Slides of the plenary talk by Samuel Lomonaco - GW Links

Fox Calc& 4-DKnots ???The Geometry of Fox’s Free Calculuswith Applications toHigher Dimensional KnotsWork in ProgressSamuel LomonacoUniversity of Maryland Baltimore County (UMBC)Email: Lomonaco@UMBC.eduWebPage: www.csee.umbc.edu/~lomonacoTalk given in PU Math Algebraic Top Seminar Apr 25, 2013Updated May 8, 2013Road Map for TalkPowerPoint slides can be found at:www.csee.umbc.edu/~lomonaco/Lectures.html• 3-D Knot Theory• Knot Diagrams, Reidemeister Moves, ,Wirtinger1Geometry of Group Presentations• Wirtinger Generalized Presentations• Geometry of Fox Free Calc• 4-D Knot TheoryHistory + Asphericity AnalogsCross Sections & MidsectionsYoshikawa Moves and Swenton Completeness• Generalized Presentations of 2-Knots ???• ConclusionKnot TheoryPlacement Problem: Knot Theory3Orientation• Ambient space SPreserving3• Group G AutoHomeo SGroup 1SPlacement#23SK ∼ KDef. 1 2if g G s.t . gK KK∼ K1 2Problem. When are two placements the same ?1 2?1

When do two Knot diagrams represent thesame or different knots ?Theorem (Reidemeister).Two knotdiagrams represent the same knot iffone can be transformed into the otherby a finite sequence of Reidemestermoves.Def.What is a knot invariant ?A knot invariant I is a mapI : Knots Mathematical Domainthat takes each knot K to a mathematicalobject I(K) such thatK ∼ K K K IConsequently, I 1 2 1 2IK K K ∼ K I 1 2 1 2The Fundamental Group 1 XKnot ExteriorX S SmallOpenTubularNbd kS3 1The Wirtinger Presentation• Fundamental Group1XKnot Invariant• Asphericity of Knots (Papakyriakopoulos) X K 1 X,1 . nX 0 for n1abc , , : r1, r2,r3GeneratorsRelatorsGroup PresentationsQuestion: When do two presentationsrepresent the same group ?Tietze Transformations:TT1Tietze 1: x : rx y : rs1: , where• yis a new symbol, and• s y 1, with Fx111 2, TT2 Tietze 2: x : r x:rs, wheremw• sConsr, i.e., s r , withj( ) 1•w Fx, 0 m, andw1• r j( ) wrj,( )w3

Group PresentationsQuestion: When do two presentationsrepresent the same group ?Theorem (TietzeTietze): Two group presentationsrepresent the same group iff there exists afinite sequence of Tietze transformationswhich transforms one into the other.The Geometry of Group PresentationsDef. An abstract Group presentationconsists of two sets• x, the set of generators,• r, the set of relators,together with an evaluation map^:r F xfrom the set of relatorsinto the freegroup on the set of symbols x.F xrThe Geometry of Group PresentationsThe Geometry of Group PresentationsExample:abc , , : r1, r2,r3, whereDef. A CW-complex is said to bemonopointed it it has only one 0-cell. 1 1r1 cbc arr23 aca b1 1 bab c1 1Proposition. Up to renaming andreordering, there exists a one-to-onecorrespondence between the set ofabstract group presentations and a setof monopointed 2-D CW-complexes.x : rKx:rThe Geometry of Group PresentationsThe CW-complexK x : yis constructedwith an initial 0-cell, denoted by , andthen iteratively attaching cells as follows:1-cells:For each generator j, adjoinan oriented 1-cellX jby attachingboth endpoints to the sole 0-cell .2-cells:For each relatororiented 2-cellmap .r kR krx x rk , adjoin anwith attachingExamplesTorus a, b : r , rababExample: P a, b : r , rababExample: 2 S D a, b: r , rbExample: 1 2 S S S a, b: r , r1Example: 1 1 2 S S : r , r , r 1, r 12 2 1 2 1 2Example: 4

Free-groupsThus, the elements of F t F x form:wt j( )F x are of theThe Geometry of the Wirtinger Presentationw1wheret wj( ) tj( ) wj( )Note. This conjugation action is a leftaction.Wirtinger Generalized PresentationThe Wirtinger 3-D CW decomposition ofthe exterior X is nothing more than thenon-abelianfree resolution:^( , , ) FF1, 2,3^ F G F a b c r r r I^cbca r 1^acab r 2^babc r 3^ 2 1rrr1 2 3 ^1F^ Please note that and .IWirtinger Generalized Presentation^( , , ) FF1, 2,3We denote the above non-abelianfreeresolution more cryptically asabc , , : r, r, r:I1 2 3^ F G F a b c r r r Iand call it a generalized presentation.FWirtinger Generalized PresentationWirtinger Generalized Presentationabcd , , , : r1, r2, r3, r3:I^bcba r 1^cbcd r 2^daca r 3rrr ra c ca ca ^1 2 4 3 I^adbd r 46

Terminology for Generalized PresentationsThe current names identities, identities ofidentities, identities of identities ofidentities, etc. are too cumbersome. Sowe instead adopt the following terminology:generatorsor1-st orderrelators(2) (3) (4)x: r : r : r : 2-ndorderrelators3-rdorderrelators4-thorderrelatorsGeneralized Tietze MovesThere is only one generalized Tietze movefor each order, namely:n-thorder generalized Tietze move T n:( n) ( n1) ( n) ( n1) T : : r : r : : r : r :nwhereand where 1and F F r( n)Generalized Presentation EquivalenceThe definition of generalized presentationequivalence is a straight forward exercisefor the audience. So is the proof of thefollowing theorem:Theorem: Two generalized presentationsare equivalent iff there is a finite sequenceof generalized Tietze moves that transformone into the other.The Geometry of Generalized PresentationsProposition. Up to renaming and reordering, thereexists a one-toto-one one correspondence between theset of finite generalized presentations and a setof monopointedCW-complexes.x : r : r (3) : r (4) : Kx : r : r (3) : r(4): The generalized tietze moves correspond tosimple homotopy moves on the associated CW-complexes.Moreover, two generalized presentations defineCW complexes of the same simple homotopy typeiff there is a finite sequence of Tieze moveswhich transforms one into the other.The Geometry of the Fox Free Calculus(2) (3) ( n)Let P x: r : r : :r be a generalized presentation, and letLet the fundamental group of .Letassociated withK KPbe the corresponding CW-complex.1G K: F x Gx : r (2)Gbe the epimorphismFinally, let be the group ring of overthe integers .GKThe Geometry of the Fox Free CalculusLetKbe the universal cover of K, and letK KKerbe the non pathwise connectedspace above K.We now use the Fox free derivatives toconstruct a chain complex C C KHence,Moreover,* *H K H C* * *H *KH* G FC*7

The Geometry of the Fox Free CalculusThe chain groups are defined as follows:• The 0-th chain groupis defined0 0 as the free FC C( )-module generated by the0-cell .• For n > 0 , the n-thchain groupnis defined as the free F -module( n)generated by the set of n-cells .nC C RR( n)The Geometry of the Fox Free CalculusThe boundary morphisms are defined asfollows:• For n = 0( )0 1C Cn = 0 ,Xxj1 Xj• For n > 0( n1) ( n)n > 0 , Cn1R CnR( n) rj R( n1)k rk R( n1) ( n)kk xWhere the Fox Free derivativesare geometrically defined as follows:/jKKerUniv. CoverCW-ComplexComplexCell Decompositions ofEpimorphism:KKKK, K,KF xG 1K(3) (4)w wXi wRj wRk wR4 (3) (4)g gXi gRj gRk gR4 X (3) (4)iRj RkR0-cells1-cells2-cells3-cellsg GwFx4-cells(2) (3) ( n)Recall x: r : r : : rand K K P.(2)If, for example, r rwith r x1x2x1x3,then the corresponding 2-cell Rin KisP x1xX2 1x X1 2x 2x1 R x 1x3X 1 x1x2x1X31 1R xx1 2X1 xX1 2 xxxX1 2 3r r r X X Xx x x1 2 31 2 3xx1 3(3)x1x3If, for example, urwith ur1 r2r1, thenthe boundary chain map of the corresponding 3-cell Uin Kis:U x r r x x r R rx Rx1 x3 x11 1 2 1 3 1 1 3 2^^u uR R r 1r 2 1 28

For completeness, we give the standardalgebraic definition of the Fox free / xjAlgebraic Def. of Fox Free Derivative / xjLet Gbe a group, and let Gdenotethe corresponding group ring over thering of integers .Def. A derivativein the group ringDGD : GGin the group ringis defined as a mapsatisfying the following condition:1)D D DD D D1) 2) o:1 2 1 2o1 2 1 21 2GGwhereis the trivializermorphism which maps each element ofto 1 of .GAlgebraic Def. of Fox Free Derivative / xjG FxDef.(Cont.)Let be the free group .Then to each free generator xj x, therecorresponds a unique derivativeF xD /xjin, called the derivative withrespect to, which has the propertyx jxixj ijj(Kroneckerdelta)2-KnotsVarious Placement Problems• 3-D Knot Theoryk:S S1 3• 4-D Knot Theoryk:S S2 4• 5-D Knot Theoryk:S S3 51-KnotS, kS3 12-KnotS, kS4 23-KnotS, kS5 34-D Knot Theory (Incomplete) HistoryRoseman SchoolKnot ProjectionsE. Artin –Spun Knots2 DifferentApproachesFox SchoolKnot Cross SectionsD. Roseman - Proj. Moves R. Fox - Cross SectionsCarter & Saito – Proj. MovesMuch more …Lomonaco - Mid SectionYoshikawa – Y-MovesSwenton/Kearton/KurlinKurlin –Y-Moves Completeness9

4-D Analog of the Apspericity of Knots ?For 1-knots,asphericity implies the exteriorX is an Eilenberg-MacLanespace, i.e.,X K 1X,1Question. Want can be said about analogof Papa’s ashphericity theorem for 2-knots?4-D Analog of the Apspericity of Knots ?S , kS4 2Def. A 2-knotis said to be quasi-aspherical (QA) if the third homology group ofthe universal cover of its exterior vanishes.Theorem. (Lomonaco) Ifis QA,then the homotopy type of its exterior X isdetermined by its algebraic 3-type, i.e., bythe triple consisting of:• X 1 • X 2 as a X 1• The first k-invariant3H 1X;2XS , kS4 2-modulekXlying inEdwin Abbott’s FlatlandThe Cross sectionalApproach to 2-Knots3-D LandThe MidsectionRepresentation2-Knots10

Edwin Abbott’s Flatland3-D LandMidsection Labeling Scheme Midsection Representation of a 2-sphereMidsection Rep of a Spun Trefoil4 2Midsection of S , k ??? P11

Reidemeister MovesMidsectionMoves on2-KnotsYoshikawa MovesThe Geometry of Group PresentationsTheorem(Yoshikawa).The Reidemeisterand Yoshikawa moves on the midsectionrepresentation of a 2-knot preserve knottype.Theorem(Swenton/Keartin/Kurlin).Two 2-knot midsections represent the same 2-knotiff there is a finite sequence of Reidemeisterand Yoshikawa moves which transforms oneinto the other.The Geometry of Group PresentationsThe WirtingerGeneralized Presentationof2-Knots ???UPOutDown12

Talk OverviewOpen Question 1This talk is a description of a researchprogram which is best summarized bythe following two questions:How does one piece together theWirtinger generalized presentations of allthe exterior cross sections of a 2-knot4 2S, kS into a generalized presentationof the exterior of the entire 2-knot???Open Question 2Conclusion ???Given a generalized presentation of the4 2exterior X of a 2-knot S, kS , howdoes one read off from this generalizedpresentation the second homotopy group X(as X -module), and the k-2 1 invariant 3??? ; kx H X X1 2The answers are …???UpcomingSpecial IssueofJournalofQuantum Information ProcessingonTopological Quantum ComputationGuest EditorsKauffman & Lomonaco14

Weird !!!This talk based on:Lomonaco, Samuel J., Jr., Five dimensional knot theory,in "Low Dimensional Topology, AMS CONM/20,Providence, Rhode Island, (1984), pp 249 - 270Lomonaco, Samuel J., Jr., The homotopy groups of knotsI. How to compute the algebraic 3-type, Pacific Journal ofMathematics, Vol. 95, No. 2 (1981), pp 349 - 390Lomonaco, Samuel J., Jr., Homology of group systemswith applications to low dimensional topology, Bulletin ofthe American Mathematics Society, Vol. 13, No. 3 (1980),pp 1049 - 1052.Lomonaco, S.J., Jr., The second homotopy group of aspun knot, Topology, Vol. 8 (1969), pp 95 - 98And also based on:Fox, R.H., A quick trip through knot theory, in“Topology of 3-Manifolds and Related Topics,”ed. by M.K. Fort, Jr., Prentice-Hall, EnglewoodsCliffs, New Jersey, (1962), 120-167.Kearton, C. & V. Kurlin, All 2-dimensional linksin 4-space live inside a universal polyhedron,Alg. Geom. Top, 8, (2008), 1223-1247.Swenton, Frank J., On a calculus for 2-knotsand surfaces in 4-space, JKTR, Vol. 10, No. 08,(2001), 1133-1141.Yoshikawa, Katsuyuki, An enumeration ofsurfaces in four-space, Osaka J. Math. 31(1994), 497-522.Some Other ReferencesArtin, Emil, Zur isotopie zweidimensionaler Flachen im R 4 , HamburgAbh. 4 (1925), 174-177.Carter, J. Scott, and Masahico Saito, Reidemeistermovesfor surfaceisotopies and their interpretations as moves to movies, J. Knot Theoryand Its Ramifications, 2 (1993), 251-284.Carter, J. Scott, Masahico Saito, “Knotted Surfaces and TheirDiagrams,” AMS, (1998).Fox, R.H., and J.W. Milnor, Singularities of 2-spheres in 4-space andequivalence of knots,Kamada, Seiichi, Surfaces in 4-Space: A view of normal forms andbraidings, in “Lectures at Knots ’96,” edited by Shin’ichi Suzuki, WorldScientific, (1997), pp. 39-71.Kawauchi, Akio, T. T. Shibuya, and S. Suziki, Descriptions on surfacesin four-space, I. Normal forms, Math. Sem. Notes, Kobe University, 10(1982), 75-125.Kawauchi, Akio, “A Survey of Knot Theory,” Birkhauser, (1990), pp.171-199.Some Other ReferencesLomonaco, Samuel J., Jr., Finitely ended knots are quasi-aspherical,"in "Algebraic and Differential Topology- Global Differential Geometry,"edited by George. M. Rassias, Teubner Publishers (Leipzig), Germany,(1984), 192 - 197.Lomonaco, Samuel J., Jr., The third homotopy group of some higherdimensional knots, in "Knots, Groups, and 3-Manifolds," (L.P.Neuwirth, ed.), Annals of Math Studies, 84, Princeton Univ. Press,(1975), 35 - 45.Lomonaco, Samuel J., Jr., The fundamental ideal and Pi2 of higherdimensional knots, AMS Proc., 88, (1973), 431 - 433.Andrews, J.J., and S.J. Lomonaco, Jr., The second homotopy group ofspun 2-spheres in 4-space, Annals of Math., 90 (1969), pp 199 - 204.Lomonaco, S.J., Jr., The second homotopy group of a spun knot,Topology, Vol. 8 (1969), pp 95 - 98.Roseman, Dennis, Reidemeistertypemoves for surfaces in four space,preprintRoseman, Dennis, Projections of knots, Fund. Math. 89 , no. 2, (1975),pp. 99-110.15