}.The Rasmussen invariant s(K) is defined as followss(K) := s min(K) + s max (K).2It turns out that s(K) = s min (K) + 1.If θ is an orientation of L then we denote by θ the reversed orientation ofeach component of L. In [10] Beliakova and Wehrli defineds(L) := min(s([s θ ] + [s θ]), s([s θ ] − [s θ])) + 1.It follows from Lemma 3.5 in [56] that the above invariant coincides with theRasmussen invariant s when L is a knot.It has many interesting properties. We list here eight of them:s(K) = s(−K), (1.1)s(K 1 #K 2 ) = s(K 1 ) + s(K 2 ), (1.2)s(K ∗ ) = −s(K), (1.3)|s(K)| ≤ 2g 4 (K), (1.4)s(K − ) ≤ s(K + ) ≤ s(K − ) + 2, (1.5)where K + and K − are knots that differ by a single crossing change: from apositive crossing in K + to a negative one in K − (positive/negative crossings areshown in Figure 1.4a).For an alternating knot K we haveFor a positive link Ls(K) = sign(K). (1.6)s(L) = w(D(L)) − o(D(L)) + 1, (1.7)11

where D(L) is some positive diagram of L, w(D(L)) is the number of positivecrossings in D(L) minus the number of negative crossings in D(L), and o(D(L))is the number of Seifert circles. When the Seifert algorithm is applied to D(K)we have the following inequality1 + w(D(K)) − o(D(K)) ≤ s(K) ≤ −1 + w(D(K)) + o(D(K)), (1.8)where D(K) is any diagram of a knot K, see [4]. In fact, it is shown in [4] thatproperty (1.8) follows from properties (1.3), (1.5) and (1.7).01 01 01α01 01 01Figure 1.3: Closure ̂α of a braid α1.1.4 Ozsvath-Szabo invariant τIn [50] Ozsvath and Szabo defined a knot invariant τ which comes from the knotFloer homology [43, 49, 55]. This invariant satisfies in particular the followingproperties:τ(K ∗ ) = −τ(K), (1.9)τ(K 1 #K 2 ) = τ(K 1 ) + τ(K 2 ), (1.10)τ(−K) = τ(K), (1.11)|τ(K)| ≤ g 4 (K), (1.12)0 ≤ τ(K + ) − τ(K − ) ≤ 1, (1.13)τ(̂α) = w(̂α) − n + 1 , (1.14)212

where α ∈ B n is a positive braid (see Definition 1.3.2), such that ̂α is a knot.By ̂α we denote the link in S 3 which is obtained in the natural way from α, seeFigure 1.3. The first five properties were proved in the paper of Ozsvath andSzabo [50], and the last property was proved by Livingston in [42].1.2 Polynomial and Vassiliev invariantsof classical and virtual linksIn this section we recall the definitions of some knot polynomials and finite typeinvariants of classical and virtual knots.1.2.1 Conway and HOMFLYPT polynomialsConway polynomialThe Conway polynomial ∇(L) is an invariant of an oriented link L, see e.g.[24, 41, 54]. It is a polynomial in variable z, which satisfies the following skeinrelation. Denote by L + , L − and L 0 a triple of links with diagrams which are+ −a0bFigure 1.4: Conway tripleidentical except for a small fragment, where L + and L − have a positive and anegative crossing respectively, and L 0 has a smoothed crossing, see Figure 1.4.Such a triple of links is called a Conway triple. The Conway polynomial ∇(L)satisfies∇(L + ) − ∇(L − ) = z∇(L 0 )It is normalized by ∇(O) = 1, where O is the unknot. A changez = t − 1 2 − t 1 2 of variables turns ∇(L) into the Alexander polynomial ∆(L). Formore information about these polynomials see [20, 40].13

HOMFLYPT polynomialThe HOMFLYPT polynomial P (L) is an invariant of an oriented link L, seee.g. [24, 41, 54]. It is a Laurent polynomial in two variables a and z whichsatisfies the following skein relation:aP (L + ) − a −1 P (L − ) = zP (L 0 ). (1.15)The HOMFLYPT is normalized by P (O m ) =( )a−a m−1, −1z where Om is anm-component unlink.Remark 1.2.1. Note that P (L)| a=1 = ∇(L) for any oriented link L in S 3 .1.2.2 Long classical and virtual linksDefinition 1.2.2. By a long m-component link diagram we mean a smoothimmersion f : R ⊔ S 1 ⊔ . . . ⊔ S 1 → R 2 such that1) it coincides with a standard embedding of y-axis outside of the unit discD 2 ⊂ R 2 ;2) each intersection point is double and transverse;3) each intersection point is endowed with classical (with a choice for underpassand overpass specified) crossing structure.Definition 1.2.3. A long link is an equivalence class of long link diagramsmodulo Reidemeister moves, see Figure 1.5.Definition 1.2.4. A virtual link diagram with m components is a generic immersionof m disjoint circles into the plane, with double points divided intoreal crossing points and virtual crossing points, with the real crossing pointsenhanced by information on overpasses and underpasses (as for classical linkdiagrams). At a virtual crossing the branches are not divided into an overpassand an underpass.A virtual crossing is denoted by. Two virtual link diagrams are equivalentif they may be transformed one to another by finite sequence of Reidemeistermoves and virtual Reidemeister moves, see Figure 1.5 and Figure 1.6respectively. Two virtual link diagrams represent the same virtual link, if onecan be obtained from the other by a sequence of these moves.14

Figure 1.5: Reidemeister movesFigure 1.6: Virtual Reidemeister movesDefinition 1.2.5. By a long virtual m-component link diagram we mean asmooth immersion f : R ⊔ S 1 ⊔ . . . ⊔ S 1 → R 2 such that1) it coincides with a standard embedding of y-axis outside of the unit discD 2 ⊂ R 2 ;2) each intersection point is double and transverse;3) each intersection point is endowed with classical (with a choice for underpassand overpass specified) or virtual crossing structure.Definition 1.2.6. A long virtual link is an equivalence class of long virtual linkdiagrams modulo Reidemeister and virtual Reidemeister moves.Definition 1.2.7. Let L 1 and L 2 two long (virtual) links, then the connectedsum L 1 #L 2 is a concatenation of L 1 and L 2 : just place a diagram of L 2 aftera diagram of L 1 , see Figure 1.7.Example 1.2.8. Figure 1.8 shows diagrams of classical, long, virtual and longvirtual trefoil knots.15

# =Figure 1.7: Connected sum of long trefoil knot and long Hopf linkFigure 1.8: Diagrams of classical, long, virtual and long virtual trefoil knotsOur motivation to study virtual links comes from the study of links in thickenedsurfaces of higher genus. Let Σ g be a surface of genus g and [0, 1] be theunit interval. The knot theory in Σ g × [0, 1] is represented by diagrams drawnon Σ g taken up to the usual Reidemeister moves transferred to diagrams on thissurface. Abstract invariants of virtual links can be interpreted as invariants oflinks that are embedded in Σ g × [0, 1] for some genus g, see e.g. [35].1.2.3 Finite type invariants of classical and virtual linksIn this subsection we recall definitions of finite type or Vassiliev invariants ofclassical and virtual knots. All these definitions are naturally extended to links.16

We start with a case of classical knots.Finite type invariants of classical knotsDefinition 1.2.9. Consider immersed closed curves in R 3 with n transversedouble points, i.e. each such immersion is an embedding when restricted tothe complement of the preimages of these double points. Equivalence classes ofsuch curves under the action of isotopies of R 3 are called singular knots with ntransverse double points.Let G be an Abelian group. A G-valued knot invariant v is extended tosingular knots by induction on the number of double points using the inductionformula:( ) ( ) ( )v = v − v . (1.16)Definition 1.2.10. A G-valued knot invariant v is said to be of finite type orVassiliev, if for some n ∈ N it vanishes for any knot K with more than n doublepoints. A minimal such n is called the degree of v.Examples 1.2.11. (i) Let c n be the coefficient of z n in the Conway polynomial∇. It follows from [7] that c n is a finite type invariant of degree n.(ii) Another family of finite type invariants is derived from the HOMFLYPTpolynomial in the following way. We make a substitution a = e h and take theTaylor expansion in h. The result will be a Laurent polynomial in z and apower series in h. Let p k,n be its coefficient at h k z n . Goussarov proved in [29]that p k,n is a finite type invariant of degree ≤ k + n.The extension of Vassiliev invariants to virtual knots is not unique. In thefollowing two subsections we recall the definitions of two different extensions offinite type invariants to virtual knots.Kauffman FTIIn [35] Kauffman defined the class of 4-valent graphs modulo rigid vertex isotopy(see [35] for the precise definition). A G-valued virtual knot invariant v isextended to this class by induction on the number of vertices using the inductionformula (1.16).Definition 1.2.12. A G-valued virtual knot invariant v is said to be of graphicalfinite type or Vassiliev invariant, if for some n ∈ N it vanishes for any 4-valentgraph Γ with more than n vertices. A minimal such n is called the degree of v.+17−

Note that this definition says nothing about the number of virtual crossingsin any such Γ.Example 1.2.13. The coefficient at x n in the power series expansion of thesubstitution a = e x of the Jones-Kauffman polynomial is known to be of Kauffmanfinite-type of degree ≤ n, see [35].Goussarov-Polyak-Viro FTIIn [30] Goussarov, Polyak and Viro introduced another extension of Vassilievinvariants to virtual links. It can be described as follows. They introduced anew kind of crossing, which is called semi-virtual. At a semi-virtual crossingthere are still over- and under-passes. In a diagram a semi-virtual crossingis shown as a real one, but surrounded by a small circle. Virtual knots areextended to knots having semi-virtual crossings. Any virtual knot invariant vcan be extended to this larger class by applying the following formal relation:( ) ( ) ( )v = v − v . (1.17)Definition 1.2.14. A virtual knot invariant v is called of GPV finite-type, iffor some n ∈ N it vanishes for any virtual knot K with more than n semi-virtualcrossings. The minimal such n is called the degree of v.Example 1.2.15. Let L be a two component virtual link. The invariantlk 1/2 (L) (lk 2/1 (L) respectively) is defined to be the sum of the signs of thereal crossings of L where the first component (respectively the second component)passes over the second one (respectively the first one). Both lk 1/2 andlk 2/1 are of GPV finite-type of degree one.Remark 1.2.16. Equations (1.16) and (1.17) imply:( ) ( ) ( )v = v − v . (1.18)It follows that for any GPV-finite type invariant, its restriction to classical knotsis a finite type invariant (of at most the same degree) in the classical sense.Note that every GPV finite-type invariant of degree ≤ n is of Kauffmanfinite-type of degree ≤ n, but the converse in general is not true, see [35].18

1.3 Braid groupsIn this section we recall a definition and some properties of braid groups.Definition 1.3.1. The full braid group B n on n strings is abstractly definedvia the following presentation:B n = ⟨σ 1 , . . . , σ n−1 | Rel i,j , if |i − j| ≥ 2, and Rel i for 1 ≤ i ≤ n − 2⟩,where Rel i,j and Rel i are, respectively, the relations σ i σ j = σ j σ i and σ i σ i+1 σ i =σ i+1 σ i σ i+1 .Definition 1.3.2. A braid α ∈ B n is called positive if it can be written as aproduct of only non-negative powers of the standard generators of B n .Each generator σ i of B n has a geometric interpretation, see Figure 1.9.Definition 1.3.3. The braid length l(γ) of γ ∈ B n is the length of the shortestword representing γ with respect to the generators σ 1 , . . . , σ n−1 .ii+1Figure 1.9: Generator σ iThe full braid group B n can be identified with the fundamental groupπ 1 (Y n , z) of the space Y n of the unordered n-tuples of distinct points in D 2 ,where z = (z 1 , . . . , z n ) ∈ Y n is a base point in Y n . Each closed path α(t) ∈ Y n ,such that α(0) = α(1) = z, is given by n-tuple of pathes (α 1 (t), . . . , α n (t)) in(D 2 ) ×n , such that α i (t) ≠ α j (t)) for all i ≠ j, α i (0) = z i and for each i thereexists unique j such that α i (1) = z j . Thus each braid [α(t)] ∈ π 1 (Y n , z) definesan element s α in a permutation group S n by setting s α (z i ) = α i (1). Thisassignment defines a surjective homomorphism B n → S n . The kernel of thishomomorphism is called the pure braid group and is denoted by P n . Note thatP n can be identified with the fundamental group π 1 (X n , z) of the space X n ofthe ordered n-tuples of distinct points in D 2 , where z = (z 1 , . . . , z n ) ∈ X n is abase point in X n .19

Proposition 1.3.4 ([18]). The center Z(B n ) of B n coincides with the centerZ(P n ) of P n . It is a cyclic group generated by the full twist (σ 1 · . . . · σ n−1 ) n .The following lemma will be needed in Section 3.1.Lemma 1.3.5. Let S be any finite generating set for P n . Then there exist tworeal positive constants K 1,S and K 2,S which depend only on S, n and σ 1 , . . . , σ n−1so thatl S (γ) ≤ K 1,S · l(γ) + K 2,S ,where l S (γ) is the length of γ with respect to S.Proof. Note that P n is a finite index subgroup of B n . Now the proof followsdirectly from [21, Corollary 24].1.4 Quasi-morphismsIn this section we recall the definition and basic properties of quasi-morphisms.For more information about quasi-morphisms see [9] and [14].Definition 1.4.1. Let G be a group. A function φ : G → R is called a quasimorphismif there exists K φ ≥ 0 such that|φ(gh) − φ(g) − φ(h)| ≤ K φfor all g, h ∈ G. The smallest of such K φ is called the defect of φ and will bedenoted by D φ .Examples 1.4.2. (i) Bounded functions and homomorphisms are quasi-morphisms.(ii) Let F n be a free group on n generators x 1 , . . . , x n . Brooks [12] showed thatthere are many quasi-morphisms on F n which are not homomorphisms. Namelylet ω be a reduced word in x 1 , . . . , x n and x −11 , . . . , x −1n . Define ρ ω : F n → Zby setting ρ ω (x) to be the difference between the number of times ω and ω −1occur as subwords of x when x is written as a reduced word in x 1 , . . . , x n andx −11 , . . . , x −1 . Then ρ ω is a quasi-morphism.nDefinition 1.4.3. A quasi-morphism φ is called homogeneous if for every g ∈ Gand m ∈ Zφ(g m ) = mφ(g).20

Any quasi-morphism φ can be homogenized: setting˜φ(g) :=limk→+∞ φ(gk )/k (1.19)we get a homogeneous (possibly trivial) quasi-morphism ˜φ. Throughout thethesis the induced homogeneous quasi-morphism obtained by the homogenizationof φ will be denoted by ˜φ.Now we recall some properties of homogeneous quasi-morphisms, see e.g.[14]:˜φ(hgh −1 ) = ˜φ(g) ∀g, h ∈ G (1.20)˜φ(gh) = ˜φ(g) + ˜φ(h) ∀g, h ∈ G such that gh = hg. (1.21)Given a group G, its first derived subgroup is the subgroup G ′ generatedby commutators [g, h] = ghg −1 h −1 of elements g, h ∈ G, i.e. each element g ′ ofG ′ can be written as a product ∏ ki=1 [g i, h i ]. The smallest integer k for whichsuch an expression exists is called the commutator length of g ′ and is denotedby comm(g ′ ). Let us define the stable commutator length of g ′ by setting||g ′ 1|| := limn→∞ n comm((g′ ) n ).Lemma 1.4.4 ([9]). Let φ : G → R be a non-trivial homogeneous quasimorphismwith a defect D φ ≠ 0. Then the stable commutator length function|| · || : G ′ → N is non-trivial.In [9], Bavard showed that if || · || is a non-trivial function, then there existsa non-trivial homogeneous quasi-morphism φ with D φ ≠ 0.1.5 Calabi homomorphismRecall that D := Diff ∞ (D 2 , ∂D 2 , area) stands for the group of area-preservingdiffeomophisms of the unit closed two dimensional disc D 2 in the Euclideanplane which are identical near the boundary ∂D 2 . In this section we recall thedefinition and properties of the Calabi homomorphism C : D → R. Denote byω the standard symplectic (i.e. the area) form on D 2 ⊂ R 2 .Definition 1.5.1. For every g ∈ D and a differential 1-form λ on D 2 , such thatdλ = ω, there exists a unique function H : D 2 → R, such that H is zero near21

the boundary and dH = g ∗ λ − λ. Define C : D → R by∫C(g) = Hω.This map is well defined and it is a homomorphism, see [5, 13], cf. [25]. It iscalled the Calabi homomorphism.D 2The celebrated theorem of Banyaga [5] states that ker(C) is a simple group.It follows that every homomorphism µ : D → R is given by µ = κ ◦ C, whereκ : R → R is a homomorphism. Note that if κ is continuous then there existsa real constant K such that µ = KC.Let H : D 2 × R → R be a C ∞ -function such that H t is zero near ∂D 2 foreach t ∈ R. Here H t := H(−, t). The Hamiltonian vector field X Ht is definedby the following equation:dH t (−) = ω(−, X Ht ).The flow ϕ t H generated by the vector field X H tis called the Hamiltonian flow.For each t the diffeomorphism ϕ t H is identity near ∂D2 and (ϕ t H )∗ ω = ω. Wesay that the diffeomorphism ϕ 1 H is the diffeomorphism generated by H. Anarea-preserving diffeomorphism ϕ is called autonomous if ϕ = ϕ 1 H for sometime-independent Hamiltonian H.Proposition 1.5.2 ([25]). Let ϕ t H be an autonomous Hamiltonian flow generatedby H : D 2 → R. Then∫C(ϕ t H) = −2t Hω.Proposition 1.5.3 ([5, 25]). The Calabi homomorphism C is not continuousin the C 0 -topology on D, but it is continuous in C 1 -topology.D 222

Chapter 2Quasi-morphisms on braidgroups defined by knotinvariantsIt is shown in [31] that the group B n admits infinitely many linearly independenthomogeneous quasi-morphisms for every integer n > 2. However none of thesequasi-morphisms are constructed geometrically. In [26] Gambaudo and Ghysgave a geometric construction of a family of quasi-morphisms sign n on groupsB n defined as follows:sign n (α) := sign(̂α),where ̂α is an n-component link obtained by closing α in the standard way (seeFigure 1.3). In [26] Gambaudo and Ghys proved that sign n is a quasi-morphismwith a defect bounded by n − 1. In this chapter we show that knot invariantsof certain type define quasi-morphisms on B n in a similar way.2.1 Main TheoremOne of the ways to find a quasi-morphism on B n is to find an R−valued invariantI of (isotopy classes of) links in S 3 so that|I(̂αβ) − I(̂α) − I(̂β)| ≤ K I ,where K I ≥ 0 depends only on I.In this section we will formulate a sufficient condition for a knot invariantunder which it yields a quasi-morphism on B n .23

We will now describe certain ways of closing braids into knots.Lemma 2.1.1. Let β ∈ B n . Then there exists a braid α β ∈ B n which satisfiesthe following properties:1. The closure of α β β is a knot.2. The closure of α β is a k-component unlink for some 1 ≤ k ≤ n.We will say that such a braid α β is a completing braid for β.Proof. Let β ∈ B n . Then there exists a braid α such that ̂αβ is a knot and̂α is a k-component link for some 1 ≤ k ≤ n. We write α as a product ofgenerators σ ±1i and perform a sequence of crossing changes (i.e. replacing someof the generators in the product by their inverses) until we are left with a braidα β such that ̂αβ = ̂α β β is a knot and ̂α β is the k-component unlink.Definition 2.1.2. Let I be a real-valued knot invariant. Let us fix some choicesof completing braids α β for every β ∈ B n . Define a functionby Î(β) := I( ̂α β β).Î : B n → R,We will show that under certain conditions Î is a quasi-morphism.Remark 2.1.3. Note that our function Î : B n → R is slightly different froman analogous quasi-morphism defined by Gambaudo-Ghys. Their definition(Î(β) := I(̂β)) requires I to be a real-valued link invariant, but in our definitionwe require I to be only a real-valued knot invariant. For example the Ozsvath-Szabo τ invariant is defined only for knots. In case I is a real-valued linkinvariant defining a quasi-morphism Î, the quasi-morphisms Î and Î differ by aconstant which depends only on n, hence their homogenizations are equal.Theorem 2.1.4. Suppose that a real-valued knot invariant I defines a homomorphismI : Conc(S 3 ) → R,such that |I(K)| ≤ cg 4 (K), where c is a real positive constant independent ofK. Then Î is a quasi-morphism on B n. Moreover, for a different sets of choicesof completing braids α β for every β ∈ B n we get a (possibly different) quasimorphismon B n such that the absolute value of its difference with Î is boundedfrom above by a constant depending only on n and therefore the homogenizationsof the two quasi-morphisms are equal.24

Proof. Take any β, γ ∈ B n and let α β , α γ , α βγ be the chosen completing braidsfor β, γ and βγ.Lemma 2.1.5. There exists a cobordism S between the knots(− ̂α βγ βγ) ∗ #( ̂α β β#̂α γ γ)andsuch that χ(S) ≥ −6n.(− ̂α βγ βγ) ∗ # ̂α βγ βγProof. If T is a cobordism between two links L 1 and L 2 , we will write L 1 T ∼ L 2 .By an observation of Baader (see [4], Section 4) for any braids µ, ν ∈ B n̂µνT∼̂µ ⊔ ̂ν,where χ(T ) = −n. Also note that since ̂α βγ , ̂α β , ̂α γ are unlinks with no morethan n components, we haveŜα1βγ ∼ ̂αβ ⊔ ̂α γ ,where χ(S 1 ) ≥ 1 − 2n.Thereforêα βγ βγ S 2∼ ̂α βγ ⊔ ̂β ⊔ ̂γ S 3∼S∼3̂αβ ⊔ ̂α γ ⊔ ̂β ⊔ ̂γ S 4∼ ̂α β β ⊔ ̂α γ γ S 5∼ ̂α β β#̂α γ γ,where χ(S 2 ) = −2n, χ(S 3 ) ≥ 1 − 2n (the cobordism S 3 is the disjoint union ofS 1 and the trivial cobordism over ̂β ⊔ ̂γ given by a disjoint union of cylinders),χ(S 4 ) = −2n and χ(S 5 ) = −1 (we take S 5 as a saddle cobordism between thedisjoint union of two knots and their connected sum). Thuŝα βγ βγ S 6∼̂α β β#̂α γ γand hence(− ̂α βγ βγ) ∗ # ̂α βγ βγ S 7∼(− ̂α βγ βγ) ∗ #( ̂α β β#̂α γ γ),whereχ(S 6 ) = χ(S 7 ) = χ(S 2 ) + χ(S 3 ) + χ(S 4 ) + χ(S 5 ) ≥ −6n,as required.25

Let us now finish the proof of the theorem. Lemma 1.1.2 implies that theknot(− ̂α βγ βγ ∗ )# ̂α βγ βγis a slice knot. Therefore using Lemma 2.1.5 we getg 4 (−( ̂α βγ βγ ∗ )#( ̂α β β#̂α γ γ)) ≤ 3n,where g 4 is the four-ball genus. We know thatfor every knot K. This yieldsApplying the equalitieswe get|I(K)| ≤ cg 4 (K) (2.1)|I(−( ̂α βγ βγ ∗ )#( ̂α β β#̂α γ γ))| ≤ c · 3n.I(−K ∗ ) = −I(K), I(K 1 #K 2 ) = I(K 1 ) + I(K 2 ), (2.2)|Î(βγ) − Î(β) − Î(γ)| ≤ 3cn,which means that Î is a quasi-morphism.Finally note that for any two choices α β and α β ′ of completing braids forβ ∈ B n one can show, similarly to the proof of Lemma 2.1.5, thatg 4 (( ̂α β β)# − ( ̂α ′ β β)∗ ) ≤ c 1 n,for some positive constant c 1 independent of β, α β , α β ′ . Combining this withequations (2.2), (2.1) we get|I( ̂α β β) − I( ̂α ′ β β)| = |I(( ̂α β β)# − ( ̂α ′ β β)∗ )| ≤ c · c 1 n.Thus different choices of completing braids yield (possibly different) quasimorphismson B n whose difference is bounded in absolute value from aboveby a constant depending only on n and therefore their homogenizations areequal.The induced homogeneous quasi-morphism is denoted by Ĩ.26

Remark 2.1.6. Note that g 4 defines a norm on Conc(S 3 ) and Theorem 2.1.4can be reformulated as follows: each element of Hom(Conc(S 3 ), R), which isLipshitz with respect to the norm defines a quasi-morphism on B n .In the next section we will present the following knot/link invariants: theOzsvath-Szabo knot invariant τ, the Rasmussen link invariant s and the linksignatures. It is well known that all of them satisfy the conditions of Theorem2.1.4. Thus by Theorem 2.1.4 they define quasi-morphisms on B n .2.2 Examples and applications2.2.1 Rasmussen quasi-morphismIn his paper [4] Baader proved that the Rasmussen link invariant s induces aquasi-morphism on the braid group B n , with a defect bounded by n + 1. Theinduced quasi-morphism was denoted by s. Alternatively, we see that s satisfiesproperties (1.1), (1.2), (1.3) and (1.4). Thus our Theorem 2.1.4 proves that sdefines a quasi-morphism on B n . The induced quasi-morphism (our definition)is denoted by ŝ. Following [26] we denote by lk : B n → Z the homomorphismfrom B n to Z by setting lk(σ ±1i ) = ±1.Theorem 2.2.1. Suppose that φ : B n → R is a quasi-morphism defined by areal-valued link invariant φ, which satisfies property (1.8), where s is substitutedby φ. Then ˜φ = lk.Proof. Take β ∈ B n . Then for all p > 0 there exists a braid α p in B n , such that̂α p β p is a knot, l(α p ) ≤ M(n) and |lk(α p )| ≤ M(n), where l is a braid lengthand M(n) is some real positive constant which depends only on n. For exampleif β ∈ P n , than we can takeα p = α = σ 1 · . . . · σ n−1for all p ∈ N, see Figure 2.1. Note thatw(D(̂α p β p )) = lk(α p β p ) and o(D(̂α p β p )) = n.It follows from the property (1.8) that1 + lk(α p β p ) − n ≤ φ(α p β p ) ≤ −1 + lk(α p β p ) + n.27

z1z2z3znzz1 2z nFigure 2.1: The braid α = σ 1 · . . . · σ n−1For all p > 0 there exist a constant K(n) > 0, such that |φ(α p )| ≤ K(n). Thus:1limp→∞ p (1 + lk(α pβ p 1) − n) ≤ limp→∞ p φ(α pβ p 1) ≤ limp→∞ p (−1 + lk(α pβ p ) + n).1Therefore lim φ(αp→∞ p pβ p ) = lk(β). Now φ is a quasi-morphism with a defect D φand so|φ(α p β p ) − φ(β p )| ≤ |φ(α p )| + D φ < K(n) + D φ .It follows thatlk(β) = limp→∞φ(α p β p )p= limp→∞φ(β p )p= ˜φ(β).Definition 2.2.2. A link diagram is called alternating if the crossings alternateunder, over, under, over, and so on as one travels along each component of thelink. A link is called alternating if it has an alternating diagram. A braidα ∈ B n is called alternating if ̂α is an alternating link.We denote by ŝign n ( ˜sign n ) the (homogeneous) quasi-morphism on B n inducedby the link invariant sign.Corollary 2.2.3. Each α ∈ B n satisfies ˜s(α) = lk(α). If γ ∈ B n is an alternatingbraid, then˜sign n (γ) = lk(γ).28

Proof. The first statement follows directly from Theorem 2.2.1.Now we will prove the second statement. For each p ∈ N the braid γ p isalternating and there exists an alternating braid α p such that l(α p ) < M(n),where ̂α p γ p is an alternating knot and M(n) is some real positive constant whichdepends only on n. It follows from property (1.6) thatŝignlim n (α p γ p )p→∞ p= limp→∞ŝ(α p γ p )p= limp→∞lk(α p γ p )p= lk(γ).Using the fact that ŝign n is a quasi-morphism we get the following equalityŝignlim n (α p γ p )p→∞ p= ˜sign n (γ).In [4] Baader defined the following quasi-morphism:s − lk + n − 1 : B n → R,where s(β) := s(̂β) (Gambaudo-Ghys definition). This quasi-morphism mapsall positive braids to zero. Therefore it descends to a quasi-morphism onB n /⟨∆ n ⟩, where ∆ 2 n is the positive generator of the center Z(B n ) of B n . In[4] Baader asked whether this quasi-morphism is bounded for n > 2. Here wegive a positive answer to this question.Proposition 2.2.4. The quasi-morphism s − lk + n − 1 : B n → R is bounded.Proof. Take any braid β ∈ B n and a braid α β , such that ̂α β is an unlink. Usingthe fact that s is a quasi-morphism with a defect bounded by n + 1, and thats(α β ) ≤ n − 1 and lk(α β ) ≤ n − 1 we get the following inequality:|s(β) − lk(β)| ≤ |s(α β β) − lk(α β β)| + 3n − 1.It follows from property (1.8) thatThis yields|s(α β β) − lk(α β β)| ≤ n − 1.|s(β) − lk(β) + n − 1| ≤ 5n − 3.The last inequality shows that the quasi-morphism s − lk + n − 1 is boundedon B n and thus also on B n /⟨∆ n ⟩.29

2.2.2 Connection between τ, s and braid numberThe invariants s and 2τ share similar properties and coincide on positive andalternating knots. It was conjectured by Rasmussen [56] that they are equal.This conjecture was disproved by Hedden and Ording [32]. In this subsectionwe show that the difference between 2τ and s is bounded by the braid numberand that ˜s = 2˜τ.Let L be an oriented link in S 3 and D(L) be any diagram of L. Recall thatin Section 1.1 we associated, to each such D(L), a family of Seifert circles.Lemma 2.2.5. Let K be an oriented knot in S 3 and D(K) be any diagram ofK. Then1 + w(D(K)) − o(D(K)) ≤ 2τ(K) ≤ −1 + w(D(K)) + o(D(K)), (2.3)where o(D(K)) is the number of Seifert circles of D(K).Proof. The proof of the lower bound in (2.3) is exactly the same as in [59],where s is substituted by 2τ. We present it for the reader’s convenience. Recallthat0 ≤ 2τ(K + ) − 2τ(K − ) ≤ 2, (2.4)τ(K ∗ ) = −τ(K). (2.5)It follows from property (1.14) that the equation 2τ(K 1 ) = s(K 1 ) is true forany positive knot K 1 . Thus for every positive knot diagram D(K 1 ) the followingequation holds:2τ(K 1 ) = w(D(K 1 )) − o(D(K 1 )) + 1. (2.6)It means that the left inequality in (2.3) is then an equality. When a positivecrossing of D(K) is changed into a negative one, the number 1 + w(D(K)) −o(D(K)) decreases by 2, while 2τ(K) decreases by at most 2, because of (2.4).Hence, the left inequality in (2.3) is preserved. The upper bound in (2.3) istrue, because of (2.5) combined with the lower bound.Proposition 2.2.6. The knot invariant τ defines a quasi-morphism on B n and2˜τ = lk.Proof. The first statement follows directly from Theorem 2.1.4. The secondstatement follows from Lemma 2.2.5 and from the proof of Theorem 2.2.1.30

Recall that every knot K in R 3 can be presented as a closure of some braidin B n . The braid number of K is the minimal such n. It is denoted by br(K).Theorem 2.2.7. Let ŝ and ̂τ be the quasi-morphisms on B n which are inducedfrom Rasmussen and Ozsvath-Szabo knot invariants. Then for every β ∈ B nthe following inequality holds |ŝ(β) − 2̂τ(β)| ≤ 2(n − 1).Proof. As we explained before both s and 2τ satisfy property (1.8). It meansthat|ŝ(β) − lk(α β β)| ≤ n − 1 and |2̂τ(β) − lk(α β β)| ≤ n − 1.Thus by the triangle inequality we have|ŝ(β) − 2̂τ(β)| ≤ 2(n − 1).Corollary 2.2.8. For every knot K the following inequality holds:|s(K) − 2τ(K)| ≤ 2(br(K) − 1).2.2.3 Signature quasi-morphismsQuasi-morphism defined by the classical signature link invariantRecall that the signature link invariant was denoted by sign and the inducedhomogeneous quasi-morphism on B n was denoted by ˜sign n .Proposition 2.2.9. 1. For every n > 2 the homogeneous quasi-morphism˜sign n : B n → R is not a homomorphism.2. The restriction of ˜sign n to P n is not a homomorphism.Proof. Claim 1. Take a braid α n ∈ B n for n > 2, where α n = σ 1 · . . . · σ n−1 .Then it follows from [47, page 148] that˜sign n (α n ) = n 2 , if n is even, and ˜sign n (α n ) = n2 − 1, if n is odd.2nIt follows that if n is even thenn2 = ˜sign n (α n ) ≠ ˜sign n (α n−1 ) + ˜sign n (σ n−1 ) = (n − 1)2 − 1+ 1,2n − 231

and if n is odd thenn 2 − 12n= ˜sign n (α n ) ≠ ˜sign n (α n−1 ) + ˜sign n (σ n−1 ) = n − 12+ 1.The proof of Claim 2 will be given in the next chapter.Let η i,n be a pure braid in P n , in which the i−th strand makes one twist inthe positive direction around previous i − 1 strands, and η − i,n is a pure braid inP n , in which the i−th strand makes one twist in the negative direction aroundprevious i − 1 strands, see Figure 2.2. Note that all of these braids commutewith each other. These braids will play a crucial role in the next chapter. Ini−1iFigure 2.2: Braid η i,n[26] Gambaudo and Ghys computed ˜sign n (η i,n ). We present this computationin detail.Lemma 2.2.10 ([26]). For all i, n ∈ N we have{i, if i is even,˜sign n (η i,n ) =i − 1, if i is odd.Proof. The torus link K(p, q) has a braid representation (σ 1 · . . . · σ p−1 ) q . NotethatK(n, np) = η p 2,n · . . . · η p n,n = (η 2,n · . . . · η n,n ) p .It follows that˜sign n (K(n, n)) = ˜sign n (η 2,n · . . . · η n,n ) =32n∑i=2˜sign n (η i,n ).

Also note that for all k, r ∈ N, such that k ≥ i and r ≥ i, the following equationholds˜sign n (η i,k ) = ˜sign n (η i,r ).It follows that˜sign n (η i,n ) = ˜sign n (K(i, i)) − ˜sign n (K(i − 1, i − 1)).It follows from [47, Theorem 7.5.1] that˜sign n (K(i, i)) = i2 − 1, if i is odd,2˜sign n (K(i, i)) = i2 2 ,if i is even.It follows that˜sign n (η i,n ) = i2 − 12−(i − 1)2, if i is odd,2Thus we get˜sign n (η i,n ) = i2 2 − (i − 1)2 − 1, if i is even.2{i, if i is even,˜sign n (η i,n ) =i − 1, if i is odd.ω-signature quasi-morphismsGambaudo and Ghys proved in [26] that sign ω induces a quasi-morphism onB n . Alternatively, Theorem 2.1.4 shows that sign ω induces a quasi-morphismon B n . The homogeneous quasi-morphism will be denoted as usual by ˜sign ω .In [27] Gambaudo and Ghys proved the following proposition.Proposition 2.2.11 ([27], Proposition 5.2). Let ω = e 2πiθ , whereθ ∈ [0, 1] ∩ Q. Then˜sign ω (σ 1 · . . . · σ n−1 ) = 2(n − 2l + 1)θ +for l−1n ≤ θ ≤ l , where 1 ≤ l ≤ n. n332l(l − 1)n

It follows from Proposition 5.2 in [27] thatfor l−1 ≤ θ ≤ l , where 1 ≤ l ≤ n.n n˜sign ω (K(n, n)) = 2n(n − 2l + 1)θ + 2l(l − 1) (2.7)Proposition 2.2.12. Let ω = e 2πiθ , where θ ∈ [0, 1] ∩ Q. Then:⎧4(i − 1)θ, if 0 ≤ θ ≤ 1 i ,˜sign ω (η i,n ) =⎪⎨4(l − 1)(1 − θ), if l − 1i≤ θ ≤ l − 1i − 1 , 2 ≤ l ≤ i − 1,4(i − 1)θ, if l − 1i − 1 ≤ θ ≤ l i , 2 ≤ l ≤ i − 1,⎪⎩4(i − 1)θ, if i − 1i≤ θ ≤ 1.(2.8)Proof. Note that˜sign ω (η i,n ) = ˜sign ω (K(i, i)) − ˜sign ω (K(i − 1, i − 1)).After a simple calculation this equality and (2.7) yield the result.Denote byAP n := ⟨η 2,n , η − 2,n, . . . , η n,n , η − n,n⟩the subgroup of P n generated by η ±1i,n , and defineV AP = {˜φ| APn : AP n → R where ˜φ is a homogeneous quasi-morphism on B n }.Proposition 2.2.13. Denote by ω(θ) = ω 2πiθ . ThenB ={˜signω ( 1 2) , 2 ˜sign ω ( 1 2) − 3 ˜sign ω ( 1 3) , . . . , (n − 1) ˜sign ω ( 1n−1) − n ˜sign ω ( 1 n)is a basis for V AP .}34

Proof. If n = 2, then ˜sign ω( 1 2) (η 2,2) = 2, and the proof follows. Let n > 2. For1 ≤ i ≤ n − 1, the equality()(n − 1) ˜sign ω( n−1) − n ˜sign 1ω((ηn)1 i,n ) = 0follows from (2.8). We also have the following equality:()(n − 1) ˜sign ω ( n−1) − n ˜sign 1ω ((ηn)1 n,n ) = −4.Note also that ˜sign ω (η i,i ) = ˜sign ω (η i,n ). For each 3 ≤ i ≤ n and for each2 ≤ j ≤ i − 1 this yields()(i − 1) ˜sign ω( i−1) − i ˜sign 1ω( 1 i )(η j,i ) = 0,and ()(i − 1) ˜sign ω ( i−1) − i ˜sign 1ω ( 1 i )(η i,i ) = −4.Thus B is a basis of V AP .35

Chapter 3Generalized Gambaudo-GhysconstructionIn this chapter we present a generalization of the Gambaudo-Ghys constructionof a family of quasi-morphisms on the group D := Diff ∞ (D 2 , ∂D 2 , area). Wepresent properties of these quasi-morphisms. Further we investigate the quasimorphismson D that are induced by s, τ and sign.3.1 Gambaudo-Ghys constructionDenote the space of Hamiltonians H : D 2 × [0, 1] → R, such that the support ofH t := H(·, t) is compact for any t, by H. We denote the space of autonomous(i.e. time-independent) Hamiltonians by H 0 .The following construction, essentially contained in Gambaudo-Ghys [26],takes a homogeneous quasi-morphism on P n and produces from it a quasimorphismon D.Let x = (x 1 , . . . , x n ) be any point in X n . Take g ∈ D and any path g t ,0 ≤ t ≤ 1, in D between Id and g. Connect z to x by a straight line in (D 2 ) n ,then act on x with the path g t , and then connect g(x) to z by the straight linein (D 2 ) n . We get a loop in (D 2 ) n . More specifically it looks as follows. Connectz i to x i by straight lines l 1,i : [ 0, 3] 1 → D 2 in the disc, then act with the path1g 3t−1 , ≤ t ≤ 2, on each x 3 3 i, and then connect g(x i ) to z i by straight linesl 2,i : [ 2, 1] → D 2 in the disc, for all 1 ≤ i ≤ n. It is a well known fact that D3coincides with the group of compactly supported Hamiltonian diffeomorphismsof D 2 (see e.g. [45]). Thus any identity-based path {h t } in D, 0 ≤ t ≤ 1, can36

e viewed as a Hamiltonian flow generated by a (time-dependent) HamiltonianH : D 2 × [0, 1] → R, such that the support of H t := H(·, t) is compact for anyt. It follows that for almost all n-tuples of different points x 1 , . . . , x n in thedisc the concatenations of the paths l 1,i : [ 0, 3] 1 → D 2 , g 3t−1 : [ 1, 2 3 3]→ D 2 andl 2,i : [ 2, 1] → D 2 , i = 1, . . . , n, yield a loop in X3 n . The homotopy type of thisloop is an element in P n , which is independent of the choice of g t because D iscontractible [23]. We will denote this element by γ(g; x).Let ˜φ be a homogeneous quasi-morphism on P n . Set∫Φ(g) = ˜φ(γ(g; x))dx, (3.1)X nwhere dx = dx 1 · . . . · dx n .Lemma 3.1.1 (c.f. [26]). The function Φ is well defined. It is a quasi-morphismon D.As an immediate corollary we get that the formula∫1˜Φ(g) := lim ˜φ(γ(g p ; x))dxp→∞ pyields a homogeneous quasi-morphism ˜Φ : D → R.X nProof. Step 1. Let us prove that |Φ(h)| < ∞ for h ∈ D.For any isotopy h t , 0 ≤ t ≤ 1, in D between h = h 1 and Id, any x ∈ X n andany 1 ≤ i, j ≤ n, i ≠ j denoteDenote byl i,j := h t(x i ) − h t (x j )∥h t (x i ) − h t (x j )∥ : [0, 1] → S1 .L i,j (x) = 1 ∫ 12π0∂∥∂t (l i,j)∥ dt, (3.2)where ∥ · ∥ is the Euclidean norm. Note that L i,j (x) is the length of the pathl i,j divided by 2π. Thus we get the following inequalityn∑2 (L i,j (x) + 4) ≥ l(γ(h; x)). (3.3)i

Take any finite generating set S of P n (there exists at least one). Note that forany homogeneous quasi-morphism ˜φ : P n → R one has()|˜φ(γ)| ≤ D ˜φ + max |˜φ(ξ)| l S (γ), (3.4)ξ∈Swhere l S (γ) is the length of a word γ with respect to S. Recall that l(γ) is thelength of γ with respect to the set {σ i } n−1i=1 . It follows from Lemma 1.3.5 thatthere exist two positive constants K 1,S and K 2,S , which are independent of γ,such thatl S (γ) ≤ K 1,S · l(γ) + K 2,S .It follows from (3.4) that|˜φ(γ(h; x))| ≤ N 1 l(γ(h; x)) + N 2 , (3.5)where N 1 = K 1,S (D ˜φ + max |˜φ(ξ)|) and N 2 = K 2,S (D ˜φ + max |˜φ(ξ)|). Inequalities(3.3) and (3.5) yield the followingξ∈S ξ∈Sinequality:It follows that|˜φ(γ(h; x))| ≤ N 1|Φ(h 1 )| ≤ 2N 1 · vol((D 2 ) n−2 )n∑i

∥Let H ∈ H be a Hamiltonian generating the flow h t , 0 ≤ t ≤ 1. Note that∂t∥ = ∥(∇H t )(h t (x))∥ ≤ max ∥(∇H t )(x)∥. Denotex∈D 2 ,t∈[0,1]∥ ∂h t(x)ThusM H :=∥ ∥∥∥ ∂∂t (l i,j)∥ ≤max ∥(∇H t )(x)∥ .x∈D 2 ,t∈[0,1]4M H∥h t (x i ) − h t (x j )∥ .Recall that h t is a Hamiltonian flow and therefore is area-preserving. Then⎛⎞∫ ∫11∂2π ∥∂t (l i,j)∥ dtdx idx j ≤ 4M ∫H⎝12π ∥x i − x j ∥ dx idx j⎠ .D 2 ×D 2 0D 2 ×D 2Now we show that∫D 2 ×D 2 1∥x i − x j ∥ dx idx jis well defined. It follows by Fubini-Tonelli theorem that∫1∥x i − x j ∥ dx idx j ≤D 2 ×D 2∫≤D 2⎛⎜⎝∫B δ (x j )1∥x i − x j ∥ dx i +∫D 2 \(D 2 ∩B δ (x j ))⎞1∥x i − x j ∥ dx ⎟i⎠ dx j ,where δ ≤ 1 is some positive constant. By using the polar coordinates on B 2 δ(x j )we get∫∫) (1∥x i − x j ∥ dx idx j < 2πδ + vol(D2 )dx j = π 2 2δ + 1 ).δδD 2 ×D 2DenoteN ′ := 4N 1 n(n + 1)⎛D 2 (⎝ vol((D2 ) n−2 )2π⎛⎝∫D 2 ×D 2 139⎞⎞∥x i − x j ∥ dx idx j⎠ + vol((D 2 ) n ) ⎠ ,

then|Φ(h)| ≤ N ′ · M H + A(X n ).Step 2. Let us prove that Φ is a quasi-morphism. Let g 1 and g 2 be elementsof D. Note that for almost all x ∈ X nγ(g 1 g 2 ; x) = γ(g 1 ; g 2 (x)) · γ(g 2 ; x).The diffeomorphisms g 1 and g 2 are area-preserving. Thus∫=∣It follows thatX n|Φ(g 1 g 2 ) − Φ(g 1 ) − Φ(g 2 )| =˜φ n (γ(g 1 g 2 ; x)) − ˜φ n (γ(g 1 ; g 2 (x))) − ˜φ n (γ(g 2 ; x))dx∣ .∫|Φ(g 1 g 2 ) − Φ(g 1 ) − Φ(g 2 )| ≤ D ˜φ .X nFrom now on the homogeneous quasi-morphism on D induced by a homogeneousquasi-morphisms ˜φ n : P n → R will be denoted by ˜Φ n .3.2 Reeb graphs and computations forautonomous diffeomorphismsIn this section we define the notion of a Morse-type Hamiltonian. We also recallthe notion of a Reeb graph. Recall that the space of autonomous Hamiltonians isdenoted by H 0 . It follows from [46, Theorem 2.7] that Morse-type Hamiltoniansform a C 1 -dense subset of H 0 . For any Hamiltonian H in this subset we presenta calculation of ˜Φ on the time-one flow of H (Theorem 3.2.4). Our result ispresented in terms of the integral of the push-forward of H to its Reeb graphagainst a certain signed measure on the graph. In the end of this section weshow that ˜Φ is continuous in C 1 -topology on the space H 0 (see Theorem 3.3.5).Definition 3.2.1. We say that a function H ∈ H 0 is of Morse-type if:1. There exists a connected open neighborhood U of ∂D 2 , such that ∂U \ ∂D 2is a smooth simple curve, H| U ≡ 0 and H has no degenerate critical points in40

D 2 \ U.2. There exists a connected open neighborhood V of ∂D 2 , such that V ⊃ Uand H| V \U has no critical points.3. The inequality H(x) ≠ H(y) holds for each two non-degenerate differentcritical points x and y.Now let us explain how we associate to H and ˜φ the Reeb graph T of Hand the signed measure µ on T .3.2.1 Angle-action coordinates and the Reeb graphDefinition 3.2.2. A charged tree T is a finite tree equipped with a signedmeasure, such that each edge has a total finite measure.Now we are ready to explain how we associate to (H, ˜φ) a charged tree(T, µ). Let H ∈ H 0 be a Morse-type function with l critical points in D 2 \ U,where U is as in Definition 3.2.1. Let c = ∂U \ ∂D 2 and suppose that the areaof the domain U is 2πε for some ε > 0. Let h 1 be a time-one flow generated byH.Step 1. Here we recall the notion of angle-action symplectic coordinates.We remove from D 2 all singular level curves inside D 2 \ U, the curve cand ∂D 2 . We get l open annuli A i (a punctured disc is also viewed as anannulus) and an open annulus A l+1 = U ◦ . Each D 2 \ A i has two connectedcomponents. We denote by CA i the component which does not contain ∂D 2 ,and by a i := ∂(CA i ) ∩ ∂A i . By Liouville-Arnold theorem (see [3]) on eachone of the annuli A i (1 ≤ i ≤ l), there exist so-called angle-action symplecticcoordinates (J i , θ i ), θ i ∈ [0, 2π], and a C ∞ -function[ area(CAi ) i :, area(CA ]i) + area(A i )→ R,2π2πsuch that on each level curve δ i in A i one has H| δi = i (J i ), where the coordinateJ i along δ i is equal to the sum of area(CA i)and the area of the annulus bounded2πby δ i and a i , divided by 2π. In these coordinates h 1 moves points on each levelcurve with a constant speed ′ i := ∂ i∂J i.41

Take the coordinates (J l+1 , θ l+1 ) on A l+1 , where J l+1 ∈ [ 12 − ε, 1 2]is the areacoordinate divided by 2π, and θ l+1 is the angle coordinate. Denote byNote that l+1 = 0, and∫D 2 H(p, q)dpdq =R i := area(A i)2πl∑∫i=1A iCR i := area(CA i).2πH(p, q)dpdq = 2πl∑i=1CR∫i +R iCR i i (J i )dJ i ,where p, q are the coordinates on R 2 .Step 2. Let H : D 2 → R be a Morse-type function with l critical points inD 2 \ U. Define the equivalence relation ∼ on points on D 2 by x ∼ y wheneverx, y are in the same connected component of a level set of H. The Reeb graphT corresponding to H is the quotient of D 2 by the relation ∼. The edges ofT come from the annuli in D 2 fibered by level loops, the valency-one verticescorrespond to the min/max critical points of H and to ∂D 2 , and the valencythreevertices correspond to the saddle critical points of H. In our case T is arooted tree with the root being the vertex of T corresponding to the domainU ⊃ ∂D 2 . Each edge e j in T corresponds to an annulus A j for each 0 ≤ j ≤ l.Example 3.2.3. Figure 3.1 shows a Reeb graph with 5 edges and 6 vertices,such that 3 of them correspond to the maximum points, 2 of them correspondto the saddle points and the root corresponds to the region bounded by thecurves c and ∂D 2 .We call a Reeb graph simple if it has only 2 vertices and one edge. Notethat if H = H(p 2 + q 2 ) is a monotone function of p 2 + q 2 such that 0 ∈ D 2 isits unique non-degenerate critical point, then T is simple.Step 3. The action coordinate on any of the annuli induces a coordinate J jon the corresponding edge e j of T . Thus each H ∈ H 0 descends to a function : T → R, where ′ = ′ j on each open edge e j . Here j ∈ {0, . . . , l}. For ahomogeneous quasi-morphism ˜φ : P n → R we define a signed measure µ on Tby settingn∑( ) ( n−i n 1dµ(J j ) := (2π) n ˜φ(η i,n )i (J j ) j) i−1i 2 − J dJ j ,i=2for each 0 ≤ j ≤ l, where J j is the coordinate on each edge e j .42

AA1 2A 1 A 2A4AA3A54A 3A 5curve cFigure 3.1: Reeb graph which corresponds to the function H with 3 maximumpoints and 2 saddle points3.2.2 Main TheoremWe will work with homogeneous quasi-morphisms on P n which are restrictionsof homogeneous quasi-morphisms on B n , or, equivalently (see e.g. [31]), withhomogeneous quasi-morphisms ˜φ : P n → R such that ˜φ(h) = ˜φ(g −1 hg) foreach g ∈ B n and each h ∈ P n . Such a quasi-morphism on P n will be calledB n -invariant. This is a technical condition−the results below are likely to holdfor all homogeneous quasi-morphisms on P n . Now we will state and prove ourmain theorem in this chapter.Theorem 3.2.4. Let H ∈ H 0 be a Morse-type function, ˜φ : P n → R a B n -invariant homogeneous quasi-morphism, and ˜Φ the corresponding homogeneousquasi-morphism on D. Then∫˜Φ(h 1 ) = ′ dµ, (3.8)where h 1 is the time-one Hamiltonian flow generated by H.T43

Proof. Step 1. Suppose that the corresponding Reeb graph T has l labelededges. Thus T is a rooted labeled tree. Recall that A l+1 corresponds to theroot of T .Take all n-tuples of points x = (x 1 , . . . , x n ) ∈ X n such that for each x i , x j ∈D 2 \ A l+1 we require that x i , x j are not critical points of H, H(x i ) ≠ H(x j ) forall i ≠ j, and i < j if the corresponding points to x i and x j in T lie on the sameedge such that the corresponding action coordinates J i and J j satisfy J i < J j .For each x i , x j ∈ A l+1 we require that the corresponding action coordinates J iand J j satisfy J i < J j .We also require that each such x defines a sequence of natural numbers{n i } l+1i=1l+1 ∑such that n i = n, and the projections of the points x 1 , . . . , x n1 on Ti=1lie on the edge e 1 , the projections of the points x n1 +1, . . . , x n1 +n 2on T lie onthe edge e 2 and so on; and the projections of the points x n1 +...+n l +1, . . . , x n onT equal to the root of T . We denote the set of such n-tuples of points withsuch an induced sequence {n i } l+1i=1 , by X T,n 1 ,...,n l+1. We denoteX T,n := ∪ X T,n1 ,...,n l+1,where the union is over the set of l + 1-tuples of numbers n 1 , . . . , n l+1 such thatl+1 ∑n i = n.i=1Let h t be a Hamiltonian flow generated by H. Take p ∈ N. Then for eachx ∈ X T,nγ(h p 1; x) = α 1,p γ 1,p (h 1 ; x) · . . . · γ l,p (h 1 ; x)α 2,p , (3.9)for some γ i,p (h 1 ; x) ∈ P n , which pairwise commute with each other, and for somebraids α 1,p and α 2,p . The braids α 1,p ∈ B n and α 2,p ∈ B n depend on x ∈ X T,n ,p ∈ N and H. For all n ∈ N there exists a constant M(n) > 0, which dependsonly on n, such that for each x ∈ X T,n as above and for each p ∈ N this constantsatisfies the following inequations: |˜φ(α 1,p )| ≤ M(n), |˜φ(α 2,p )| ≤ M(n). Eachγ i,p (h 1 ; x) depends on the integer p, the function H, the pointsx n1 +n 2 +...+n i−1 +1, . . . , x n1 +n 2 +...+n i,and on the position of the points whose projections on T do not lie on the edgee i .Let η i,k,n be a pure braid in P n , in which the (i + k)-th strand makes onetwist in the positive direction around previous i − 1 strands, and let η − i,k,n be a44

kiFigure 3.2: The braid η i,k,nn−k−ipure braid in P n , in which the (i + k)-th strand makes one twist in the negativedirection around previous i − 1 strands, see Figure 3.2.If k = 0, then, by definition, η i,0,n = η i,n . We setAP l,n = {η i,k,n , η − i,k,n , | 2 ≤ i + k ≤ n}.Then AP l,n generates a subgroup of P n . Note that each braid η i,k,n ∈ AP l,n isconjugate in B n to the braid η i,n . Thus ˜φ(η i,k,n ) = ˜φ(η i,n ) (here we use the factthat ˜φ is B n -invariant). Note that each γ i,p (h 1 ; x) is equal to η j i,k,nfor some j,where i, j, k, n depend on p, x and H.For β ∈ P k we define β k,m ∈ P k+m to be the image of β under the standardinclusion of P k into P m+k . It means that β k,m = β ⊔ Id m , where the identitybraid Id m ∈ P m is placed on the right of the braid β ∈ P k . Denoteβ(i, p) := γ i,p (h 1 ; x).Note that each γ i,p (h 1 ; x) is conjugate in B n to the braid β(i, p) ni ,n−n i. Thusthe conjugacy class in B n of each γ i,p (h 1 ; x) depends only on the integer p, thefunction H and the points x n1 +n 2 +...+n i−1 +1, . . . , x n1 +n 2 +...+n i.Note that any homogeneous quasi-morphism restricts to a homomorphismon an abelian subgroup. Recall that ˜φ is a B n -invariant quasi-morphism. Itfollows that for each x ∈ X T,n1limp→∞ p ˜φ(γ(hp 1; x)) =∑l+1i=1limp→∞1p ˜φ(γ i,p(h 1 ; x)) (3.10)45

and∫˜Φ(h 1 ) =X n∑l+1n!1∑l+1limp→∞ p ˜φ(γ(hp 1; x))dx = n!n∑n−n∑ 1i=1 n 1 =0 n 2 =0n−n∑1 −...−n l· · ·n l+1 =0∫i=1X T,n∫X T,n1 ,...,n l+11limp→∞ p ˜φ(γ i,p(h 1 ; x))dx =1limp→∞ p ˜φ(β(i, p) n i ,n−n i)dx. (3.11)We also see that for each fixed n in!n∑n−n∑ 1n 1 =0 n 2 =0n−n∑1 −...−n l· · ·n l+1 =0∫X T,n1 ,...,n l+1( nn i)vol(X Ai ,n i)1limp→∞ p ˜φ(β(i, p) n i ,n−n i)dx =∫X Ai ,n i1limp→∞ p ˜φ(β(i, p) n i ,n−n i)dx ni , (3.12)where dx ni = dx n1 +n 2 +...+n i−1 +1 · . . . · dx n1 +n 2 +...+n i−1 +n i, A i is an annulus whichcorresponds to an edge e i , X Ai ,n i(respectively X Ai ,n i) is the space of ordered n i -tuples of different points in A i (respectively (n − n i ) tuples of points in D 2 \ A i )such that this n i -tuple (respectively (n−n i )-tuple) is a part of x ∈ X T,n . Denotethe integral in the right-hand side in (3.12) by I i . Thus equations (3.11) and(3.12) tell us that in order to compute ˜Φ(h 1 ) it is enough to know the value I ifor each 1 ≤ i ≤ l + 1. We will compute I i by using angle-action coordinateson the domain A i .Step 2. We prove this theorem by induction on the number l of edges ofthe corresponding Reeb graph. If l = 1, then the proof is exactly the same asin [26, Lemma 5.2]. The induction hypothesis states that our theorem is truefor every Reeb graph T with less than l edges.Let T be the Reeb graph of H which has l edges, and denote by e l theedge which connects the root of T to other vertex v l . Let k H and l − k H − 1be the number of edges in two connected components T 1 and T 2 of the graphT \ {e l , root, v l }. We cut the disc into l + 1 pieces, by cutting it along it criticallevel curves, which passes through the critical points, and by cutting it along acurve c. We get l +1 open annuli. Denote the k H open annuli, corresponding toedges of T 1 , by A 1 , . . . , A kH , and denote the l−k H −1 open annuli, correspondingto edges of T 2 , by A kH +1, . . . , A l−1 . The open annulus A l corresponds to an edge46

A lA l +1curve cFigure 3.3: Curve c together with annuli A l and A l+1e l , i.e. the curve c is a connected component of ∂A l . We denote by A l+1 theother open annulus adjacent to c, see Figure 3.3.Let (J i , θ i ) be angle-action coordinates on A i for 1 ≤ i ≤ l, and let (J l+1 , θ l+1 )be the coordinates on A l+1 , where J l+1 is the area coordinate divided by 2π andθ l+1 is the angle coordinate. Denote by ε the area of A 0 divided by 2π. Nowwe can rewrite (3.11) as follows:∑l+1˜Φ(h 1 ) = n!(2π) nwhere Seq = {n i } l+1i=1n∑n−n∑ 1i=1 n 1 =0 n 2 =0· · ·n−n∑1 −...−n ln l+1 =0∫∫· · ·D Seq,l1limp→∞ p ˜φ(β(i, p) n i ,n−n i)dJ,l+1 ∑is a sequence of natural numbers such thati=1(3.13)n i = n andD Seq,l ⊂ [ 0, 1 2] ×nis the domain which is determined by the following inequalities:CR i < J i,1 < . . . < J i,ni < CR i + R i for each 1 ≤ i ≤ l − 1,R 1 + . . . + R l−1 < J l,1 < . . . < J l,nl < 1 2 − ε,12 − ε < J l+1,1 < . . . < J l+1,nl+1 < 1 2 ,47

and dJ = dJ 1,1 · . . . · dJ 1,n1 · . . . · dJ l+1,nl +1 · . . . · dJ l+1,n . We have to multiply by(2π) n in the formula above, because there are n angle coordinates.Step 3. Let H 1 = H| A1 ∪...∪A kHand H 2 = H| AkH +1∪...∪A l−1be Morse-typefunctions with Reeb graphs T 1 and T 2 respectively. We set(kH)(∑l−1) ∑R 1,l := R i , R 2,l := R i , R := R 1,l + R 2,l .i=1i=k H +1Denote( ) n − k˜φ i,k (J j ) := ˜φ(η i,n )i (J j ) i−1 (R 1,l − J j )in−k−i ∂∂J j(J j ),Ψ n (A j ) := (2π) nn∑( ) ( CR k ∫ j +R jn−kn 11,l)k 2 − R ∑˜φ i,k (J j )dJ jk=0CR jfor 1 ≤ j ≤ k H , and denote( ) n − k˜φ i,k (J j ) := ˜φ(η i,n )i (J j ) i−1 (R 2,l − J j )iΨ n (A j ) := (2π) ni=0n−k−i ∂∂J j(J j ),n∑( ) ( CR k ∫ j +R jn−kn 12,l)k 2 − R ∑˜φ i,k (J j )dJ jk=0CR jfor k H + 1 ≤ j ≤ l − 1. We denote( ) n − d˜φ i,k,d (J l ) = ˜φ(η i,n ) i(i − 1) · . . .i( ) n−d−i 1. . . · (i + k + d − n)(J l − R) i+k+d−n−1 2 − J ∂l − ε (J l ),∂J lwhere i + k + d − n ≥ 1 and setΨ n (A l ) := (2π) nn∑d=0( nd)ε d n−d∑k=1R n−d−k(n − d − k)!12 −ε ∫Ri=0∑n−di=n−d−k+1˜φ i,k,d (J l )dJ l .48

It follows from the induction hypothesis and (3.13) that˜Φ n (h 1 ) =l∑Ψ n (A j ).j=1An easy computation shows thatCR∫j +R jΨ n (A j ) = (2π) nfor 1 ≤ j ≤ l − 1, andCR j1∫2 −εΨ n (A l ) = (2π) nRn∑i=2n∑i=2( ) ( ) n−i n 1˜φ(η i,n )i (J j ) i−1i 2 − J ∂j (J j )dJ j∂J j( ) ( ) n−i n 1˜φ(η i,n )i (J l ) i−1i 2 − J ∂l (J l )dJ l .∂J lIt follows that∫˜Φ(h 1 ) =T ′ dµ.3.3 Properties of induced quasi-morphisms onthe group DIn this section we prove some properties the quasi-morphisms on D induced byquasi-morphisms on P n .Proposition 3.3.1. For each n ≥ 2 the homogeneous quasi-morphism ˜Φ n ,defined by a homogeneous quasi-morphism ˜φ : P n → R, is not continuous in theC 0 -topology on D.Proof. The case n = 2 was proved by Gambaudo and Ghys in [25]. Let n > 2.For each k ∈ N, k > 2, we define H k : D 2 → R as follows:1. H k (p, q) = h k (J), where J = p2 +q 2.22. h k (J) = 0 for J > 1. For each k there exist 0 < ε k k < 1 such that h k k(J) = C k49

for J ∈ [0, ε k ], where C k ≠ 0. The function h k (J) has no critical points in (ε k , 1 k ).3. The integral1∫20(2π) nn∑i=2( ) ( ) n−i n 1˜φ n (η i,n )i (J) i−1i 2 − J h ′ k(J)dJ = N k ,where N k ≥ 1 for all k > 2. We denote by h k,1 the time one Hamiltonian flowof H k . It is obvious that h k,1 converges to Id in C 0 -topology. It follows fromthe proof of Theorem 3.2.4 that ˜Φ n (h k,1 ) = N k ≥ 1. But ˜Φ n (Id) = 0.The goal of the remaining part of this section is to show that if H k −−−→ Hk→∞in C 1 -topology, thenlim ˜Φ n (h 1,k ) = ˜Φ n (h 1 ).k→∞Here H k , H ∈ H 0 , and h 1,k , h 1 are Hamiltonian time-one flows generated by H kand H respectively.Let g ∈ D and ψ t : [0, 1] → D be such that ψ 0 = Id and ψ 1 = g. We denoteG({ψ t }) :=∫D 2 ×D 2 12π∫ 10∂∥∂t( )∥ψt (x) − ψ t (y) ∥∥∥dtdxdy.∥ψ t (x) − ψ t (y)∥Lemma 1 in [28] shows that G({ψ t }) is well defined. DenoteG(g) := inf G({ψ t }),where the infimum is taken over all isotopies in D joining the identity with g.Lemma 2 in [28] states that for all g and h in DThus the following limit exists:G(gh) ≤ G(g) + G(h).G(g i )L(g) = lim .i→∞ iProposition 3.3.2. Let g ∈ D, and let ˜φ : P n → R be a homogeneous quasimorphism.Then ∣ ∣∣˜Φn (g)∣∣ ≤ K 1 L(g),where K 1 > 0 is independent of g.50

Proof. Apply equation (3.7) in Lemma 3.1.1 to g p instead of h 1 and take theinfimum of the expression in the right-hand side over all the paths connectingthe identity with g p . The resulting infimum has the form K 1 G(g p ) + K 2 , whereK 1 and K 2 are positive constants independent of g. Thus|Φ(g p )| ≤ K 1 G(g p ) + K 2 ,for any p. Dividing both sides by p and taking the limit as p → +∞ we get therequired inequality.Now we will define a metric on D. A tangent vector to D at a point g is aHamiltonian vector field Y g (see [22]) whose L 2 -norm is defined by⎛∫∥Y g ∥ 2 := ⎝D 2⎞∥Y g (x)∥ 2 dx⎠where ∥.∥ stands for the standard Euclidean norm. For any path ψ t in D whichconnects g and h, the length of ψ t is given by the formula:We denotel({ψ t }) =∫ 10dψ t∥ dt ∥ dt.2d 2 (g, h) := inf l({ψ t }),where the infimum is taken over all paths in D joining g with h. One can showthat d 2 is a distance function. Thus (D, d 2 ) is a metric space. Note that d isright invariant: d 2 (g, h) = d 2 (gf, hf).Corollary 3.3.3. Let g ∈ D, and let ˜φ : P n → R be a homogeneous quasimorphism.Then|˜Φ n (g)| ≤ K 3 d 2 (Id, g),where K 3 > 0 depends only on n.Proof. It follows from [28, Theorem 1] that for any g ∈ D there exists a universalconstant K > 0, such thatL(g) ≤ Kd 2 (Id, g).Now take K 3 = K · K 1 . The statement follows from Proposition 3.3.2.5112,

Lemma 3.3.4. Let F ∈ H 0 . Then for any ε > 0 and p ∈ N there exists δ p > 0,such that if H ∈ H 0 is δ p -close to F in C 1 -topology, thend 2 (f p 1 , h p 1) < ε,where f t and h t are the Hamiltonian flows generated by F and H.Proof. For the convenience we normalize the area of D 2 to be 1. First, we willshow that for all p ∈ N there exists δ 1,p > 0 such thatNote thatmax ∥∇Fx∈D 2 (x) − ∇H (x) ∥ < δ 1,p ⇒ d 2 (f p 1 , h p 1) < ε.d 2 (f p 1 , h p 1) = d 2 (Id, f p 1 h −p1 ) ≤ l({f p t h −pt }).It follows from [51, Proposition 1.4.D] that∂(f p t h −p∂tt )(x) = p · (sgrad(F ) − sgrad(Hf −pt )) (fpt h−p t (x)) ,where sgrad is the symplectic gradient. Thus∂(f p t h −pt )∥∥ (x)∥∥(sgrad(F∂t ∥ = p · ) − sgrad(Hf−pt∥= p · ∥(∇F − ∇(Hf −p)) (fp ∥ ,tt h−p t (x)))) (fpt h−p t (x))Note that f t is an autonomous Hamiltonian flow. Thus F f t (x) = F (x) for allx ∈ D 2 and t ∈ R. It follows that ∀x ∈ D 2 and ∀p ∈ Z∇F (h−pt (x))(Df−pWe get the following inequality:∫ 1d 2 (f p 1 , h p 1) ≤ p0⎛∫⎝D 2t∥(Df −pt ) (fp) (fpt h−p tt h−p t(x)) = ∇F (f p t h−p t (x)) .(x)) ∥2 ∥∇F (h−pt∥⎞ 12(x)) − ∇H (h −pt (x)) ∥2 ⎠ dt.DenoteM Dft :=max ∥(Dfx∈D 2 t −1 ) (x) ∥.,t∈[0,1]52

TakeWe get the following inequalityδ 1,p =εp(M Dft ) p .d 2 (f p 1 , h p 1) ≤ p(M Dft ) p max ∥∇F (x) − ∇H(x)∥ ≤ ε.x∈D2 Denote by δ p = δ 1,p2 . We have shown that if H is δ p-close to F in C 1 -topology,then d 2 (f p 1 , h p 1) < ε.Theorem 3.3.5. Let F ∈ H 0 . Then for any ε > 0 there exists δ > 0, such thatif H ∈ H 0 is δ-close to F in C 1 -topology then:∣∣˜Φ n (f 1 ) − ˜Φ n (h 1 ) ∣ ≤ ε,where f 1 and h 1 are the time-one Hamiltonian flows generated by F and H.Proof. Let ε > 0, let D˜Φnbe the defect of the homogeneous quasi-morphism˜Φ n : D → R,and let K 3 be the constant which was defined in Corollary 3.3.3. Take p ∈ Nsuch that D˜Φn +K 3< ε. It follows from Lemma 3.3.4 that there exists δp p > 0,such that if H is δ p -close to F in C 1 -topology, then d 2 (f p 1 , h p 1) < 1. Thus we getthe following inequality:∣∣˜Φ n (f 1 ) − ˜Φ n (h 1 ) ∣ = 1 ∣∣˜Φ n (f p 1 ) −p˜Φ ∣n (h p + ∣˜Φ D˜Φn n (f p 1 h −p1 ) ∣1) ∣ ≤.pIt follows from Corollary 3.3.3 that∣∣˜Φ n (f p 1 h −p1 ) ∣ ≤ K 3 d 2 (Id, f p 1 h −p1 ) = K 3 d 2 (f p 1 , h p 1) < K 3 .Thus ∣ ∣∣˜Φn (f 1 ) − ˜Φ n (h 1 )∣∣ < D˜Φn + K 3p< ε.53

3.4 Induced quasi-morphisms and Calabihomomorphism3.4.1 Rasmussen quasi-morphisms on DLet ˜s be the homogeneous quasi-morphism on P n induced by the Rasmussenknot invariant s. The induced family of homogeneous quasi-morphisms on D iswell defined for every integer n ≥ 2, this family is denoted by ˜Ras n .Lemma 3.4.1. Let x, a ∈ R. Then for every n ∈ Nn(n − 1)x(x + a) n−2 =n∑i=2( ) ni(i − 1) x i−1 a n−i . (3.14)iProof. Denote by f(x) = (x + a) n and by f ′ the derivative of f. The prooffollows from the fact that n(n − 1)x(x + a) n−2 = xf ′′ (x) and that(x + a) n =n∑i=0( ni)x i a n−i .Theorem 3.4.2. For every g ∈ D˜Ras n (g) = 2n(n − 1)π n−1 · C(g). (3.15)Proof. The theorem of Banyaga [5] states that ker(C) is a simple group, whereC : D → R is a Calabi homomorphism. Thus there exists a homomorphismκ : R → R such that for all g ∈ Dκ(C(g)) = ˜Ras n (g). (3.16)Restricting ˜Ras n and C on a 1-parametric subgroup of D, on which they do notvanish, one easily sees that κ is continuous. It means thatM n · C(g) = ˜Ras n (g), (3.17)where M n is some real constant which depends only on n.54

Take H : D 2 → R, such that H equals zero near the boundary, and suchthat H has one maximum point and has no other critical points. Denote by h 1the time-one Hamiltonian flow generated by H. Recall that∫C(h 1 ) = −2 H.D 2ThusC(h 1 ) = −4π1∫20(J)dJ,where J is the symplectic action coordinate. It follows from Corollary 2.2.3that˜s(η i,n ) = lk(η i,n ) = 2(i − 1).It follows from Lemma 3.4.1 thatn∑( ) ( ) n−i n 1(2π) n ˜s(η i,n )i J i−1i 2 − J = 8π n n(n − 1)J.i=2It follows from Theorem 3.2.4 thatNote that ( 1 ) = 0. Thus2˜Ras n (h 1 ) = 8π n n(n − 1)˜Ras n (h 1 ) = −8π n n(n − 1)It follows that M n = 2n(n − 1)π n−1 .1∫201∫20J ′ (J)dJ.(J)dJ = 2n(n − 1)π n−1 C(h 1 ).3.4.2 Signature quasi-morphisms and CalabihomomorphismRecall that ˜sign n : B n → R is a homogeneous quasi-morphism for all n ≥ 2.The induced family of homogeneous quasi-morphisms is denoted by ˜Sign n,D 2.55

In [26] Gambaudo and Ghys proved that all of them are non-trivial and linearlyindependent. In the case where the Reeb graph of H is simple they showed that∫˜Sign n,D 2(h 1 ) = ′ dµ.Proposition 3.4.3. For each natural number n we have∫(˜Sign n,D 2(h 1 ) = nπ ) n 1 + 4(n − 1)J − (1 − 4J)n−1 ′ (J)dJ, (3.18)Twhere h 1 is the time-one Hamiltonian flow defined by a Morse-type HamiltonianH.First we prove the following combinatorial equality.Lemma 3.4.4. Let x, a ∈ R. Then for every n ∈ Nn∑i=2n2( ) ( ) n−i n 1˜sign n (η i,n )i x i−1i 2 − x =((x + a) n−1 + 2(n − 1)x(x + a) n−2 − (a − x) n−1) .Proof. Recall that in Lemma 2.2.10 we proved that{i, if i is even,˜sign n (η i,n ) =i − 1, if i is odd.It follows thatn∑( ) n˜sign n (η i,n )i x i−1 a n−i =ii=2n∑i=2( ) ni(i − 1) x i−1 a n−i + 1 i 2n(n − 1)x(x + a) n−2 + n 2n∑i=1T( ) n (1i) − (−1)i−1x i−1 a n−i =i((x + a) n−1 − (a − x) n−1) .56

Proof of Proposition 3.4.3. We change variables x = J and a = 1 −J in Lemma23.4.4. It follows thatn∑( ) ( ) n−i n 1˜sign n (η i,n )i J i−1i 2 − J =i=2( (1 ) n−1 ( ) n−1 ( ) ) n−1n1 1+ 4(n − 1)J −2 22 2 − 2J . (3.19)Now the proof follows easily from Theorem 3.2.4 and (3.19).We are finally ready to prove the second claim in Proposition 2.2.9.Proof of Proposition 2.2.9. Claim 2. Suppose that ˜sign n : P n → R is a homomorphismfor n > 2. Then ˜Sign n,D 2 : D → R is a homomorphism. LetC : D → R be the Calabi homomorphism. Using the theorem of Banyaga [5],which states that ker(C) is a simple group, we get that there exists a homomorphismκ : R → R such that˜Sign n,D 2 = κC.Restricting ˜Sign n,D 2 and C on a 1-parametric subgroup of D, on which they donot vanish, one easily sees that κ is continuous. Hence κ(y) = K 1,n y for ally ∈ R, and˜Sign n,D 2 = K 1,n C,where K 1,n is some real constant which depends only on n. But using Proposition3.4.3, we see that this is impossible for n > 2.Theorem 3.4.5. For each g ∈ D generated by an autonomous Hamiltonianlimn→∞˜Sign n,D 2(g)π n−1 n(n − 1) = C(g).Proof. Step 1. Suppose that g is generated by a Morse-type HamiltonianH ∈ H 0 . It follows from (3.18) thatlimn→∞˜Sign n,D 2(g)π n−1 n(n − 1) = C(g).Step 2. Suppose that g is generated by any H ∈ H 0 . Then there exists aMorse-type function F ∈ H 0 , which is C 1 -close to H (see [46, Theorem 2.7]).Now the statement follows from Theorem 3.3.5.57

3.5 Open problemsThe following three open problems arise naturally in connection with our results.Problem 1. Does there exist a real-valued knot invariant I, independentfrom the ω-signatures, τ and the Rasmussen invariant s, such that I satisfies theconditions of Theorem 2.1.4, and such that the homogenization of the inducedquasi-morphism Î : P n → R is non-trivial?Problem 2. Surprisingly, the proof of the second claim in Proposition 2.2.9– dealing with a quasi-morphism on P n – involves the Calabi homomorphismon D. It would be interesting to find a direct proof of this claim.Problem 3. The following problem is motivated by a question posed tous by L.Polterovich. Denote by D aut ⊂ D the set of area-preserving diffeomorphismsgenerated by autonomous Hamiltonians.Observe that, by our results in subsection about signature quasi-morphisms,one can find a linear combination Φ of homogeneous quasi-morphisms ˜sign ω onB n so that the corresponding homogeneous quasi-morphism ˜Φ : D → R vanisheson D aut .On the other hand, using a construction from [26] (completely different fromthe one described in this paper) one can construct more homogeneous quasimorphisms˜Ψ α on D that vanish on D aut .It would be interesting to check whether the quasi-morphism ˜Φ is non-trivialand, if so, whether it is a linear combination of the quasi-morphisms ˜Ψ α .So far we have not been able to compute explicitly the value of any of theGambaudo-Ghys quasi-morphisms coming from quasi-morphisms on B n on anyarea-preserving diffeomorphism that is not generated by a Hamiltonian flowpreserving a foliation.58

Chapter 4Link invariants via countingsurfacesA Gauss diagram is a simple, combinatorial way to present a knot. It is knownthat any Vassiliev invariant may be obtained from a Gauss diagram formulathat involves counting (with signs and multiplicities) subdiagrams of certaincombinatorial types. These formulas generalize the calculation of a linkingnumber by counting signs of crossings in a link diagram. Until recently, explicitformulas of this type were known only for few invariants of low degrees.In this chapter we present simple formulas for two infinite families of invariantsin terms of counting surfaces of a certain genus and number of boundarycomponents in a Gauss diagram. The first family of invariants is a Chmutov-CapKhoury-Rossi family generalized to classical and long virtual links witharbitrary number of components. The resulting invariants are identified withcoefficients of the Conway polynomial. The second family of invariants is identifiedwith certain derivatives of the HOMFLYPT polynomial. In the remainingpart of the thesis we consider only oriented links.4.1 Gauss diagrams and arrow diagramsIn this section we recall a notion of Gauss diagrams, arrow diagrams and Gaussdiagram formulas. We then define a special type of arrow diagrams whichwill be used to define Gauss diagram formulas for coefficients of the Conwaypolynomial, and for coefficients of some other polynomial derived from theHOMFLYPT polynomial.59

4.1.1 Gauss diagrams of classical and virtual linksGauss diagrams (see e.g. [30], [53]) provide a simple combinatorial way toencode classical and virtual links.Definition 4.1.1. Given a classical (possibly framed) link diagram D, considera collection of oriented circles parameterizing it. Unite two preimages of everycrossing of D in a pair and connect them by an arrow, pointing from the overpassingpreimage to the underpassing one. To each arrow we assign a sign (localwrithe) of the corresponding crossing. The result is called the Gauss diagramG corresponding to D.We consider Gauss diagrams up to an orientation-preserving diffeomorphismsof the circles. In figures we will always draw circles of the Gauss diagramwith a counter-clockwise orientation.Example 4.1.2. Diagrams of the trefoil knot and the Hopf link, together withthe corresponding Gauss diagrams, are shown in the following picture.+++A classical link can be uniquely reconstructed from the corresponding Gaussdiagram [30]. Many fundamental knot invariants, such as the knot group andthe Alexander polynomial, may be easily obtained directly from the Gaussdiagram. We are going to work with based Gauss diagrams, i.e. Gauss diagramswith a base point (different from the endpoints of the arrows) on one of thecircles. If we cut a based circle at the base point, we will get a Gauss diagramof a long link, see Figure 4.1.++++++Figure 4.1: Diagrams of based and long classical Hopf links together with theassociated Gauss diagrams60

+++Figure 4.2: Virtual trefoil and a virtual Hopf link with the corresponding GaussdiagramsNote that not every collection of circles with signed arrows is realizable asa Gauss diagram of a classical link, see Figure 4.2.The Gauss diagram of a virtual link diagram is constructed in the sameway as for a classical link diagram, but all virtual crossings are disregarded,see Figures 4.2 and 4.3. Each non-realizable Gauss diagram with a base pointrepresents a long virtual link, see Figure 4.3.++++Figure 4.3: Based and long virtual trefoil with the corresponding Gauss diagramsTwo Gauss diagrams represent isotopic classical/virtual links (long links)if and only if they are related by a finite number of Reidemeister moves forGauss diagrams (applied away from the base point) shown in Figure 4.4, whereε = ±1. See e.g. [16, 48, 52].Two Gauss diagrams represent isotopic classical framed links if and only ifthey are related by a finite number of Reidemeister moves for framed Gaussdiagrams. It suffices to consider Ω 2 and Ω 3 of Figure 4.4 and substitute the61

Ω 1 : ε Ω 2 :ε−εΩ 3 :++++++Figure 4.4: Reidemeister moves of Gauss diagramsmove Ω 1 byεεΩ F 1 : −ε−εNote that segments involved in Ω 2 or Ω 3 may lie on different components of thelink and the order in which they are traced along the link may be arbitrary.4.1.2 Arrow diagrams and Gauss diagram formulasAn arrow diagram is a modification of a notion of a Gauss diagram, in whichwe forget about realizability and signs of arrows, see Figure 4.5. In other words,Figure 4.5: Connected arrow diagramsan arrow diagram consists of a number of oriented circles with several arrowsconnecting pairs of distinct points on them. We consider these diagrams upto orientation-preserving diffeomorphisms of the circles. An arrow diagram isbased, if a base point (different from the end points of the arrows) is markedon one of the circles. An arrow diagram is connected, if it is connected as62

a graph. Further we will consider only based connected arrow diagrams, sowe will omit mentioning these requirements throughout this chapter, unless amisunderstanding is likely to occur. In figures we will always draw the circlesof an arrow diagram with a counter-clockwise orientation.M. Polyak and O. Viro suggested [53] the following approach to computelink invariants using Gauss diagrams.Definition 4.1.3. Let A be an arrow diagram with m circles and let G be abased Gauss diagram of an m-component oriented (long, virtual) link. A homomorphismϕ : A → G is an orientation preserving homeomorphism betweeneach circle of A and each circle of G, which maps a base point of A to the basepoint of G and induces an injective map of arrows of A to the arrows of G.The set of arrows in Im(ϕ) is called a state of G induced by ϕ and is denotedby S(ϕ). The sign of ϕ is defined as sign(ϕ) = ∏ α∈S(ϕ)sign(α). A set of allhomomorphisms ϕ : A → G is denoted by Hom(A, G).Note that since the circles of A are mapped to circles of G, a state S of Gdetermines both the arrow diagram A and the map ϕ : A → G with S = S(ϕ).Definition 4.1.4. A pairing between an arrow diagram A and G is defined by∑⟨A, G⟩ = sign(ϕ).ϕ∈Hom(A,G)For an arbitrary arrow diagram A the pairing ⟨A, G⟩ does not represent a linkinvariant, i.e. it depends on the choice of a Gauss diagram of a link. However,for some special linear combinations of arrow diagrams the result is independentof the choice of G, i.e. does not change under the Reidemeister moves for Gaussdiagrams. Using a slightly modified definition of arrow diagrams Goussarov,Polyak and Viro showed in [30] that each real-valued Vassiliev invariant of longknots may be obtained this way. In particular, all coefficients of the Conwaypolynomial may be obtained using suitable combinations of arrow diagrams.4.1.3 Surfaces corresponding to arrow diagramsGiven an arrow diagram A, we define an oriented surface Σ(A) as follows.Firstly, replace each circle of A with an oriented disk bounding this circle.Secondly, glue 1-handles to boundaries of these disks using each arrow as a coreof a ribbon. See Figure 4.6.63

Figure 4.6: Constructing a surface from an arrow diagramDefinition 4.1.5. By the genus and the number of boundary components of anarrow diagram A we mean the genus and the number of boundary componentsof Σ(A).Remark 4.1.6. Let A be an arrow diagram with m circles and n arrows. Thenthe Euler characteristic χ of Σ(A) equals to χ(Σ(A)) = m−n. If A is connected,n ≥ m − 1. If A has one boundary component, n ≠ m(mod2).Example 4.1.7. The arrow diagram with one circle in Figure 4.5 is of genusone, while the other arrow diagram in the same figure is of genus zero. Both ofthem have one boundary component.Further we will work only with based connected arrow diagrams with oneor two boundary components.4.1.4 Ascending and descending arrow diagramsIn this subsection we define a special type of arrow diagrams with one and twoboundary components.Definition 4.1.8. Let A be a based arrow diagram with one boundary component.As we go along the boundary of Σ(A) starting from the base point,we pass on the boundary of each ribbon twice: once in the direction of itscore arrow, and once in the opposite direction. A is ascending (respectively,descending) if we pass each ribbon of Σ(A) first time in the direction oppositeto its core arrow (respectively, in the direction of its core arrow).Remark 4.1.9. In order to define the notion of ascending (descending) arrowdiagrams we used the fact that all arrow diagrams are based and connected.The position of the base point in a connected arrow diagram is essential todefine an order of the passage.Example 4.1.10. Arrow diagrams presented below are ascending (a), descending(b) and neither ascending nor descending (c).64

a b cDenote by A n,m (respectively, D n,m ) the set of all ascending (respectively,descending) arrow diagrams with n arrows, m circles and one boundary component.Example 4.1.11. The sets A 2,1 and D 2,1 are presented below.A 2,1 := and D 2,1 :=Definition 4.1.12. Let G be any Gauss diagram with m circles. We setA n,m (G) :=∑⟨A, G⟩ D n,m (G) := ∑⟨A, G⟩A∈A n,m A∈D n,mand define the following polynomials:∇ asc (G) :=∞∑A n,m (G)z n ∇ des (G) :=n=0∞∑D n,m (G)z nThese polynomials will play an important role in the next section. Now wegeneralize a notion of ascending (descending) arrow diagram to arrow diagramswith two boundary components.Definition 4.1.13. Let A be an arrow diagram with two boundary components.As we go along the component of ∂Σ(A) starting from the base point, we passon the boundary of each ribbon once or twice. We call core arrows, whichwe pass only in one direction, the separating arrows. Now we place anotherstarting point • on the second component of ∂Σ(A) near the first separatingarrow which we encounter in the passage, and start going along this componentof ∂Σ(A). A is ascending (respectively, descending) if we pass each ribbon ofΣ(A) first time in the direction opposite to its core arrow (respectively, in thedirection).n=065

Example 4.1.14. Arrow diagrams below have two boundary components. Diagram(a) is ascending and diagram (b) is descending. Separating arrows areshown in bold.aDenote by A 2 n,m (respectively, D 2 n,m) the set of all ascending (respectively,descending) arrow diagrams with n arrows, m circles and two boundary components.Example 4.1.15. All diagrams in the set A 2 2,2 are presented below.bLet G be any Gauss diagram with m circles. We setA 2 n,m(G) :=∑⟨A, G⟩ Dn,m(G) 2 := ∑⟨A, G⟩.A∈A 2 n,mA∈D 2 n,mA state S(ϕ) corresponding to ϕ : A → G for an ascending (respectivelydescending) diagram A with one or two boundary components will be also calledascending (respectively descending). It is useful to reformulate this notion interms of a tracing of G.Definition 4.1.16. Given ϕ : A → G, a passage along the boundary of thesurface Σ(A) induces a tracing of G: we follow an arc of a circle of G startingfrom the base point until we hit an arrow in S(ϕ), turn to this arrow, thencontinue on another arc of G following the orientation and so on, until wereturn to the base point. In case of two boundary components we repeat thesame procedure starting near the image of the first separating arrow. Then astate S(ϕ) is ascending (respectively descending), if we approach every arrowin the tracing first time at its head (respectively at its tail).4.1.5 Separating statesIn this subsection we define a notion of a separating state. This notion will beextensively used in the following sections.66

1 12 22 2 1 1iijja b cFigure 4.7: Ascending and descending labelingDefinition 4.1.17. Let G be a based Gauss diagram. An ascending (respectivelydescending) separating state S of G is a state S of G, together with alabeling of arcs of G (i.e., intervals of circles of G between endpoints of arrows)by 1 and 2 such that:1. Each arc near α ∈ S is labeled as in Figure 4.7a (respectively in Figure4.7b).2. Each arc near α /∈ S is labeled as in Figure 4.7c.3. An arc with a base point is labeled by 1.Every separating (ascending or descending) state S in G defines a new Gaussdiagram G S with labeled circles as follows:We smooth each arrow in G which belongs to S, as shown in Figure 4.8, anddenote resulting smoothed Gauss diagram by G S . Each circle in G S is labeledby i, if it contains an arc labeled by i.Figure 4.8: Smoothing of an arrowNow we return to arrow diagrams with two boundary components. LetA ∈ A 2 n,m or A ∈ Dn,m. 2 We denote by σ(A) the set of separating arrows in Aand label the arcs of circles in A by 1 if the corresponding arc belongs to thefirst boundary component of Σ(A) and by 2 otherwise.67

Note that each homomorphism ϕ : A → G induces an ascending or descendingseparating state S of G, by taking S = ϕ(σ(A)) and labeling each arc of Gby the same label as the corresponding arc of A.Definition 4.1.18. Let G be a based Gauss diagram with m circles. Let Sbe an ascending (respectively descending) separating state of G, A ∈ A 2 n,m(respectively A ∈ D 2 n,m), and ϕ : A → G. We say that ϕ is S-admissible, ifan ascending (respectively descending) separating state induced by ϕ coincideswith S.Definition 4.1.19. Let S be an ascending (respectively descending) separatingstate of G, and A ∈ A 2 n,m (respectively A ∈ Dn,m). 2 In each case we define anS-pairing ⟨A, G⟩ S by:⟨A, G⟩ S :=∑sign(ϕ),ϕ:A→Gwhere the summation is over all S-admissible ϕ : A → G. We setA 2 n,m(G) S :=∑⟨A, G⟩ S and Dn,m(G) 2 S := ∑⟨A, G⟩ S .A∈A 2 n,mA∈D 2 n,m4.2 Counting surfaces with one boundarycomponentIn this section we review Gauss diagram formulas for coefficients of the Conwaypolynomial ∇ obtained in [15, Theorem 3.5] for classical knots and 2-componentclassical links.Theorem 4.2.1 ([15]). Let G be a Gauss diagram of a classical knot or 2-component classical link L.where ∇(L) is the Conway polynomial.∇ asc (G) = ∇ des (G) = ∇(L), (4.1)Let G be a Gauss diagram with m circles. We give a direct proof of theinvariance of both ∇ asc (G) and ∇ des (G) under the Reidemeister moves which donot involve a base point. This allows us to extend this result to m-componentclassical links and also to define two different generalizations of the Conwaypolynomial to long virtual links. At the end of this section we present someproperties of these polynomials.68

4.2.1 Invariants of long linksIn this subsection we generalize the result of [15] to m-component (classical orvirtual) long links.Theorem 4.2.2. Let G be a Gauss diagram of an m-component (classical orvirtual) long link L. Then ∇ asc (G) and ∇ des (G) define polynomial invariantsof long links, i.e. do not depend on the choice of G.Proof. We will prove that ∇ asc (G) is an invariant of an underlying link. Theproof for ∇ des (G) is the same. It suffices to show that A n,m (G) is invariant underthe Reidemeister moves Ω 1 , Ω 2 and Ω 3 of Figure 4.4 applied away from the basepoint. Let G and ˜G be two Gauss diagrams that differ by an application of Ω 1 ,so that ˜G has one additional isolated arrow α on one of the circles. Ascendingstates of G are in bijective correspondence with ascending states of ˜G which donot contain α. But α cannot not be in the image of ϕ : A → G with A ∈ A n,m ,because A should have one boundary component. ThusA n,m (G) = A n,m ( ˜G).Let G and ˜G be two Gauss diagrams that differ by an application of Ω 2 , sothat ˜G has two additional arrows α ε and α −ε , see Figure 4.4. Ascending statesof G are in bijective correspondence with ascending states of ˜G which do notcontain α ±ε . Note that both α ε and α −ε can not be in the image of ϕ : A → Gwith A ∈ A n,m because A has one boundary component. Ascending states of ˜Gwhich contain one of α ±ε come in pairs S ∪ α ε and S ∪ α −ε with opposite signs,thus cancel out in A n,m ( ˜G). HenceA n,m (G) = A n,m ( ˜G).Let G and ˜G be two Gauss diagrams that differ by an application of Ω 3 , asshown in Figure 4.4 (G is on the left and ˜G is on the right).Then there is a bijective correspondence between ascending states of G and˜G. This correspondence preserves the signs and the combinatorics of the orderin which the tracing enters and leaves the neighborhood of these arrows. Thetable below summarizes this correspondence.69

112112323321323323112112112323321323323112211211113232313233223211211113232313233223For a better understanding of this table, let us explain one of the cases indetails. Denote the top, left, and right arrows in the fragment by α t , α l , andα r respectively.Consider a state S ∪α l ∪α r of G which contains two arrows of the fragment.The order of tracing the fragment depends on S. Only two orders of tracingmay give an ascending state:1in1out1in1outStates of G :2out2in3in3outor3out3in2in2out1in1out1in1outStates of ˜G :2out2in3in3outor3out3in2in2outHere the three consecutive entries and exits from the fragment are indicatedby 1in, 1out, 2in, 2out, 3in, 3out. In the first case, the corresponding state of˜G is S ∪ α t ∪ α r . Note that the pattern of entries and exits from the fragment70

is indeed the same as in G. In the second case, the corresponding state of ˜G isS ∪ α t ∪ α l . The pattern of entries and exits is again the same as in G.4.2.2 Properties of A n,m (G) and D n,m (G)Theorem 4.2.3. Let G + , G − and G 0 be Gauss diagrams which differ only inthe fragment shown in Figure 4.9. Then∇ asc (G + ) − ∇ asc (G − ) = z∇ asc (G 0 ) ∇ des (G + ) − ∇ des (G − ) = z∇ des (G 0 ).(4.2)+ −Figure 4.9: A Conway triple of Gauss diagramsProof. Again we will prove this theorem for ∇ asc . Denote by m and m 0 thenumber of circles in G ± and G 0 , respectively. It is enough to prove that forevery n ∈ NA n,m (G + ) − A n,m (G − ) = A n−1,m0 (G 0 ) (4.3)Denote the arrows of G + and G − appearing in the fragment in Figure 4.9 byα + and α − , respectively. The proof of the skein relation is the same as in [15].All ascending states of G ± which do not contain α ± cancel out in (4.3) in pairs.Each ascending state S ∪ α ± of G ± corresponds to a unique ascending state Sof G 0 , and vice versa: if S is an ascending state of G 0 , then (depending on theorder of the fragments in the tracing) exactly one of S ∪ α + and S ∪ α − willgive an ascending state on either G + or G − .It is easy to see that both A n,m (G) and D n,m (G) depend on the positionof the base point when G is a Gauss diagram associated with a virtual linkdiagram. Let G and Ĝ be two Gauss diagrams shown in Figure 4.10. ThenA 2,1 (G) = 1, A 2,1 (Ĝ) = 0, D 2,1(G) = 0, D 2,1 (Ĝ) = 1.However, for classical links this is not the case. Our next theorem statesthat in the case of classical links, both ∇ asc (G) and ∇ des (G) are independentof the position of the base point.71

G =+ +G^=+ +Figure 4.10: Dependence on a basepointTheorem 4.2.4. Let G be a based Gauss diagram of an m-component classicallink L. Both ∇ asc (G) and ∇ des (G) are independent of the position of the basepoint.Proof. We will prove the independence of A n,m (G) of the position of the basepoint, the proof for D n,m (G) is the same. We prove this statement by inductionon the number of arrows in G.If G has no arrows, there is nothing to prove. Now let us assume that thestatement holds for any (classical) Gauss diagram with less than k arrows, andlet G be a Gauss diagram with k arrows. If k < n, then A n,m (G) = 0 and weare done, so we may assume that k ≥ n. Suppose that the base point lies onthe i-th component of G. We should prove that we may move the base pointacross any arrowhead or arrowtail on the i-th component, and to shift it to anyother, say, j-th, component. Denote by G 1 , . . . , G 7 Gauss diagrams which differonly in a fragment which looks likei j i j i j i j i j i j i jG 1 G 2 G 3 G 4 G 5 G 6 G 7respectively. It suffices to prove that we have:A n,m (G 1 ) = A n,m (G 2 ) , A n,m (G 3 ) = A n,m (G 4 ) , A n,m (G 3 ) = A n,m (G 5 ) (4.4)Denote by α the arrow appearing in the fragment above. The first equalityis immediate; indeed, in G 1 and G 2 there are no ascending states with oneboundary component which contain α (and all other ascending states are in abijective correspondence).72

To prove the second equation in (4.4), note that there is a bijection betweenascending states of A n,m (G 3 ) and A n,m (G 4 ) which do not contain α; the remainingascending states of G 3 and G 4 look exactly like the ones of G 6 and G 7 ,respectively; thus we haveA n,m (G 3 ) − A n,m (G 4 ) = A n−1,m0 (G 6 ) − A n−1,m0 (G 7 ) = 0,where m 0 = m±1 is the number of circles in G 6 and G 7 and the second equalityholds by the induction hypothesis.The proof of the last equality in (4.4) is more complicated. We will use aninner induction on the number r of arrows which have only their arrowtail onthe i-th component.If r = 0, then A n,m (G 3 ) = A n,m (G 5 ) = 0. Indeed, for r = 0 there are noascending states in G 5 (since we cannot reach the i-th circle), so A n,m (G 5 ) = 0.Also, since r = 0, i-th component of the link is under all other components, sowe may move it apart by a finite sequence of Reidemeister moves Ω 2 and Ω 3applied away from the base point, converting G 3 into a Gauss diagram withan isolated i-th component (here we use the fact that G 3 is associated with aclassical link diagram); thus A n,m (G 3 ) = 0 by Theorem 4.2.2.Let’s establish the step of induction. On both G 3 and G 5 pick the samearrow, which has only its arrowtail on the i-th component, and apply the skeinrelation of Theorem 4.2.3 to simplify the corresponding Gauss diagrams. Diagramson the right-hand side of the skein relation have less than k crossings,so the right-hand sides are equal by the induction on k; the remaining terms inthe left-hand side are also equal by induction on r.Corollary 4.2.5. Let G be a Gauss diagram of an m-component classical linkL. Then∇ asc (G) = ∇ des (G) = ∇(L),i.e. for every n ≥ 0A n,m (G) = D n,m (G) = c n (L). (4.5)Proof. By Theorems 4.2.2 – 4.2.4, A n,m (G) and D n,m (G) are link invariantswhich satisfy the same skein relation as c n (L). It remains to compare theirnormalization, i.e. their values on the unknot O. A standard Gauss diagramG(O) of an unknot consists of one circle with no arrows. ThusD 0,1 (G(O)) = A 0,1 (G(O)) = 1, and D n,m (G(O)) = A n,m (G(O)) = 0 otherwise.Hence A n,m (G) and D n,m (G) have the same normalization as c n (L) and thecorollary follows.73

Example 4.2.6. Consider a 3-component link L and the Gauss diagram G ofL shown below:4−−−−1 23The only ascending state of G is {3, 4}. It sign is +. Thus c 2 (L) = 1 andc n (L) = 0 for all n ≠ 2, so ∇(L) = z 2 .4.2.3 Alexander-Conway polynomials of long virtual linksIn this subsection we study properties of the polynomials ∇ asc and ∇ des for longvirtual links, and compare them with other existing constructions.Let L be a classical or long virtual link and G be any Gauss diagram ofL. Polynomials ∇ asc (L) := ∇ asc (G) and ∇ des (L) := ∇ des (G) were defined in[15], but the proof that they are well defined for classical links with more than 2components and for long virtual links was not presented. By Theorem 4.2.2 andCorollary 4.2.5 ∇ asc (L) and ∇ des (L) are invariants of L, and if L is a classicallink∇ asc (L) = ∇ des (L) = ∇(L).Note that for long virtual links it may happen that∇ asc (L) ≠ ∇ des (L).For example, let G be a Gauss diagram of the long virtual Hopf link L shownin Figure 4.2. Then ∇ asc (L) = z, but ∇ des (L) = 0. We denote by∇ asc−des (L) := ∇ asc (L) − ∇ des (L).This polynomial vanishes on classical links but, as we will see below, may beused to distinguish virtual links from classical links.Let D be a diagram of a (long) virtual link L and G its corresponding Gaussdiagram. Pick a classical crossing on D. A move on D and the correspondingmove on G shown in Figure 4.11 is called the virtualization move.Theorem 4.2.7. Let L and L 1 be long virtual links. Then the following holds.a) ∇ asc (L#L 1 ) = ∇ asc (L)∇ asc (L 1 ) , ∇ des (L#L 1 ) = ∇ des (L)∇ des (L 1 ).74

εεFigure 4.11: The virtualization moveb) Non-trivial coefficients of ∇ des and ∇ asc are not invariant under the virtualizationmovec) Coefficients of ∇ des (L) and ∇ asc (L) are Vassiliev invariants in the senseof both GPV [30] and Kauffman [35].Proof. The proof of Part a) is straightforward and follows from the definitionof ∇ asc and ∇ des .Part b). Consider even n first. Let n = 2k and let G and G 1 be Gaussdiagrams of long virtual knots shown in Figure 4.12a and 4.12b respectively.Note that G and G 1 differ by an application of the virtualization move. BothG and G 1 have n + 1 arrows. It is easy to check thatA n,1 (G) = −1 D n,1 (G) = 1, but A n,1 (G 1 ) = D n,1 (G 1 ) = 0.To prove the statement for odd n, add a Hopf-linked unknot to the abovediagrams.Part c) It is enough to prove that A n,m (G) and D n,m (G) are GPV finite typeinvariants, because any GPV finite type invariant is automatically of Kauffmanfinite type. But [30, page 12] implies, that any invariant given by an arrowdiagram formula with n arrows is GPV finite type of degree n.Remark 4.2.8. It follows from [19, Theorem 1], that Part b) in Theorem 4.2.7follows from Part c).Let K T be a virtual knot with a virtual diagram shown in Figure 4.13. Thisknot is called the Kishino knot. It has attracted attention for its remarkableproperty that it is a connected sum of two diagrams of the trivial knot; it hastrivial Jones polynomial, Z ′ K T(t, y) = 0 (for the definition of Z ′ see [58]), and75

−−2k2k−−++abFigure 4.12: Diagrams of long virtual knots that differ by an application of thevirtualization movethe virtual knot group of K T is isomorphic to Z, see [38]. It was first proved tobe non-classical in [38]. We show that K T is non-classical using the polynomials∇ asc , ∇ des and ∇ asc−des .Figure 4.13: A diagram of the Kishino knotProposition 4.2.9. Polynomials ∇ asc , ∇ des and ∇ asc−des detect the fact thatK T is a non-classical knot.Proof. Recall that for any Gauss diagram G of a classical knot all these polynomialsare independent of the position of the base point. Consider two Gaussdiagrams G and Ĝ of K T which differ only by the position of the base point,see Figure 4.14.We have∇ asc (G) = 1 − 2z 2 + z 4 ∇ des (G) = 1 ∇ asc−des (G) = −2z 2 + z 4 but∇ asc (Ĝ) = 1 ∇ des(Ĝ) = 1 − 2z2 + z 4 ∇ asc−des (Ĝ) = 2z2 − z 4 .76

^G = G−−=−+ ++ +−Figure 4.14: Gauss diagrams of the Kishino knotComparison with other constructions of Alexander-Conwaypolynomials of virtual linksIn [58] Sawollek associated to every link diagram D of a virtual link L a Laurentpolynomial Z D ′ (t, y) in two variables t, y. He proved that Z′ D (t, y) is an invariantof virtual links up to multiplication by powers of t ±1 , and that it vanishes onclassical links. He also showed that Z D ′ (t, y) satisfies the following skein relation:Z ′ D +(t, y) − Z ′ D −(t, y) = (t −1 − t)Z ′ D 0(t, y).It is obvious that both ∇ asc (L) and ∇ des (L) are crucially different from Z L ′ (t, y)because both of them do not vanish on classical links, but one can suspectthat ∇ asc−des (L) coincides with Z L ′ (t, y) after a possible renormalization and achange of variables z = t −1 − t. Sawollek proved the following theorem:Theorem 4.2.10 ([58]). Let D, D 1 , D 2 be virtual link diagrams and let D 1 ⊔D 2denote the disconnected sum of the diagrams D 1 and D 2 . ThenZ ′ D 1 ⊔D 2(t, y) = Z ′ D 1(t, y)Z ′ D 2(t, y).Note that ∇ asc−des (L ⊔ L) = 0 for any long virtual link L, but Z L⊔L ′ (t, y) =(Z L ′ (t, y))2 . Thus ∇ asc−des (L) is also drastically different from Z L ′ (t, y).Another generalizations of Alexander polynomials to virtual links are derivedfrom the virtual and extended virtual link groups, see [58] and [60, 61]respectively.1. Following [58] we denote by ∆ L (t) a polynomial which is derived from thevirtual link group of a link L. It is well defined up to sign and multiplicationby powers of t ±1 . For every virtual link diagram D the associated polynomialis denoted by ∆ D (t). In contrast to the classical Alexander polynomial, theAlexander polynomial of [58] for virtual links does not satisfy any linear skeinrelation as stated in the next theorem:77

Theorem 4.2.11 ([58]). For any normalization A D (t) of the polynomial ∆ D (t),i.e., A D (t) = ε D t n D∆ D (t) with some ε D ∈ {−1, 1} and n D ∈ Z, the equationp 1 (t)A D+ (t) + p 2 (t)A D− (t) + p 3 (t)A D0 (t) = 0 with p 1 (t), p 2 (t), p 3 (t) ∈ Z[t ±1 ] hasonly the trivial solution p 1 (t) = p 2 (t) = p 3 (t) = 0.Since ∇ asc , ∇ des and ∇ asc−des satisfy the Conway skein relation, it followsthat all of them are crucially different from ∆ D (t) of [58].2. Let L be a virtual link. The polynomial ∆ 1 (L) of [60, 61] is a polynomialin variables v, u and is well defined up to multiplication by powers of (uv) ±1 .If L is a classical link, then ∆ 1 (L) is equal to the Alexander polynomial in thevariable uv. It follows that ∇ asc−des is different from ∆ 1 , because ∆ 1 is notidentically zero on the family of classical links.Given a virtual link L, we denote by L ∗ the mirror image of L, i.e. a linkobtained by inverting the sign of each classical crossing in a diagram of L. Thefollowing corollary was proved in [61].Corollary 4.2.12 ([61], Corollary 5.2). Let L be a virtual m-component link.Then∆ 1 (L)(u, v) = (−1) m ∆ 1 (L ∗ )(v, u)up to multiplication by powers of (uv) ±1 .In particular, for any virtual knot ∆ 1 (K)(u, v) = −∆ 1 (K ∗ )(v, u).Consider a mirror pair of long virtual knots K and K ∗ with Gauss diagramsG and G ∗ shown in Figure 4.15.G= G * =+ + − −Figure 4.15: A mirror pair of Gauss diagramsThen ∇ asc (K) = ∇ des (K ∗ ) = 1 + z 2 , ∇ asc (K ∗ ) = ∇ des (K) = 1. Thus both ∇ ascand ∇ des are crucially different from ∆ 1 (L) in [60, 61].Another way to see that both ∇ asc and ∇ des are different from ∆ 1 is byfinding a virtual knot K, such that both ∇ asc and ∇ des detect that this knotis non-classical, but ∆ 1 (K) = 1 (so ∆ 1 does not distinguish this knot from theunknot). An example of such a knot K, together with a pair G and Ĝ of its78

G=−−−−G^=−−−−Figure 4.16: A virtual knot and two of its based Gauss diagramsGauss diagrams which differ by the position of the base point, is given in Figure4.16. It was shown in [61] that ∆ 1 (K) = 1.We have∇ asc (G) = 1 + z 2 ∇ des (G) = 1 + z 2 but∇ asc (Ĝ) = 1 + 2z2 + z 4 ∇ des (Ĝ) = 1.It follows that both ∇ asc and ∇ des show that K is non-classical.Finally, another generalization of the Alexander polynomial (related to thepolynomial Z ′ of [58]) to long virtual knots was presented in [1]. It is a Laurentpolynomial ζ in a variable t over the following ring T = Z[p, p −1 , q, q −1 ]/((p −1)(p − q), (q − 1)(p − q)). This polynomial also vanishes on classical knots andthus ζ significantly differs from ∇ asc and ∇ des .Question. Is it possible to derive ∇ asc−des from ζ?4.3 Counting surfaces with two boundarycomponentsIn this section we present a new infinite family of Gauss diagram formulas,which correspond to counting of orientable surfaces with two boundary components.At the end of this section we identify the resulting invariants withcertain derivatives of the HOMFLYPT polynomial.4.3.1 Link invariants and diagrams with two boundarycomponentsIn this subsection we define invariants of classical links using ascending anddescending arrow diagrams with two boundary components.79

Recall that for every Gauss diagram G we defined notions of ascending anddescending separating states of G, see Subsection 4.1.5. Also, for every ascending(respectively descending) separating state S of G and for an arrow diagramA ∈ A 2 n,m (respectively A ∈ Dn,m) 2 we defined, in the same subsection, a notionof S-admissible pairing ⟨A, G⟩ S . Note that every ascending (respectivelydescending) separating state S of G defines two Gauss diagrams G ′ S and G′′ S asfollows: G ′ S (respectively G′′ S ) consists of all circles of G S labeled by 1 (respectivelyby 2), and its arrows are arrows of G with both ends on these circles. Allarrows with ends on circles of G S with different labels are removed. The basepoint on G ′ is the base point ∗ of G. The base point on G ′′ is placed near thefirst arrow in S which we encounter as we walk on G starting from ∗.If G is a Gauss diagram of a classical link L, then G ′ S and G′′ S correspond toclassical links L ′ S and L′′ S , which are defined as follows. We smooth all crossingswhich correspond to arrows in S, as shown below:We obtain a diagram of a smoothed link L S with labeling of components inducedfrom the labeling of circles of G S . Denote by L ′ S and L′′ S sublinks which consistof components labeled by 1 and 2 respectively.It follows from Corollary 4.2.5 that for every n ≥ 0 we haveA n,m ′(G ′ S) = D n,m ′(G ′ S) = c n (L ′ S) and A n,m ′′(G ′′ S) = D n,m ′′(G ′′ S) = c n (L ′′ S),where m ′ and m ′′ are the number of components of L ′ S and L′′ S respectively.Using this together with the definition of A 2 n,m(G) S and D 2 n,m(G) S in Subsection4.1.5 we getLemma 4.3.1. Let G be a Gauss diagram of an m-component link L. Thenfor every n ≥ 0 and an ascending (respectively descending) separating state Sof G we haven−|S|∑A 2 n,m(G) S = sign(S) c i (L ′ S)c n−|S|−i (L ′′ S)i=0n−|S|∑Dn,m(G) 2 S = sign(S) c i (L ′ S)c n−|S|−i (L ′′ S)i=0andrespectively.Here |S| is the number of arrows in S and sign(S) = ∏ α∈S sign(α).80

Summing over all ascending (descending) separating states S, we obtainCorollary 4.3.2. Let G be any Gauss diagram of an m-component link L.Then for every n ≥ 0 we haveA 2 n,m(G) =D 2 n,m(G) =n∑∑k=1 S,|S|=kn∑∑k=1 S,|S|=k∑n−ksign(S) c i (L ′ S)c n−k−i (L ′′ S)i=0∑n−ksign(S) c i (L ′ S)c n−k−i (L ′′ S)where the second summation is over all ascending and descending separatingstates S of G respectively.It turns out that both A 2 n,m(G) and D 2 n,m(G) are invariant under Ω 2 and Ω 3 :Theorem 4.3.3. Let G be any Gauss diagram of an m-component link L. ThenA 2 n,m(G) and D 2 n,m(G) are invariant under Reidemeister moves Ω 2 and Ω 3 whichdo not involve the base point.Proof. We will prove the invariance of A 2 n,m(G); the proof for D 2 n,m(G) is thesame.Let G and ˜G be two Gauss diagrams that differ by an application of Ω 2 ,so that ˜G has two additional arrows α ε and α −ε , see Figure 4.4. Ascendingstates of G are in bijective correspondence with ascending states of ˜G whichdo not contain α ±ε . Note that α ε and α −ε can not be both in the image ofϕ : A → G with A ∈ A 2 n,m, because A is an ascending diagram with twoboundary components. Ascending states of ˜G which contain one of α ±ε comein pairs S ∪ α ε and S ∪ α −ε with opposite signs, thus cancel out in A 2 n,m( ˜G).HenceA 2 n,m(G) = A 2 n,m( ˜G).Now, let G and ˜G be two Gauss diagrams that differ by an application ofΩ 3 , see Figure 4.4 (G is on the left and ˜G is on the right). Denote the top,left, and right arrows in the fragment by α t , α l , and α r respectively. There is abijective correspondence between ascending separating states of G and ˜G, suchthat none of the arrows α r , α l and α t belong to these states. Indeed, we mayidentify separating states of G and ˜G which have the same arrows and the samelabeling of arcs. For any such separating state S we have A 2 n,m(G) S = A 2 n,m( ˜G) Sby Lemma 4.3.1 and Theorem 4.2.2.81i=0

122GG ~2 1 12 21111SL S~ S12 1 111~SLL SL ~ SFigure 4.17: Identifying ascending separating states containing α lAn ascending separating state of G or ˜G may contain either exactly onearrow of the fragment i.e. α r or α l or α t , or it may contain both arrows α r andα l . There is a bijective correspondence between ascending separating states ofG and ˜G which contain α l . Two possible cases of this correspondence (whichdiffer by the labeling) are shown in Figure 4.17 and Figure 4.18.G1222 11222SL SL SG ~12 1 2222 21~SL ~ SL ~ SFigure 4.18: Identifying other ascending separating states containing α lIn both cases, links L ′ S and L′′ S constructed from G and ˜G are isotopic,thus by Lemma 4.3.1 A 2 n,m(G) S = A 2 n,m( ˜G) ˜S.The situation with ascendingseparating states which contain α r is completely similar and is omitted.The correspondence of ascending separating states which contain α l ∪ α r or82

GG1 11 12 22 2 21 12 22 22 1 2S t L t L tS lr L lr L lrG ~1 1222 2 2 2 2~S t~L t~L tFigure 4.19: Comparison of ascending separating states of G and ˜Gα t is more complicated. One of the two possible cases is summarized in Figure4.19. Links ˜L ′ t, L ′ t and L ′ lr are isotopic, thusc i ( ˜L ′ t) = c i (L ′ t) = c i (L ′ lr).Moreover, for c i (˜L ′′t ) we have:( ) ((c i + + = c i + −)+c i−1+)= c i( )+c i−1(+)The first equality is the skein relation of Theorem 4.2.3, and the second equalityholds by the invariance of c i under Ω 2 . Hencec i (˜L ′′t ) = c i (L ′′t ) + c i−1 (L ′′lr).Denote by k the number of arrows in S t . Note that the number of arrows in ˜S tand S lr is k and k + 1, respectively. Thus∑n−ki=0∑n−kc i (˜L ′ t)c n−k−i (˜L ′′t ) =i=0n−k−1∑c i (L ′ t)c n−k−i (L ′′t ) +83i=0c i (L ′ lr)c n−k−i−1 (L ′′lr).

G1 11 12 2S t1 1 1L tL tG ~ G ~1 1222 21 1 112 1 1 121 2 1~S t~S lr~L t~L lr~L t~L lrFigure 4.20: Comparison of other ascending separating states of G and ˜GNote that sign( ˜S t ) = sign(S t ) = sign(S lr ), thus by Lemma 4.3.1A 2 n,m( ˜G) ˜St= A 2 n,m(G) St + A 2 n,m(G) Slr .The second possible case (which differs by labeling) is shown in Figure 4.20.Abusing the notation we again denote the corresponding ascending separatingstates by S t , ˜S t , ˜S lr . Links L ′′ ′′ ′′t , ˜L t and ˜L lr are isotopic. Applying the skeinrelation for c i (L ′ t) similarly to the above, in this case we getA 2 n,m(G) St = A 2 n,m( ˜G) ˜St+ A 2 n,m( ˜G) ˜Slr.Our next step is to study the behavior of A 2 n,m(G) and D 2 n,m(G) under anapplication of Reidemeister move Ω 1 . Both A 2 n,m(G) and D 2 n,m(G) change underΩ 1 , see Example 4.3.4.Example 4.3.4. Let G, G 1 and G 2 be Gauss diagrams of an unknot shown inFigures 4.21a, 4.21b and 4.21c respectively. Then D 2 1,1(G) = 0, butD 2 1,1(G 1 ) = 1; and A 2 1,1(G) = 0, but A 2 1,1(G 2 ) = −1.84

+ −a b cFigure 4.21: Gauss diagrams G, G 1 and G 2However, this problem is easy to solve. Denote byand letAD n,m (G) := A 2 n,m(G) + D 2 n,m(G)I n,m (G) := AD n,m (G) − w(G)c n−1 (L),where w(G) is the writhe of G, i.e. the sum of signs of all arrows in G.Theorem 4.3.5. Let G be a Gauss diagram of a long m-component link L.Then I n,m (G) is an invariant of an underlying link L, i.e. is independent of achoice of G.Proof. By Theorem 4.3.3, it remains to prove the invariance of I n,m under Ω 1(applied away from the base point). Let ˜G and G be two Gauss diagrams whichare related by an application of Ω 1 , such that ˜G contains a new isolated arrow α.Then α contributes either a new ascending or a new descending separating state{a}, depending on whether we meet its head or tail first on the passage from thebase point. Contribution of this state to AD n,m ( ˜G) is either sign(α)A n−1,m (G),or sign(α)D n−1,m (G); butsign(α)A n−1,m (G) = sign(α)c n−1 (L) = sign(α)D n−1,m (G)by Corollary 4.2.5, thusI n,m ( ˜G) − I n,m (G) = (sign(α) − w( ˜G) + w(G))c n−1 (L).It remains to note that w( ˜G) − w(G) = sign(α).Our next step is to study dependence of A 2 n,m(G) and D 2 n,m(G) on the positionof the base point. The example below shows that each of them dependson the base point.85

Example 4.3.6. Let G and Ĝ be two Gauss diagrams of the right-handedtrefoil shown below. Then A 2 3,1(G) = 0 and D3,1(G) 2 2= 1, but A3,1(Ĝ) = 1 andD 2 3,1(Ĝ) = 0. + +G=+G^ =++ +It turns out, however, that the sum A 2 n,m(G) + Dn,m(G) 2 = AD n,m (G) doesnot depend on the base point. Indeed, let G and Ĝ be two Gauss diagramswhich differ only by a position of their base points. Let Ŝ be an ascendingseparating state of Ĝ. If an arc which contains the base point ∗ of G is labeledby 1, then S = Ŝ is an ascending separating state of G and by Lemma 4.3.1 wehave A 2 n,m(G) S = A 2 n,m(Ĝ) Ŝ. If an arc which contains the base point ∗ of G islabeled by 2, we consider a descending separating state S of G which has thesame arrows as Ŝ, but opposite labels. By Lemma 4.3.1, D2 n,m(G) S = A 2 n,m(Ĝ) Ŝ .We repeat this process, replacing ascending separating states with descendingand A with D. Summing up by separating states, in view of Corollary 4.3.2 weobtain the followingTheorem 4.3.7. Let G be a based Gauss diagram of an m-component link L.Then AD n,m (G) is independent of the position of the base point.It has two important corollaries.Corollary 4.3.8. Let G be any Gauss diagram of an m-component framed linkL. Then AD n (L) = AD n,m (G) is an invariant of an underlying framed link, i.edoes not depend on G.Proof. It is easy to see that, if Gauss diagrams G and ˜G differ by Ω F 1 move,then AD n,m (G) = AD n,m ( ˜G). Now the proof follows directly from Theorems4.3.3 and 4.3.7.Corollary 4.3.9. Let G be any Gauss diagram of an m-component link L. ThenI n (L) := I n,m (G) is an invariant of an underlying link L, i.e. is independent ofa choice of G.4.3.2 Properties of I nIn this subsection we establish the skein relation for I n,m . Then we identify I n,mwith coefficients of a certain polynomial, which is derived from the HOMFLYPTpolynomial.86

Skein relationIn this part we establish the skein relation for AD n,m (G).Theorem 4.3.10. Let L + , L − , L 0 be a Conway triple of links with the correspondingConway triple G + , G − , G 0 of Gauss diagrams, see Figures 1.4 and4.9. Denote the number of circles of G ± and G 0 by m and m 0 , respectively.ThenAD n,m (G + ) − AD n,m (G − ) =⎧⎪⎨ AD n−1,m0 (G 0 ) , if m 0 = m − 1⎪⎩ AD n−1,m0 (G 0 ) + ∑ ∑c i (L ′ )c n−i−1 (L 0 L ′ ) if m 0 = m + 1 (4.6)n−1L ′ ⊂L 0 i=0where the summation is over all sublinks L ′ of L 0 which contain exactly onenew component resulting from the smoothing.Proof. Denote the arrows of G + and G − appearing in Figure 4.9 by α + and α − ,respectively.Let us look at labels of ascending separating states of G ± and G 0 on four arcsof the shown fragment. If labels of all four arcs are the same, we may identifystates of G ± and G 0 with the same arrows and labels of arcs, see Figure 4.22a.Lemma 4.3.1 and Theorem 4.2.3 imply, that for every such state SA 2 n,m(G + ) S − A 2 n,m(G − ) S = A 2 n−1,m 0(G 0 ) S .If labels on two arcs near the head of α ± coincide, but differ from labels nearthe tail of α ± , by Lemma 4.3.1 we have A 2 n,m(G + ) S − A 2 n,m(G − ) S = 0 for anysuch state S of G ± , and there is no corresponding state of G 0 . See Figure 4.22b(for i ≠ j).There are two further cases when labels of two arcs near the head of α ± aredifferent. Such a state S of G 0 corresponds either to an ascending separatingstate S ∪α + of G + , or to an ascending separating state S ∪α − of G − , see Figure4.22c. By Lemma 4.3.1 we have A 2 n,m(G + ) S∪α+ = A 2 n−1,m 0(G 0 ) S in the first caseand A 2 n,m(G − ) S∪α− = −A 2 n−1,m 0(G 0 ) S in the second case.If m 0 = m − 1, there are no other ascending separating states of any ofthe diagrams and, repeating this computation for descending separating states,summing over states and using Corollary 4.3.2, we obtain the first equality in(4.6).87

iiii+iiii+jj2 2 2 + 21 1 1 1iiaii_iiii_bjj1 1 1 _ 12 2 2 2cFigure 4.22: Correspondence of separating states of G 0 and G ±If m 0 = m + 1, both ends of α ± are on the same circle of G ± and there isan additional contribution to AD n,m (G ± ) of separating states S = {α ± } of G ± ,which contain only the arrow α ± (and some labeling of arcs) 1 . Such separatingstates correspond to labeling all circles of G 0 by 1, 2 so that the based circle islabeled by 1, and two new components of G 0 resulting from the smoothing havedifferent labels. Denote by L ′ the sublink labeled by 1. The case of descendingseparating states {α ± } is similar. By Corollary 4.3.2, the contribution of thesestates to AD n,m (G + ) − AD n,m (G − ) equalsand the theorem follows.∑n−1L ′ ⊂L 0 i=0∑c i (L ′ )c n−i−1 (L 0 L ′ )Identification of the invariant I nIn this subsubsection we identify I n with certain derivatives of the HOMFLYPTpolynomial.Let P (L) be the HOMFLYPT polynomial of a link L. We denote by P ′ a(L)the first derivative of P (L) w.r.t. a. Then P ′ a(L)| a=1 is a polynomial in thevariable z. We denote by p n (L) the coefficient of z n in zP ′ a(L)| a=1 .Theorem 4.3.11. Let L be an m-component link. Then for every n ≥ 0I n (L) = p n (L) − C n (L), (4.7)1 These states have no counterpart in G 0 , since such a separating state of G 0 should beempty and the corresponding surface disconnected.88

where C n (L) is defined byC n (L) := ∑ L ′ ⊂Ln∑c i (L ′ )c n−i (L L ′ ),and the summation is over all proper sublinks L ′ of L.i=0Proof. It is enough to show that I n (L)+C n (L) and p n (L) satisfy the same skeinrelation and take the same value on unlinks with any number of components.The skein relation for zP ′ a(L)| a=1 follows directly from the skein relationfor P (L). Differentiating this skein relation w.r.t. a, substituting a = 1, andmultiplying by z we obtainzP ′ a(L + )| a=1 − zP ′ a(L − )| a=1 + zP (L + )| a=1 + zP (L − )| a=1 = z 2 P ′ a(L 0 )| a=1 .Note that P (L)| a=1 = ∇(L) is the Conway polynomial of L. ThuszP ′ a(L + )| a=1 − zP ′ a(L − )| a=1 + z∇(L + ) + z∇(L − ) = z 2 P ′ a(L 0 )| a=1 .Taking the n-th coefficient, we getp n (L + ) − p n (L − ) + c n−1 (L + ) + c n−1 (L − ) = p n−1 (L 0 ). (4.8)The skein relation for C n is obtained directly from the Conway skein relation.It depends on the number m 0 of the components in L 0 :⎧⎪⎨ C n−1 (L 0 ), if m 0 = m − 1C n (L + )−C n (L − ) =⎪⎩ 2 ∑ ∑c i (L ′ )c n−i−1 (L 0 L ′ (4.9)) if m 0 = m + 1,n−1L ′ ⊂L 0 i=0where the summation is over all sublinks L ′ of L 0 which contain both newcomponents appearing after the smoothing. Now Theorem 4.3.10 and equality(4.9) yieldAD n,m (G + ) − AD n,m (G − ) + C n (L + ) − C n (L − ) = AD n−1,m0 (G 0 ) + C n−1 (L 0 ).Deducting w(G 0 )c n−2 (L 0 ) from both sides of this equation and noticing thatw(G 0 ) = w(G + ) − 1 = w(G − ) + 1 and c n−1 (L + ) − c n−1 (L − ) = c n−2 (L 0 ), sow(G 0 )c n−2 (L 0 ) = w(G + )c n−1 (L + ) − w(G + )c n−1 (L + ) − c n−1 (L + ) − c n−1 (L − ),89

we obtain the desired skein relation for I n (L) + C n (L):(I n (L + ) + C n (L + )) − (I n (L − ) + C n (L − ))+c n−1 (L + ) + c n−1 (L − ) =I n−1 (L 0 ) + C n−1 (L 0 ).It remains to compare values of I n (L) + C n (L) and p n (L) on an m-componentunlink O m . From the definition of I n (L) we get I n (O m ) = AD n,m (O m ) = 0 forany n and m. Also, the equality p n (O m ) = C n (O m ) holds for any n and m, sincep 0 (O 2 ) = C 0 (O 2 ) = 2 and p n (O m ) = C n (O m ) = 0 otherwise. This concludes theproof of the theorem.Example 4.3.12. Let G be a Gauss diagram of a link H 2 shown in Figure4.23. Let us calculate C n (H 2 ) and I n (H 2 ). Both components of H 2 are trivial,41 2 3++++Figure 4.23: Link H 2 and its Gauss diagramso C 0 (H 2 ) = 2 and C n (H 2 ) = 0 for n ≠ 0. The only ascending states of Gare {1, 2}, {1, 4}, and {3, 4}; the only descending state of G is {2, 3}. ThusAD 2,2 (G) = 4. Note that c 1 (H 2 ) = 2 and c n (H 2 ) = 0 for n ≠ 1, thusI 2 (H 2 ) = 4 − 4 · 2 = −4 and I n (H 2 ) = 0 for n ≠ 2. Indeed, one may check thatP (H 2 ) = a −3 z −1 − a −5 z −1 + a −3 z + a −1 z, so zP ′ a(H 2 )| a=1 = 2 − 4z 2 .4.3.3 The case of knotsIn this subsection we define for every n ≥ 0 another two invariants I A,n andI D,n of classical knots.Definition 4.3.13. Let G be a based Gauss diagram of a knot K. We go onthe circle of G starting from the base point until we return to the base point.Denote by w A (G) (respectively w D (G)) sum of signs of all arrows of G whichwe pass first at the arrowhead (respectively arrowtail).Theorem 4.3.14. Let G be any based Gauss diagram of a knot K. Then forevery n ≥ 0 bothI A,n (G) := A 2 n,1(G) − w A (G) · c n−1 (K),I D,n (G) := D 2 n,1(G) − w D (G) · c n−1 (K)90

are invariants of a knot K.These invariants will be denoted by I A,n (K) and I D,n (K) respectively.Proof. We will prove the invariance of I A,n (G); the proof for I D,n (G) is thesame.A well-known fact in knot theory is that for classical knots, theories of closedand long knots are equivalent. Thus it suffices to prove the invariance of I A,n (G)under Reidemeister moves Ω 1 – Ω 3 applied away from the base point. Note thatboth w A (G) and A 2 n,1(G) are invariant under Ω 2 and Ω 3 (see Lemma 4.3.3). Itremains to prove the invariance of I A,n (G) under Ω 1 .Let G and ˜G be two based Gauss diagrams that differ by an application ofΩ 1 , so that ˜G has an additional isolated arrow α either as in Figure 4.24a, oras in Figure 4.24b.εεabFigure 4.24: Two versions of the Reidemeister move Ω 1In the first case, A 2 n,1( ˜G) = A 2 n,1(G) and w A ( ˜G) = w A (G), thus we haveI A,n (G) = I A,n ( ˜G).In the second case, by Corollary 4.2.5 we getA 2 n,1( ˜G) = A 2 n,1(G) + εc n−1 (K).We also have w A ( ˜G) = w A (G) + ε, and thus again I A,n (G) = I A,n ( ˜G).Note that for every G we have w(G) = w A (G) + w D (G); also, for knots onehas I n (K) = p n (K).Corollary 4.3.15. For every knot KI n (K) = I A,n (K) + I D,n (K) = p n (K).91

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