Knot Theory by Computer SERIES on KNOTS and EVERYTHING
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Complete Invariant Graphs of Alternating Knots
Complete invariant graphs of alternating knots Christian Soulié First submission: April 2004 (revision 1) Abstract : Chord diagrams and related enlacement graphs of alternating knots are enhanced to obtain complete invariant graphs including chirality detection. Moreover, the equivalence by common enlacement graph is specified and the neighborhood graph is defined for general purpose and for special application to the knots. I - Introduction : Chord diagrams are enhanced to integrate the state sum of all flype moves and then produce an invariant graph for alternating knots. By adding local writhe attribute to these graphs, chiral types of knots are distinguished. The resulting chord-weighted graph is a complete invariant of alternating knots. Condensed chord diagrams and condensed enlacement graphs are introduced and a new type of graph of general purpose is defined : the neighborhood graph. The enlacement graph is enriched by local writhe and chord orientation. Hence this enhanced graph distinguishes mutant alternating knots. As invariant by flype it is also invariant for all alternating knots. The equivalence class of knots with the same enlacement graph is fully specified and extended mutation with flype of tangles is defined. On this way, two enhanced graphs are proposed as complete invariants of alternating knots. I - Introduction II - Definitions and condensed graphs II-1 Knots II-2 Sign of crossing points II-3 Chord diagrams II-4 Enlacement graphs II-5 Condensed graphs III - Realizability and construction III - 1 Realizability -
Oriented Pair (S 3,S1); Two Knots Are Regarded As
S-EQUIVALENCE OF KNOTS C. KEARTON Abstract. S-equivalence of classical knots is investigated, as well as its rela- tionship with mutation and the unknotting number. Furthermore, we identify the kernel of Bredon’s double suspension map, and give a geometric relation between slice and algebraically slice knots. Finally, we show that every knot is S-equivalent to a prime knot. 1. Introduction An oriented knot k is a smooth (or PL) oriented pair S3,S1; two knots are regarded as the same if there is an orientation preserving diffeomorphism sending one onto the other. An unoriented knot k is defined in the same way, but without regard to the orientation of S1. Every oriented knot is spanned by an oriented surface, a Seifert surface, and this gives rise to a matrix of linking numbers called a Seifert matrix. Any two Seifert matrices of the same knot are S-equivalent: the definition of S-equivalence is given in, for example, [14, 21, 11]. It is the equivalence relation generated by ambient surgery on a Seifert surface of the knot. In [19], two oriented knots are defined to be S-equivalent if their Seifert matrices are S- equivalent, and the following result is proved. Theorem 1. Two oriented knots are S-equivalent if and only if they are related by a sequence of doubled-delta moves shown in Figure 1. .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .. .... .... .... .... .... .... .... .... .... ... -
Mutant Knots and Intersection Graphs 1 Introduction
Mutant knots and intersection graphs S. V. CHMUTOV S. K. LANDO We prove that if a finite order knot invariant does not distinguish mutant knots, then the corresponding weight system depends on the intersection graph of a chord diagram rather than on the diagram itself. Conversely, if we have a weight system depending only on the intersection graphs of chord diagrams, then the composition of such a weight system with the Kontsevich invariant determines a knot invariant that does not distinguish mutant knots. Thus, an equivalence between finite order invariants not distinguishing mutants and weight systems depending on intersections graphs only is established. We discuss relationship between our results and certain Lie algebra weight systems. 57M15; 57M25 1 Introduction Below, we use standard notions of the theory of finite order, or Vassiliev, invariants of knots in 3-space; their definitions can be found, for example, in [6] or [14], and we recall them briefly in Section 2. All knots are assumed to be oriented. Two knots are said to be mutant if they differ by a rotation of a tangle with four endpoints about either a vertical axis, or a horizontal axis, or an axis perpendicular to the paper. If necessary, the orientation inside the tangle may be replaced by the opposite one. Here is a famous example of mutant knots, the Conway (11n34) knot C of genus 3, and Kinoshita–Terasaka (11n42) knot KT of genus 2 (see [1]). C = KT = Note that the change of the orientation of a knot can be achieved by a mutation in the complement to a trivial tangle. -
Hyperbolic Structures from Link Diagrams
Hyperbolic Structures from Link Diagrams Anastasiia Tsvietkova, joint work with Morwen Thistlethwaite Louisiana State University/University of Tennessee Anastasiia Tsvietkova, joint work with MorwenHyperbolic Thistlethwaite Structures (UTK) from Link Diagrams 1/23 Every knot can be uniquely decomposed as a knot sum of prime knots (H. Schubert). It also true for non-split links (i.e. if there is no a 2–sphere in the complement separating the link). Of the 14 prime knots up to 7 crossings, 3 are non-hyperbolic. Of the 1,701,935 prime knots up to 16 crossings, 32 are non-hyperbolic. Of the 8,053,378 prime knots with 17 crossings, 30 are non-hyperbolic. Motivation: knots Anastasiia Tsvietkova, joint work with MorwenHyperbolic Thistlethwaite Structures (UTK) from Link Diagrams 2/23 Of the 14 prime knots up to 7 crossings, 3 are non-hyperbolic. Of the 1,701,935 prime knots up to 16 crossings, 32 are non-hyperbolic. Of the 8,053,378 prime knots with 17 crossings, 30 are non-hyperbolic. Motivation: knots Every knot can be uniquely decomposed as a knot sum of prime knots (H. Schubert). It also true for non-split links (i.e. if there is no a 2–sphere in the complement separating the link). Anastasiia Tsvietkova, joint work with MorwenHyperbolic Thistlethwaite Structures (UTK) from Link Diagrams 2/23 Of the 1,701,935 prime knots up to 16 crossings, 32 are non-hyperbolic. Of the 8,053,378 prime knots with 17 crossings, 30 are non-hyperbolic. Motivation: knots Every knot can be uniquely decomposed as a knot sum of prime knots (H. -
Some Generalizations of Satoh's Tube
SOME GENERALIZATIONS OF SATOH'S TUBE MAP BLAKE K. WINTER Abstract. Satoh has defined a map from virtual knots to ribbon surfaces embedded in S4. Herein, we generalize this map to virtual m-links, and use this to construct generalizations of welded and extended welded knots to higher dimensions. This also allows us to construct two new geometric pictures of virtual m-links, including 1-links. 1. Introduction In [13], Satoh defined a map from virtual link diagrams to ribbon surfaces knotted in S4 or R4. This map, which he called T ube, preserves the knot group and the knot quandle (although not the longitude). Furthermore, Satoh showed that if K =∼ K0 as virtual knots, T ube(K) is isotopic to T ube(K0). In fact, we can make a stronger statement: if K and K0 are equivalent under what Satoh called w- equivalence, then T ube(K) is isotopic to T ube(K0) via a stable equivalence of ribbon links without band slides. For virtual links, w-equivalence is the same as welded equivalence, explained below. In addition to virtual links, Satoh also defined virtual arc diagrams and defined the T ube map on such diagrams. On the other hand, in [15, 14], the notion of virtual knots was extended to arbitrary dimensions, building on the geometric description of virtual knots given by Kuperberg, [7]. It is therefore natural to ask whether Satoh's idea for the T ube map can be generalized into higher dimensions, as well as inquiring into a generalization of the virtual arc diagrams to higher dimensions. -
A Symmetry Motivated Link Table
Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 15 August 2018 doi:10.20944/preprints201808.0265.v1 Peer-reviewed version available at Symmetry 2018, 10, 604; doi:10.3390/sym10110604 Article A Symmetry Motivated Link Table Shawn Witte1, Michelle Flanner2 and Mariel Vazquez1,2 1 UC Davis Mathematics 2 UC Davis Microbiology and Molecular Genetics * Correspondence: [email protected] Abstract: Proper identification of oriented knots and 2-component links requires a precise link 1 nomenclature. Motivated by questions arising in DNA topology, this study aims to produce a 2 nomenclature unambiguous with respect to link symmetries. For knots, this involves distinguishing 3 a knot type from its mirror image. In the case of 2-component links, there are up to sixteen possible 4 symmetry types for each topology. The study revisits the methods previously used to disambiguate 5 chiral knots and extends them to oriented 2-component links with up to nine crossings. Monte Carlo 6 simulations are used to report on writhe, a geometric indicator of chirality. There are ninety-two 7 prime 2-component links with up to nine crossings. Guided by geometrical data, linking number and 8 the symmetry groups of 2-component links, a canonical link diagram for each link type is proposed. 9 2 2 2 2 2 2 All diagrams but six were unambiguously chosen (815, 95, 934, 935, 939, and 941). We include complete 10 tables for prime knots with up to ten crossings and prime links with up to nine crossings. We also 11 prove a result on the behavior of the writhe under local lattice moves. -
Tait's Flyping Conjecture for 4-Regular Graphs
CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector Journal of Combinatorial Theory, Series B 95 (2005) 318–332 www.elsevier.com/locate/jctb Tait’s flyping conjecture for 4-regular graphs Jörg Sawollek Fachbereich Mathematik, Universität Dortmund, 44221 Dortmund, Germany Received 8 July 1998 Available online 19 July 2005 Abstract Tait’s flyping conjecture, stating that two reduced, alternating, prime link diagrams can be connected by a finite sequence of flypes, is extended to reduced, alternating, prime diagrams of 4-regular graphs in S3. The proof of this version of the flyping conjecture is based on the fact that the equivalence classes with respect to ambient isotopy and rigid vertex isotopy of graph embeddings are identical on the class of diagrams considered. © 2005 Elsevier Inc. All rights reserved. Keywords: Knotted graph; Alternating diagram; Flyping conjecture 0. Introduction Very early in the history of knot theory attention has been paid to alternating diagrams of knots and links. At the end of the 19th century Tait [21] stated several famous conjectures on alternating link diagrams that could not be verified for about a century. The conjectures concerning minimal crossing numbers of reduced, alternating link diagrams [15, Theorems A, B] have been proved independently by Thistlethwaite [22], Murasugi [15], and Kauffman [6]. Tait’s flyping conjecture, claiming that two reduced, alternating, prime diagrams of a given link can be connected by a finite sequence of so-called flypes (see [4, p. 311] for Tait’s original terminology), has been shown by Menasco and Thistlethwaite [14], and for a special case, namely, for well-connected diagrams, also by Schrijver [20]. -
An Introduction to Knot Theory and the Knot Group
AN INTRODUCTION TO KNOT THEORY AND THE KNOT GROUP LARSEN LINOV Abstract. This paper for the University of Chicago Math REU is an expos- itory introduction to knot theory. In the first section, definitions are given for knots and for fundamental concepts and examples in knot theory, and motivation is given for the second section. The second section applies the fun- damental group from algebraic topology to knots as a means to approach the basic problem of knot theory, and several important examples are given as well as a general method of computation for knot diagrams. This paper assumes knowledge in basic algebraic and general topology as well as group theory. Contents 1. Knots and Links 1 1.1. Examples of Knots 2 1.2. Links 3 1.3. Knot Invariants 4 2. Knot Groups and the Wirtinger Presentation 5 2.1. Preliminary Examples 5 2.2. The Wirtinger Presentation 6 2.3. Knot Groups for Torus Knots 9 Acknowledgements 10 References 10 1. Knots and Links We open with a definition: Definition 1.1. A knot is an embedding of the circle S1 in R3. The intuitive meaning behind a knot can be directly discerned from its name, as can the motivation for the concept. A mathematical knot is just like a knot of string in the real world, except that it has no thickness, is fixed in space, and most importantly forms a closed loop, without any loose ends. For mathematical con- venience, R3 in the definition is often replaced with its one-point compactification S3. Of course, knots in the real world are not fixed in space, and there is no interesting difference between, say, two knots that differ only by a translation. -
Categorified Invariants and the Braid Group
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 143, Number 7, July 2015, Pages 2801–2814 S 0002-9939(2015)12482-3 Article electronically published on February 26, 2015 CATEGORIFIED INVARIANTS AND THE BRAID GROUP JOHN A. BALDWIN AND J. ELISENDA GRIGSBY (Communicated by Daniel Ruberman) Abstract. We investigate two “categorified” braid conjugacy class invariants, one coming from Khovanov homology and the other from Heegaard Floer ho- mology. We prove that each yields a solution to the word problem but not the conjugacy problem in the braid group. In particular, our proof in the Khovanov case is completely combinatorial. 1. Introduction Recall that the n-strand braid group Bn admits the presentation σiσj = σj σi if |i − j|≥2, Bn = σ1,...,σn−1 , σiσj σi = σjσiσj if |i − j| =1 where σi corresponds to a positive half twist between the ith and (i + 1)st strands. Given a word w in the generators σ1,...,σn−1 and their inverses, we will denote by σ(w) the corresponding braid in Bn. Also, we will write σ ∼ σ if σ and σ are conjugate elements of Bn. As with any group described in terms of generators and relations, it is natural to look for combinatorial solutions to the word and conjugacy problems for the braid group: (1) Word problem: Given words w, w as above, is σ(w)=σ(w)? (2) Conjugacy problem: Given words w, w as above, is σ(w) ∼ σ(w)? The fastest known algorithms for solving Problems (1) and (2) exploit the Gar- side structure(s) of the braid group (cf. -
Hyperbolic Structures from Link Diagrams
University of Tennessee, Knoxville TRACE: Tennessee Research and Creative Exchange Doctoral Dissertations Graduate School 5-2012 Hyperbolic Structures from Link Diagrams Anastasiia Tsvietkova [email protected] Follow this and additional works at: https://trace.tennessee.edu/utk_graddiss Part of the Geometry and Topology Commons Recommended Citation Tsvietkova, Anastasiia, "Hyperbolic Structures from Link Diagrams. " PhD diss., University of Tennessee, 2012. https://trace.tennessee.edu/utk_graddiss/1361 This Dissertation is brought to you for free and open access by the Graduate School at TRACE: Tennessee Research and Creative Exchange. It has been accepted for inclusion in Doctoral Dissertations by an authorized administrator of TRACE: Tennessee Research and Creative Exchange. For more information, please contact [email protected]. To the Graduate Council: I am submitting herewith a dissertation written by Anastasiia Tsvietkova entitled "Hyperbolic Structures from Link Diagrams." I have examined the final electronic copy of this dissertation for form and content and recommend that it be accepted in partial fulfillment of the equirr ements for the degree of Doctor of Philosophy, with a major in Mathematics. Morwen B. Thistlethwaite, Major Professor We have read this dissertation and recommend its acceptance: Conrad P. Plaut, James Conant, Michael Berry Accepted for the Council: Carolyn R. Hodges Vice Provost and Dean of the Graduate School (Original signatures are on file with official studentecor r ds.) Hyperbolic Structures from Link Diagrams A Dissertation Presented for the Doctor of Philosophy Degree The University of Tennessee, Knoxville Anastasiia Tsvietkova May 2012 Copyright ©2012 by Anastasiia Tsvietkova. All rights reserved. ii Acknowledgements I am deeply thankful to Morwen Thistlethwaite, whose thoughtful guidance and generous advice made this research possible. -
Knots with Unknotting Number 1 and Essential Conway Spheres 1
Algebraic & Geometric Topology 6 (2006) 2051–2116 2051 arXiv version: fonts, pagination and layout may vary from AGT published version Knots with unknotting number 1 and essential Conway spheres CMCAGORDON JOHN LUECKE For a knot K in S3 , let T(K) be the characteristic toric sub-orbifold of the orbifold (S3; K) as defined by Bonahon–Siebenmann. If K has unknotting number one, we show that an unknotting arc for K can always be found which is disjoint from T(K), unless either K is an EM–knot (of Eudave-Munoz)˜ or (S3; K) contains an EM– tangle after cutting along T(K). As a consequence, we describe exactly which large algebraic knots (ie, algebraic in the sense of Conway and containing an essential Conway sphere) have unknotting number one and give a practical procedure for deciding this (as well as determining an unknotting crossing). Among the knots up to 11 crossings in Conway’s table which are obviously large algebraic by virtue of their description in the Conway notation, we determine which have unknotting number one. Combined with the work of Ozsvath–Szab´ o,´ this determines the knots with 10 or fewer crossings that have unknotting number one. We show that an alternating, large algebraic knot with unknotting number one can always be unknotted in an alternating diagram. As part of the above work, we determine the hyperbolic knots in a solid torus which admit a non-integral, toroidal Dehn surgery. Finally, we show that having unknotting number one is invariant under mutation. 57N10; 57M25 1 Introduction Montesinos showed [24] that if a knot K has unknotting number 1 then its double branched cover M can be obtained by a half-integral Dehn surgery on some knot K∗ in S3 . -
Absolutely Exotic Compact 4-Manifolds
ABSOLUTELY EXOTIC COMPACT 4-MANIFOLDS 1 2 SELMAN AKBULUT AND DANIEL RUBERMAN Abstract. We show how to construct absolutely exotic smooth struc- tures on compact 4-manifolds with boundary, including contractible manifolds. In particular, we prove that any compact smooth 4-manifold W with boundary that admits a relatively exotic structure contains a pair of codimension-zero submanifolds homotopy equivalent to W that are absolutely exotic copies of each other. In this context, absolute means that the exotic structure is not relative to a particular parame- terization of the boundary. Our examples are constructed by modifying a relatively exotic manifold by adding an invertible homology cobor- dism along its boundary. Applying this technique to corks (contractible manifolds with a diffeomorphism of the boundary that does not extend to a diffeomorphism of the interior) gives examples of absolutely exotic smooth structures on contractible 4-manifolds. 1. Introduction One goal of 4-dimensional topology is to find exotic smooth structures on the simplest of closed 4-manifolds, such as S4 and CP2. Amongst mani- folds with boundary, there are very simple exotic structures coming from the phenomenon of corks, which are relatively exotic contractible mani- arXiv:1410.1461v3 [math.GT] 11 Dec 2014 folds discovered by the first-named author [2]. More specifically, a cork is a compact smooth contractible manifold W together with a diffeomor- phism f : ∂W → ∂W which does not extend to a self-diffeomorphism of W , although it does extend to a self-homeomorphism F : W → W . This gives an exotic smooth structure on W relative to its boundary, namely the pullback smooth structure by F.