Intrinsic Properties of Spatial Graphs - Denison University

Introduction Linking and Knotting Other Intrinsic Properties ConclusionIntrinsic Properties of Spatial GraphsBlake MellorLoyola Marymount UniversityThe UnKnot Conference, 2012Blake MellorIntrinsic Properties of Spatial Graphs

Introduction Linking and Knotting Other Intrinsic Properties ConclusionIntroductionKnot theory: embeddings of a closed loop in 3-dimensionalspace.Spatial Graphs: embeddings of more complicated graphs in3-space.Question: What properties of spatial graphs are intrinsic to theabstract graph?Examples: Intrinsically linked, intrinsically knotted. Others?Blake MellorIntrinsic Properties of Spatial Graphs

Introduction Linking and Knotting Other Intrinsic Properties ConclusionComplex Linking and KnottingIn the last talk, you heard about what is known about whichgraphs are intrinsically knotted and linked.We can also ask how complicated these links and knots needto be. In 2002, Flapan proved:Theorem (Flapan, 2002)For every n, there is an m such that every embedding of K m inR 3 contains a link with |lk(J, K )| ≥ n and a knot with crossingnumber at least n.Blake MellorIntrinsic Properties of Spatial Graphs

Introduction Linking and Knotting Other Intrinsic Properties ConclusionComplex Linking and KnottingMore recently, we have:Theorem (Flapan, M, Naimi, 2008)For any n ∈ N, there’s a graph such that every embedding of itcontains a link with n knotted components each with crossingnumber ≥ n which are all pairwise linked with |lk(J, K )| ≥ n.So there are graphs such thatevery embedding contains alink at least as complicated asthe one shown to the right.Blake MellorIntrinsic Properties of Spatial Graphs

Introduction Linking and Knotting Other Intrinsic Properties ConclusionHowever, the graphs constructed to prove this theorem areunreasonably large.QuestionWhat is the smallest m such that every embedding of K m in R 3contains a link with |lk(J, K )| ≥ 2? A knot with crossing numberat least 4?Blake MellorIntrinsic Properties of Spatial Graphs

Introduction Linking and Knotting Other Intrinsic Properties ConclusionCounting Links and KnotsRather than asking how complicated the links and knots in aspatial graph must be, we could instead ask how many theremust be.Blake MellorIntrinsic Properties of Spatial Graphs

Introduction Linking and Knotting Other Intrinsic Properties ConclusionCounting Links and KnotsRather than asking how complicated the links and knots in aspatial graph must be, we could instead ask how many theremust be.Conway and Gordon found an embedding of K 6 with exactlyone link, and an embedding of K 7 with exactly one (trefoil) knot.But larger graphs will be forced to contain many links and knots.Blake MellorIntrinsic Properties of Spatial Graphs

Introduction Linking and Knotting Other Intrinsic Properties ConclusionCounting Links and KnotsRather than asking how complicated the links and knots in aspatial graph must be, we could instead ask how many theremust be.Conway and Gordon found an embedding of K 6 with exactlyone link, and an embedding of K 7 with exactly one (trefoil) knot.But larger graphs will be forced to contain many links and knots.Theorem (Blain*, Bowlin*, Foisy, Hendricks*, LaCombe*, 2007)For n > 8, every embedding of K n contains at leastknotted Hamiltonian cycles.(n − 1)!7!Blake MellorIntrinsic Properties of Spatial Graphs

Introduction Linking and Knotting Other Intrinsic Properties ConclusionCounting Links and Knots in Complete Partite GraphsFleming and M (2009) gave upper and lower bounds for theminimum number of links in complete partite graphs on 7 or 8vertices.Abrams* and M (2010) extended this to counting links incomplete partite graphs on 9 vertices and knots in completepartite graphs on 8 vertices.In most cases, the bounds are not sharp, and there isconsiderable room for improvement. The table below gives afew results.graph K 7 K 3,3,1,1 K 8 K 4,4,1 K 9mnl 21 25 217 ≤ mnl ≤ 305 74 ≤ 3987mnk 1 1 15 ≤ mnk ≤ 29 ?? ??Blake MellorIntrinsic Properties of Spatial Graphs

Introduction Linking and Knotting Other Intrinsic Properties ConclusionA Minimal Embedding of K 8 ?Here is an embedding of K 8 with 305 links and 29 knots.Is it minimal?Blake MellorIntrinsic Properties of Spatial Graphs

Introduction Linking and Knotting Other Intrinsic Properties ConclusionBook embeddingsIn a book embedding, a graph is embedded in a “book” of halfplanes joined along a “spine” (a line), so that the vertices are allembedded on the spine, and the edges in each page do notcross. A book embedding is minimal if it uses the smallestpossible number of pages.Blake MellorIntrinsic Properties of Spatial Graphs

Introduction Linking and Knotting Other Intrinsic Properties ConclusionBook embeddingsIn a book embedding, a graph is embedded in a “book” of halfplanes joined along a “spine” (a line), so that the vertices are allembedded on the spine, and the edges in each page do notcross. A book embedding is minimal if it uses the smallestpossible number of pages.b 4b 3b 2Here is a book embedding forthe complete bipartite graphK 4,4 . It also realizes theminimal number of links inK 4,4 .b 1a 1a 2 a 3a 4Blake MellorIntrinsic Properties of Spatial Graphs

Introduction Linking and Knotting Other Intrinsic Properties ConclusionBook embeddingsMinimal book embeddings of graphs are good candidates forembeddings with the simplest possible knotting and linkingbehavior.Blake MellorIntrinsic Properties of Spatial Graphs

Introduction Linking and Knotting Other Intrinsic Properties ConclusionBook embeddingsMinimal book embeddings of graphs are good candidates forembeddings with the simplest possible knotting and linkingbehavior.Otsuki (1996) defined a canonical book representation of acomplete graph K n , and showed that it minimized the number ofpairs of linked 3-cycles and the number of knotted 7-cycles.Fleming and M (2009) showed it also minimized the number oflinks of a 3-cycle and a 4-cycle.Blake MellorIntrinsic Properties of Spatial Graphs

Introduction Linking and Knotting Other Intrinsic Properties ConclusionBook embeddingsMost recently, Politano* and Rowland (2011) counted andclassified all the knots in the canonical book embeddings of K nfor n ≤ 11.graph K 7 K 8 K 9 K 10 K 11# knots 1 29 577 9991 165,102Blake MellorIntrinsic Properties of Spatial Graphs

Introduction Linking and Knotting Other Intrinsic Properties ConclusionBook embeddingsMost recently, Politano* and Rowland (2011) counted andclassified all the knots in the canonical book embeddings of K nfor n ≤ 11.graph K 7 K 8 K 9 K 10 K 11# knots 1 29 577 9991 165,102QuestionDoes the canonical book embedding of K n realize the minimalnumber of links and/or knots for K n ?Blake MellorIntrinsic Properties of Spatial Graphs

Introduction Linking and Knotting Other Intrinsic Properties ConclusionStraight-edge EmbeddingsIn a straight-edge embedding, every edge is embedded as astraight line.Theorem (Negami, 1991)Given a link L, there is a number R(L) such that whenevern ≥ R(L), every straight-edge embedding of K n contains a linkequivalent to L.Blake MellorIntrinsic Properties of Spatial Graphs

Introduction Linking and Knotting Other Intrinsic Properties ConclusionStraight-edge EmbeddingsIn a straight-edge embedding, every edge is embedded as astraight line.Theorem (Negami, 1991)Given a link L, there is a number R(L) such that whenevern ≥ R(L), every straight-edge embedding of K n contains a linkequivalent to L.Which links can be realized in straight-edge embeddings ofparticular graphs?Blake MellorIntrinsic Properties of Spatial Graphs

Introduction Linking and Knotting Other Intrinsic Properties ConclusionStraight-edge EmbeddingsIn a straight-edge embedding, every edge is embedded as astraight line.Theorem (Negami, 1991)Given a link L, there is a number R(L) such that whenevern ≥ R(L), every straight-edge embedding of K n contains a linkequivalent to L.Which links can be realized in straight-edge embeddings ofparticular graphs?Hughes* (2007) and Huh and Jeon (2008) proved that everystraight-edge embedding of K 6 has one or three pairs of linkedtriangles; Ludwig and Arbisi* (2010) investigated links in K 7 .Blake MellorIntrinsic Properties of Spatial Graphs

Introduction Linking and Knotting Other Intrinsic Properties ConclusionStraight-edge EmbeddingsRamirez Alfonsin (1999) used oriented matroids to prove thatevery straight-edge embedding of K 7 contains a trefoil knot.This is an approach which lends itself to computerinvestigation, so it is possible to tackle larger graphs than ispossible with a geometric approach.Blake MellorIntrinsic Properties of Spatial Graphs

Introduction Linking and Knotting Other Intrinsic Properties ConclusionStraight-edge EmbeddingsRamirez Alfonsin (1999) used oriented matroids to prove thatevery straight-edge embedding of K 7 contains a trefoil knot.This is an approach which lends itself to computerinvestigation, so it is possible to tackle larger graphs than ispossible with a geometric approach.Recently, Naimi and Pavelescu (2012) have used orientedmatroids to prove that every straight-edge embedding of K 9contains a link of three components, and that straight-edgeembeddings of K 3,3,1 contain 1, 2, 3, 4, or 5 non-trivialtwo-component links.Blake MellorIntrinsic Properties of Spatial Graphs

Introduction Linking and Knotting Other Intrinsic Properties ConclusionStraight-edge EmbeddingsRamirez Alfonsin (1999) used oriented matroids to prove thatevery straight-edge embedding of K 7 contains a trefoil knot.This is an approach which lends itself to computerinvestigation, so it is possible to tackle larger graphs than ispossible with a geometric approach.Recently, Naimi and Pavelescu (2012) have used orientedmatroids to prove that every straight-edge embedding of K 9contains a link of three components, and that straight-edgeembeddings of K 3,3,1 contain 1, 2, 3, 4, or 5 non-trivialtwo-component links.What are the results of extending these methods to othergraphs?Blake MellorIntrinsic Properties of Spatial Graphs

Introduction Linking and Knotting Other Intrinsic Properties ConclusionLinking and Knotting in other ManifoldsDefinition (Flapan, Howards, Lawrence, M, 2006)An embedding of a graph G in a 3-manifold M is unknotted ifevery cycle in G bounds a disk in M, and it is unlinked if everypair of disjoint cycles in G bound disjoint disks in M.Blake MellorIntrinsic Properties of Spatial Graphs

Introduction Linking and Knotting Other Intrinsic Properties ConclusionLinking and Knotting in other ManifoldsDefinition (Flapan, Howards, Lawrence, M, 2006)An embedding of a graph G in a 3-manifold M is unknotted ifevery cycle in G bounds a disk in M, and it is unlinked if everypair of disjoint cycles in G bound disjoint disks in M.Theorem (Flapan, Howards, Lawrence, M, 2006)For any 3-manifold M, a graph G is intrinsically linked (resp.knotted) in M if and only if it is intrinsically linked (resp. knotted)in S 3 .Blake MellorIntrinsic Properties of Spatial Graphs

Introduction Linking and Knotting Other Intrinsic Properties ConclusionLinking in RP 3Definition (Foisy et. al.*, 2008)An embedding of a graph G in a 3-manifold M is unlinked if forevery pair of disjoint cycles L 1 and L 2 , both cycles are unknotsand there is a 3-ball A such that L 1 ⊂ A and L 2 ⊂ A c .In particular, in RP 3 , there are two types of unknot: one ishomologically trivial, and the other is not.Blake MellorIntrinsic Properties of Spatial Graphs

Introduction Linking and Knotting Other Intrinsic Properties ConclusionLinking in RP 3Definition (Foisy et. al.*, 2008)An embedding of a graph G in a 3-manifold M is unlinked if forevery pair of disjoint cycles L 1 and L 2 , both cycles are unknotsand there is a 3-ball A such that L 1 ⊂ A and L 2 ⊂ A c .In particular, in RP 3 , there are two types of unknot: one ishomologically trivial, and the other is not.Theorem (Foisy et. al.*, 2008)There are at least 594 minor-minimal intrinsically linked graphsin RP 3 .Blake MellorIntrinsic Properties of Spatial Graphs

Introduction Linking and Knotting Other Intrinsic Properties ConclusionLinking in RP 3Moreover, some graphs which are intrinsically linked in S 3 arenot intrinsically linked in RP 3 . Here is an unlinked embeddingof K 6 in RP 3 .What might happen in other manifolds?Blake MellorIntrinsic Properties of Spatial Graphs

Introduction Linking and Knotting Other Intrinsic Properties ConclusionTopological Symmetry GroupsThe topological symmetry group was introduced by Jon Simonin 1986 to study the symmetries of molecules whose structureis not rigid. It can be used to study the symmetries of any graphembedded in S 3 .DefinitionLet γ be an abstract graph with automorphism group Aut(γ).Let Γ be an embedding of γ in S 3 . The (orientation preserving)topological symmetry group of Γ, denoted TSG + (Γ), is thesubgroup of Aut(γ) induced by orientation preservinghomeomorphisms of the pair (S 3 , Γ).Blake MellorIntrinsic Properties of Spatial Graphs

Introduction Linking and Knotting Other Intrinsic Properties ConclusionTopological Symmetry GroupsThe topological symmetry groups of complete graphs havebeen completely classified (Flapan, M, Naimi, 2012). Forexample:Theorem (Flapan, M, Naimi, 2012)A complete graph K m with m ≥ 4 has an embedding Γ in S 3such that TSG + (Γ) = S 4 if and only if m ≡ 0, 4, 8, 12, 20(mod 24).Blake MellorIntrinsic Properties of Spatial Graphs

Introduction Linking and Knotting Other Intrinsic Properties ConclusionTopological Symmetry GroupsThe topological symmetry groups of complete graphs havebeen completely classified (Flapan, M, Naimi, 2012). Forexample:Theorem (Flapan, M, Naimi, 2012)A complete graph K m with m ≥ 4 has an embedding Γ in S 3such that TSG + (Γ) = S 4 if and only if m ≡ 0, 4, 8, 12, 20(mod 24).Some work has been done on topological symmetry groups ofbipartite complete graphs (M, 2010; Hake*, M, Pittluck*, 2012);but many more groups are possible than with complete graphs,and there is much still to be done. Very little is known aboutother families of graphs.Blake MellorIntrinsic Properties of Spatial Graphs

Introduction Linking and Knotting Other Intrinsic Properties ConclusionAn embedding of K 20 with TSG + = S 4Blake MellorIntrinsic Properties of Spatial Graphs

Introduction Linking and Knotting Other Intrinsic Properties ConclusionIntrinsically composite?Given a knot K , let p(K ) be the minimal number of factors in adecomposition of the knot into a connected sum of prime knots.Are their graphs for which every embedding has a knot withlarge p(K )? No!Blake MellorIntrinsic Properties of Spatial Graphs

Introduction Linking and Knotting Other Intrinsic Properties ConclusionIntrinsically composite?Given a knot K , let p(K ) be the minimal number of factors in adecomposition of the knot into a connected sum of prime knots.Are their graphs for which every embedding has a knot withlarge p(K )? No!Theorem (Flapan, Howards, 2009)Every graph has an embedding in which every knotted cycle isa hyperbolic knot.Since all hyperbolic knots are prime, this means every graphhas an embedding in which all knots are prime knots, withp(K ) = 1.Blake MellorIntrinsic Properties of Spatial Graphs

Introduction Linking and Knotting Other Intrinsic Properties ConclusionColoring knotstricoloringBlake MellorIntrinsic Properties of Spatial Graphs

Introduction Linking and Knotting Other Intrinsic Properties ConclusionColoring knots0431tricoloringmod 5 coloringBlake MellorIntrinsic Properties of Spatial Graphs

Introduction Linking and Knotting Other Intrinsic Properties ConclusionColoring knots0zy431xtricoloringmod 5 coloring2z - x - y = 0 (mod p)Blake MellorIntrinsic Properties of Spatial Graphs

Introduction Linking and Knotting Other Intrinsic Properties ConclusionColoring graphs(Graphs must have all vertices of even degree.)023211111Blake MellorIntrinsic Properties of Spatial Graphs

Introduction Linking and Knotting Other Intrinsic Properties ConclusionColoring graphs(Graphs must have all vertices of even degree.)0x 1x n2321x 2x 3x n-1111x - x + x - ... - x = 01 2 3 n1Blake MellorIntrinsic Properties of Spatial Graphs

Introduction Linking and Knotting Other Intrinsic Properties ConclusionColoring graphs(Graphs must have all vertices of even degree.)0x 1x n2321x 2x 3x n-1111x - x + x - ... - x = 01 2 3 n1Are there graphs which are “intrinsically colorable mod p”?Blake MellorIntrinsic Properties of Spatial Graphs

Introduction Linking and Knotting Other Intrinsic Properties ConclusionColoring matricesSystem of linear equations: variable for each arc; equation foreach crossing and vertex.⎡ ⎤⎡⎤ x−1 2 −1 0 0 0 0 0 0 1 ⎡ ⎤0 −1 2 −1 0 0 0 0 0x 200 0 −1 0 2 −1 0 0 0x 300 −1 0 2 −1 0 0 0 0x 400 0 0 0 0 −1 2 −1 0x 5=0⎢ 0 0 0 2 0 0 −1 −1 0x 60⎥⎣ 0 0 0 −1 0 0 0 2 −1⎦⎢x 7⎢0⎥⎥ ⎣⎣x ⎦ 0⎦1 0 0 0 −1 0 1 0 −1 80x 9Matrix has c + e columns and c + v rows, where c, e, v are thenumbers of crossings, edges, vertices. Colorings are non-trivialsolutions where at least two colors used.Blake MellorIntrinsic Properties of Spatial Graphs

Introduction Linking and Knotting Other Intrinsic Properties ConclusionDeterminant of a graphObservation: Any graph with e > v is intrinsically colorablewith at least e − v independent non-trivial colorings.Define the determinant of an embedded graph as the greatestcommon divisor of the determinants of all(c + v − 1) × (c + v − 1) minors.Pre-Theorem: An embedded graph has e − v + 1 independentnon-trivial colorings mod p if and only if p divides thedeterminant.We are currently working to extend this to define an Alexanderpolynomial for graphs. (Joint work with Terry Kong*, AlecLewald*, Vadim Pigrish*.)Blake MellorIntrinsic Properties of Spatial Graphs

Introduction Linking and Knotting Other Intrinsic Properties ConclusionIntrinsic PropertiesThere are many other questions one could ask about intrinsicproperties of spatial graphs. One is asking which propertiesare, in fact, intrinsic: for example, the property of containingcomposite knots is not intrinsic.QuestionWhich measures of complexity for knots and/or links areintrinsically large? I.e. for any n there is a graph such that everyembedding of the graph contains a knot and/or link for whichthe measure is larger than n?For each such intrinsically large property, we can ask the samequestions we have asked about the linking number of links,crossing number of knots, etc.Blake MellorIntrinsic Properties of Spatial Graphs

Introduction Linking and Knotting Other Intrinsic Properties ConclusionExtrinsic PropertiesTheta graph 2-bouquet graph handcuff graphRather than looking at intrinsic properties of graphs, we couldask how to distinguish embeddings of a particular graph.Important examples are the θ graphs, bouquet graphs and thehandcuff graph. Various invariants, such as the Yamadapolynomial, have been developed to distinguish embeddings.Any question asked about knots can also be asked aboutspatial graphs, so the opportunities are endless!Blake MellorIntrinsic Properties of Spatial Graphs

Introduction Linking and Knotting Other Intrinsic Properties ConclusionReferences(Authors marked with an * worked on the research asundergraduates.)L. Abrams*, B. Mellor, L. Trott*: Counting links and knots incomplete graphs, preprint (2010)P. Blain*, G. Bowlin*, J. Foisy, J. Hendricks*, J. LaCombe*:Knotted Hamiltonian cycles in spatial embeddings ofcomplete graphs, New York J. Math. 13 (2007) 11–16J. Bustamante*, J. Federman*, J. Foisy, K. Kozai*, K.Matthews*, K. McNamara*, E. Stark*, K. Trickey*:Intrinsically linked graphs in projective space, Alg. Geom.Top. 9 (2009) 1255–1274E. Flapan: Intrinsic knotting and linking of completegraphs, Alg. Geom. Top. 2 (2002) 371–380Blake MellorIntrinsic Properties of Spatial Graphs

Introduction Linking and Knotting Other Intrinsic Properties ConclusionReferencesE. Flapan, H. Howards, D. Lawrence, B. Mellor: Intrinsiclinking and knotting of graphs in arbitrary 3-manifolds, Alg.Geom. Top. 6 (2006) 1025–1035E. Flapan, H. Howards: Every graph has an embedding inS 3 containing no non-hyperbolic knot, Proc. AMS 137(2009) 4275–4285E. Flapan, B. Mellor, R. Naimi: Intrinsic linking and knottingare arbitrarily complex, Fund. Math. 201 (2008) 131–148E. Flapan, B. Mellor, R. Naimi, M. Yoshizawa*:Classification of topological symmetry groups of K n ,preprint (2012)T. Fleming, B. Mellor: Counting links in complete graphs,Osaka J. Math. 46 (2009) 1–29Blake MellorIntrinsic Properties of Spatial Graphs

Introduction Linking and Knotting Other Intrinsic Properties ConclusionReferencesK. Hake*, B. Mellor, M. Pittluck*: Topological symmetrygroups of complete bipartite graphs, in preparationC. Hughes*: Linked triangle pairs in a straight edgeembedding of K 6 , Pi Mu Epsilon J. 12 (2006) 213–218Y. Huh, C.B. Jeon: Knots and links in linear embeddings ofK 6 , J. Korean Math. Soc. 44 (2007) 661–671L. Ludwig, P. Arbisi*: Linking in straight-edge embeddingsof K 7 , J. Knot Theory Ramif. 19 (2010) 1431–1447B. Mellor: Complete bipartite graphs whose topologicalsymmetry groups are polyhedral, preprint (2011)Blake MellorIntrinsic Properties of Spatial Graphs

Introduction Linking and Knotting Other Intrinsic Properties ConclusionReferencesR. Naimi, E. Pavelescu: On the number of links in a linearlyembedded K 3,3,1 , preprint (2012)R. Naimi, E. Pavelescu: Linear embeddings of K 9 are triplelinked, preprint (2012)S. Negami: Ramsey theorems for knots, links and spatialgraphs, Trans. AMS 324 (1991) 527–541A. Politano*, D. Rowland: Knots in the canonical bookrepresentation of complete graphs, preprint (2011)J.L. Ramirez Alfonsin: Spatial graphs and orientedmatroids: the trefoil, Disc. Comp. Geom 22 (1999)149–158Blake MellorIntrinsic Properties of Spatial Graphs

Introduction Linking and Knotting Other Intrinsic Properties ConclusionThank YouThank you all for coming to this talk.Thanks to Lew Ludwig and Colin Adams for organizing theconference and giving me the opportunity to speak.Any questions?bmellor@lmu.eduhttp://myweb.lmu.edu/bmellorBlake MellorIntrinsic Properties of Spatial Graphs