Dylan P. Thurston- The algebra of knotted trivalent graphs and Turaev’s shadow world

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  ISSN 1464-8997 (on line) 1464-8989 (printed) 337 (eometry & Topology /onographs Volume 4: Invariants of knots and 3-manifolds (Kyoto 2001) Pages 337–362 The algebra of knotted trivalent graphs and Turaev’s shadow world Dylan P. Thurston Abstract Knotted trivalent graphs (KTGs) form a rich algebra with a few simple operations: connected sum, unzip, and bubbling. With these operations, KTGs are generated by the unknotted tetrahedron and M¨ ob
  ISSN 1464-8997 (on line) 1464-8989 (printed) 337 G eometry &  T   opology  M onographs Volume 4: Invariants of knots and 3-manifolds (Kyoto 2001)Pages 337–362 The algebra of knotted trivalent graphsand Turaev’s shadow world Dylan P. Thurston Abstract Knotted trivalent graphs (KTGs) form a rich algebra with afew simple operations: connected sum, unzip, and bubbling. With theseoperations, KTGs are generated by the unknotted tetrahedron and M¨obiusstrips. Many previously known representations of knots, including knot di-agrams and non-associative tangles, can be turned into KTG presentationsin a natural way.Often two sequences of KTG operations produce the same output on allinputs. These “elementary” relations can be subtle: for instance, thereis a planar algebra of KTGs with a distinguished cycle. Studying theserelations naturally leads us to Turaev’s shadow surfaces , a combinatorialrepresentation of 3-manifolds based on simple 2-spines of 4-manifolds. Weconsider the knotted trivalent graphs as the boundary of a such a simplespine of the 4-ball, and to consider a Morse-theoretic sweepout of the spineas a “movie” of the knotted graph as it evolves according to the KTGoperations. For every KTG presentation of a knot we can construct such amovie. Two sequences of KTG operations that yield the same surface aretopologically equivalent, although the converse is not quite true. AMS Classification 57M25; 57M20, 57Q40 Keywords Knotted trivalent graphs, shadow surfaces, spines, simple 2-polyhedra, graph operations 1 Introduction In this paper we study the algebra of  knotted trivalent graphs (KTGs). Aknotted trivalent graph is a framed 1 embedding of a trivalent graph into R 3 ,modulo isotopy. These KTGs support some simple operations, forming analgebra-like structure. Every knot may be presented as a sequence of KTGoperations starting with elementary graphs. Thus we may use KTGs as a novelrepresentation of knot theory via generators and relations. 1 See Section2for the precise notion of framing we use. Published 3 February 2004: c  G eometry &  T   opology  P  ublications  338 Dylan P. Thurston We may compare a KTG presentation with other representations of knots, suchas: ã Planar knot diagrams; ã Braid closures; ã n -bridge representations; ã Pretzel representations; ã Rational tangles and algebraic knots [5]; ã Parenthesized tangles [2,11]; and ã Curves in a book with three pages 2 [7].These representations of knots all take an inherently 3-dimensional object (aknot) and squash it into 2 dimensions (as in a knot diagram) or sometimes eveninto 1 dimension (as in a parenthesized tangle). The algebra of KTGs dealsmore directly with knots as 3-dimensional objects, while strictly generalizingall of the above representations of knots, in the sense that, for instance, anyknot diagram can be turned into a sequence of operations on KTGs in a naturalway. 3 We will illustrate how KTGs generalize knot diagrams in Section4.The next step in studying the algebra of KTGs is to find the relations in thealgebra, and, in particular, the elementary  relations (Section5): pairs of se-quences of operations that produce the same output on all inputs. Further justification for calling these relations “elementary” comes from the fact thatin other spaces that support the same operations the elementary relations areautomatically satisfied, while other relations give us non-trivial equations tosolve. Tracing out the track of the KTG as it evolves through a sequence of operations, we construct a movie surface (Section6), a decorated simple 2-polyhedron. If two different sequences of operations generate the same moviesurface, then they are universally equal. The converse is not quite true.In fact, movie surfaces are a special case of  shadow diagrams , as we brieflydiscuss in Section7.Shadow diagrams for 3-manifolds and for 4-manifoldsbounded by 3-manifolds were introduced by Turaev. They were initially intro-duced to describe links inside circle bundles over a surface [15]. The constructionwas later generalized to allow descriptions of all 3-manifolds [14,16]. From aquantum topology point of view, shadow diagrams encapsulate the algebra of  2 A book with three pages looks like this: 3 Here “algebra” is used in the universal algebra sense of a set supporting someoperations G eometry &  T   opology  M onographs , Volume 4 (2002)  The algebra of knotted trivalent graphs and Turaev’s shadow world 339 quantum 6j-symbols in a concise way; more classically, a shadow diagram of a3-manifold can be thought of as an analogue of a pair of pants decompositionof a surface.In summary, every KTG presentation of a knot gives a certain abstract surfacerepresenting that knot complement. Thus it turns out that our intrinsically 3-dimensional representation of knot theory can also be encoded in 2-dimensionalterms. The relation between the various spaces and constructions is summarizedin Figure1. Framed links - exterior Framed link exteriors - surgery Closed 3-manifolds presentation  6 by making S  1 -bundle  6 by making S  1 -bundle  6 Sequences of KTG operations - timeevolution Collapsibleshadow surfaces - cap off  ∂  Closed shadow surfacesFigure 1: A summary of the relations between links, 3-manifolds, KTGs and shadowsurfaces This paper is an exposition of results that were, at least implicitly, previouslyknown. Rather, our aim is to give an exposition of the relationship betweenshadow surfaces and knotted trivalent graphs. In a future paper, we will provesome theorems that came out of this work, including the relationship betweenhyperbolic volume and minimal complexity shadow diagrams representing aknot. In addition, the unifying framework of KTGs is part of an ongoingproject to find new combinatorial presentations of knots, which may be morealgebraically manageable than the full-blown algebra of KTGs. See Section8for more on both of these. 1.1 Acknowledgements A very large part of the credit for this work must go to Dror Bar-Natan, withwhom I have had a long, productive, and fun collaboration. It was a greatexperience coming up with the algebra of KTGs with him and puzzling overthe wide variety of mysterious relations that we found together. In this papers,Sections2through5are joint work with him. In addition, I would like to thank Riccardo Benedetti, Francesco Costantino, Robion Kirby, Tomotada Ohtsuki,Carlo Petronio, A. Referee, Dale Rolfsen, Chung-chieh Shan, Vladimir Turaev,and Genevieve Walsh for a great many helpful conversations and comments. G eometry &  T   opology  M onographs , Volume 4 (2002)  340 Dylan P. Thurston This work was supported by an NSF Postdoctoral Research Fellowship, a JSPSPostdoctoral Research Fellowship, and by BSF grant #1998119. 2 The space of knotted trivalent graphs There are relatively few operations on knots. Traditional operations, like con-nect sum, cabling, or general satellites, when applied to non-trivial knots, neveryield hyperbolic knots, but almost all knots (by any reasonable measure of com-plexity) are hyperbolic. Thus typical knot operations are no good for breakingthe vast majority of knots down into simpler pieces. To fix this, we will passto the larger space of knotted trivalent graphs. This space will allow moreoperations; enough to generate all knots from a few simple generators.“Graphs” in this paper might more properly be called “1-dimensional com-plexes”; that is, they may have multiple edges, self loops, and circle compo-nents. Definition 1 A framed graph  or fat graph  is a thickening of an ordinary graphinto a surface (not necessarily oriented): the vertices are turned into disks andthe edges are turned into bands attaching to the disks. More abstractly, aframed graph is a 1-dimensional simplicial complex Γ together with an embed-ding Γ ֒ → Σ of Γ into a surface Σ as a spine; more combinatorially, a framedgraph is a graph with a cyclic ordering on the edges incident to each vertex and1 or − 1 on each edge (representing a straight or flipped connection, respec-tively), modulo reversing the ordering at a vertex and negating the elements onthe adjoining edges. +1  ; − 1  ; ; The notion of spines comes from PL topology: a spine of a simplicial complex Y  is a subcomplex X  of  Y  onto which Y  collapses, where collapsing means suc-cessively removing pairs of a k -simplex ∆ k and a ( k +1)-simplex ∆ k +1 , where∆ k +1 is the unique ( k + 1)-simplex having ∆ k on its boundary.For instance, the two framed graphs with spine Γ = S  1 are the annulus andthe M¨obius band. Definition 2 A knotted trivalent graph  (KTG) is a trivalent framed graph Γembedded (as a surface) into R 3 , considered up to isotopy. G eometry &  T   opology  M onographs , Volume 4 (2002)
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