DOCSLIB.ORG

Hyperbolic 3-Manifolds Dehn Surgery Approach to the Figure-Eight Knot

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Hyperbolic 3-Manifolds Dehn Surgery Approach to the Figure-Eight Knot Hyperbolic 3-Manifolds Dehn Surgery Approach to The Figure-Eight Knot Complement Sarah Dubicki, '11 Math 490: Senior Independent Study Advisor Dr. Barbara Nimershiem May 5, 2011 Abstract This project provides an examination of knot complements and their rela- tionship to hyperbolic 3-manifolds. It begins with the study of knot theory, specifically knot complements in S3. The knots we will be focusing on are those whose complements are hyperbolic, which includes a class of knots known as the twist knots, and in particular we will look at the hyperbolic structure of the link complement for the first even twist knot, the figure-eight knot. We then take the Dehn surgery approach to obtaining the structure of the figure-eight knot complement. By performing Dehn surgery on the Whitehead link, we are able to obtain the figure-eight knot. This project concludes by examining the Dehn surgery on the Whitehead link that yields the figure-eight knot. The per- formance of Dehn surgery on the hyperbolic structure of the Whitehead link leads us to the hyperbolic structure of the figure-eight knot complement. 1 I. Background Information We begin in knot theory with the definition of a knot. We say that a knot is a closed curve in S3 that is nowhere self-intersecting. In knot theory, there exist three categories of knots: torus knots, satellite knots, and hyperbolic knots. A (p,q)-torus knot is a closed loop around an unknotted torus with p meridians and q longitudes, where p and q are relatively prime. A satellite knot is essen- tially a knot inside a knotted torus, and a hyperbolic knot is a knot such that its complement in S3 has a hyperbolic structure. These categories are defined such that every knot falls into exactly one category, as seen in [2] p 119. We are interested in the hyperbolic category of knots. A subcategory of the hyperbolic knots is the class of twist knots with the trefoil knot removed. A twist knot is a twisted closed loop with any number of half-twists added. Figure 1: A closed loop with no twists added, from [12]. Figure 2: The figure-eight knot (the second twist knot): a closed loop with two half twists added, from [11]. 2 This paper will also involve work with the Whitehead Link. The standard projection of this link is as shown in Figure 3. However, this paper will use the projection as shown in Figure 4 for the link, achieved through a series of identity-preserving Reidemeister moves. Figure 3: The standard projection of the Whitehead Link. Figure 4: The alternate projection of the Whitehead Link. II. Twist Knot Complement - Adams's Approach In the first chapter of his Ph.D. dissertation [1], entitled Explicit Hyper- bolic Structures on Link complements, Adams provides an algorithm that can be applied to a hyperbolic link in S3 to yield the hyperbolic structure of its complement (in S3). Adams's method begins by putting a link, L, with n cross- ings in a regular projection where L lies on the projection except at a crossing. At a crossing, L lies on the surface of a sphere, which we call Si, where i is the 3 crossing number and 1 ≤ i ≤ n. The 3-cell in S − L bounded by Si is Bi. As 3 such, Adams thinks of the complement of L in S3 in terms of two subsets of S3: n C1, which is the half space above the projection plane with L and [i=1(Int(Bi)) removed, and C2, which is the half space below the projection plane with L n and [i=1(Int(Bi)) removed. The union of C1 and C2, C1 [ C2, is the space 3 n 3 fS − L − [i=1(Int(Bi))g. This is nearly the complement of L in S , we need to find a way to fill in the missing Bi's. Adams notes that Bi is homeomorphic to an octahedron, which he calls Oi. He then defines four disks in Oi with which we determine four 3-cells in Oi, each bounded by two disks and two faces of the octahedron. Each of these 3-cells shares the vertical line segment from the overstrand to the understrand. C1 shares 4 faces of Oi each with two of the 3-cells, and C2 shares 4 faces of Oi 0 each with the two remaining 3-cells. We obtain a new space C1 by gluing the 0 first two 3-cells along the four faces they share with C1 and C2 by gluing the remaining two 3-cells along the four faces they share with C2. Identifying the 0 0 boundaries of C1 and C2 using previous identifications fills in the missing Bi's and gives us a space homeomorphic to S3 − L. We can think of this process as pinching the top 3-cell around the overstrand so it touches itself along the edge from the overstrand to the understrand and pinching the bottom 3-cell around the understrand so that it touches itself in the same edge. This process deforms our link projection so that when it is flattened to a plane, it becomes a directed graph which corresponds to a poly- 0 0 hedron. The boundary of C1 and the boundary C2 of each correspond to a 0 polyhedron-determining directed graph. Flattening C1 to a plane, we have the original link projection whose understrand arcs at a crossing are labeled with the number of that crossing and directed away from the crossing. Flattening 0 C2 to a plane, we have the original link projection whose overstrand arcs at a crossing are labeled with the number of that crossing and directed toward the 4 crossing. Arcs labeled with the same number will be identified such that arrow directions match in S3 − L, and two edges with the same edge number have the 0 0 same edge type. Our directed graphs divide the boundaries of C1 and C2 into 0 regions that designate faces of our polyhedra. Each face in C1 is identified to a 0 face in C2 and edge types around both faces must match exactly. Adams also shows that digons (faces with only two edges) may be collapsed to a single edge. Following this algorithm, we find a polyhedron that determines the hyperbolic structure on the complement of L in S3. III. Figure-Eight Complement - Adams's Approach We now look at the explicit case of constructing the complement of the figure-eight knot, and begin by laying out two projections of the knot, as in Figure 5 (one will correspond to the \top" tetrahedron and the other will cor- respond to the \bottom" tetrahedron). We will label each crossing, and direct strands at the crossings. For the pro- Figure 5: Projections of the figure-eight knot. jection that corresponds to the \top" tetrahedron, we direct the under-strand at each crossing toward the crossing. For the projection that corresponds to the \bottom" tetrahedron, we direct the over-strand at each crossing toward the crossing. See Figure 6. 5 We will now designate an orientation on each of the strands, noting that Figure 6: Directed projections of the figure-eight knot. the digons may be collapsed to single lines. Note that there will be two edge types in this case. We will identify edge type 1 with the 1-strand, and we will identify edge type 2 with the 3-strand. This yields the tetrahedra as shown in Figure 7. These tetrahedra will be used to determine to the hyperbolic structure of Figure 7: The two directed graphs above correspond the bottom two tetrahedra with directed edges. the figure eight knot complement. We would like to know the angle measures of 6 these tetrahedra, so we will embed them as ideal tetrahedra with one vertex at 1, one at 0, one at 1, and the last at z, where z 2 C. A dihedral angle in our tetrahedron is the angle between two faces of our tetrahedron. We label these i i i angles z1; z2; z3 in a clockwise direction, where i = 1 for the top tetrahedron and i = 2 for the bottom tetrahedron (see Figure 8), noting that opposite dihedral angles are equal, a result in [3] p 345. Further, the transformation f1(z) = z corresponds to putting the vertex on the bottom face of the tetrahedron whose z−1 angle measure is z1 at the origin, and the transformations f2(z) = z and 1 f3(z) = 1−z correspond to putting the other two vertices at the origin while i i preserving the tetrahedron. Thus, we have zj = fj(z ). Figure 8: Tetrahedra embedded as ideal tetrahedra. We know that the sum of dihedral angles about any vertex is π, as explained in [3] p 345. Further, we know that the angle sum around an edge of one of these tetrahedra after all identifications are made must be 2π, as a complete- ness requirement outlined in Poincar´e'sPolyhedron Theorem (for the reader who wishes to see more on this theorem, Maskit provides a well-detailed out- line in section IV of [6]). We can see the completed identifications around two different edges of these tetrahedra in Figure 9 7 Figure 9: The completed identification about the edge through z1 and 1, edge 1, is on the left, and the completed identification about the edge through vertex z2 and 1, edge 2, is on the right. We have the following system of edge equations 1 2 1 2 1 2 z1 z2 z1 z1 z2 z1 = 1 (1) 1 2 1 2 1 2 z2 z3 z3 z2 z3 z3 = 1 (2) Here, the first equation corresponds to the identifications about edge 1 and the second equation corresponds to the identifications about edge 2.
Load more

Recommended publications
  • Jones Polynomial for Graphs of Twist Knots
    Available at Applications and Applied http://pvamu.edu/aam Mathematics: Appl. Appl. Math. An International Journal ISSN: 1932-9466 (AAM) Vol. 14, Issue 2 (December 2019), pp. 1269 – 1278 Jones Polynomial for Graphs of Twist Knots 1Abdulgani ¸Sahinand 2Bünyamin ¸Sahin 1Faculty of Science and Letters 2Faculty of Science Department Department of Mathematics of Mathematics Agrı˘ Ibrahim˙ Çeçen University Selçuk University Postcode 04100 Postcode 42130 Agrı,˘ Turkey Konya, Turkey [email protected] [email protected] Received: January 1, 2019; Accepted: March 16, 2019 Abstract We frequently encounter knots in the flow of our daily life. Either we knot a tie or we tie a knot on our shoes. We can even see a fisherman knotting the rope of his boat. Of course, the knot as a mathematical model is not that simple. These are the reflections of knots embedded in three- dimensional space in our daily lives. In fact, the studies on knots are meant to create a complete classification of them. This has been achieved for a large number of knots today. But we cannot say that it has been terminated yet. There are various effective instruments while carrying out all these studies. One of these effective tools is graphs. Graphs are have made a great contribution to the development of algebraic topology. Along with this support, knot theory has taken an important place in low dimensional manifold topology. In 1984, Jones introduced a new polynomial for knots. The discovery of that polynomial opened a new era in knot theory. In a short time, this polynomial was defined by algebraic arguments and its combinatorial definition was made.
    [Show full text]
  • Gluing Maps and Cobordism Maps for Sutured Monopole Floer Homology
    Gluing maps and cobordism maps for sutured monopole Floer homology Zhenkun Li Abstract The naturality of sutured monopole Floer homology, which was introduced by Kronheimer and Mrowka [17], is an important ques- tion and is partially answered by Baldwin and Sivek [1]. In this paper we construct the cobordism maps for sutured monopole Floer homology, thus improve its naturality. The construction can be carried out for sutured instantons as well. In the paper we also con- struct gluing maps in sutured monopoles and sutured instantons. Contents 1 Introduction 3 1.1 Maintheoremsandbackgrounds. 3 1.2 Outlineoftheproof....................... 6 1.3 Futurequestions ........................ 9 2 Prelimilaries 11 2.1 Monopole Floer homology for 3´manifold . 11 2.2 SuturedmonopoleFloerhomology . 12 2.3 The naturality of sutured monopole Floer homology . 14 3 Handle gluing maps and cancelations 21 3.1 Prelimilary discussions . 21 arXiv:1810.13071v3 [math.GT] 15 Jul 2019 3.2 Constructions of handle gluing maps . 26 3.3 Basicpropertiesofhandleattachingmaps . 31 4 The general gluing maps 45 1 Zhenkun Li CONTENTS 5 The cobordism maps 50 5.1 Constructions and functoriality . 50 5.2 Duality and turning cobordism around . 52 6 A brief discussion on Instanton 58 2 Zhenkun Li 1 INTRODUCTION 1 Introduction 1.1 Main theorems and backgrounds Sutured manifold is a powerful tool introduced by Gabai [6] in 1983, to study the topology of 3-manifolds. In 2010, the construction of monopole Floer homology was carried out on balanced sutured manifold by Kron- heimer and Mrowka [17]. The combination of Floer theories and sutured manifolds has many important applications.
    [Show full text]
  • Knots: a Handout for Mathcircles
    Knots: a handout for mathcircles Mladen Bestvina February 2003 1 Knots Informally, a knot is a knotted loop of string. You can create one easily enough in one of the following ways: • Take an extension cord, tie a knot in it, and then plug one end into the other. • Let your cat play with a ball of yarn for a while. Then find the two ends (good luck!) and tie them together. This is usually a very complicated knot. • Draw a diagram such as those pictured below. Such a diagram is a called a knot diagram or a knot projection. Trefoil and the figure 8 knot 1 The above two knots are the world's simplest knots. At the end of the handout you can see many more pictures of knots (from Robert Scharein's web site). The same picture contains many links as well. A link consists of several loops of string. Some links are so famous that they have names. For 2 2 3 example, 21 is the Hopf link, 51 is the Whitehead link, and 62 are the Bor- romean rings. They have the feature that individual strings (or components in mathematical parlance) are untangled (or unknotted) but you can't pull the strings apart without cutting. A bit of terminology: A crossing is a place where the knot crosses itself. The first number in knot's \name" is the number of crossings. Can you figure out the meaning of the other number(s)? 2 Reidemeister moves There are many knot diagrams representing the same knot. For example, both diagrams below represent the unknot.
    [Show full text]
  • Jones Polynomials, Volume, and Essential Knot Surfaces
    KNOTS IN POLAND III BANACH CENTER PUBLICATIONS, VOLUME 100 INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA 2013 JONES POLYNOMIALS, VOLUME AND ESSENTIAL KNOT SURFACES: A SURVEY DAVID FUTER Department of Mathematics, Temple University Philadelphia, PA 19122, USA E-mail: [email protected] EFSTRATIA KALFAGIANNI Department of Mathematics, Michigan State University East Lansing, MI 48824, USA E-mail: [email protected] JESSICA S. PURCELL Department of Mathematics, Brigham Young University Provo, UT 84602, USA E-mail: [email protected] Abstract. This paper is a brief overview of recent results by the authors relating colored Jones polynomials to geometric topology. The proofs of these results appear in the papers [18, 19], while this survey focuses on the main ideas and examples. Introduction. To every knot in S3 there corresponds a 3-manifold, namely the knot complement. This 3-manifold decomposes along tori into geometric pieces, where the most typical scenario is that all of S3 r K supports a complete hyperbolic metric [43]. Incompressible surfaces embedded in S3 r K play a crucial role in understanding its classical geometric and topological invariants. The quantum knot invariants, including the Jones polynomial and its relatives, the colored Jones polynomials, have their roots in representation theory and physics [28, 46], 2010 Mathematics Subject Classification:57M25,57M50,57N10. D.F. is supported in part by NSF grant DMS–1007221. E.K. is supported in part by NSF grants DMS–0805942 and DMS–1105843. J.P. is supported in part by NSF grant DMS–1007437 and a Sloan Research Fellowship. The paper is in final form and no version of it will be published elsewhere.
    [Show full text]
  • Arxiv:1501.00726V2 [Math.GT] 19 Sep 2016 3 2 for Each N, Ln Is Assembled from a Tangle S in B , N Copies of a Tangle T in S × I, and the Mirror Image S of S
    HIDDEN SYMMETRIES VIA HIDDEN EXTENSIONS ERIC CHESEBRO AND JASON DEBLOIS Abstract. This paper introduces a new approach to finding knots and links with hidden symmetries using \hidden extensions", a class of hidden symme- tries defined here. We exhibit a family of tangle complements in the ball whose boundaries have symmetries with hidden extensions, then we further extend these to hidden symmetries of some hyperbolic link complements. A hidden symmetry of a manifold M is a homeomorphism of finite-degree covers of M that does not descend to an automorphism of M. By deep work of Mar- gulis, hidden symmetries characterize the arithmetic manifolds among all locally symmetric ones: a locally symmetric manifold is arithmetic if and only if it has infinitely many \non-equivalent" hidden symmetries (see [13, Ch. 6]; cf. [9]). Among hyperbolic knot complements in S3 only that of the figure-eight is arith- metic [10], and the only other knot complements known to possess hidden sym- metries are the two \dodecahedral knots" constructed by Aitchison{Rubinstein [1]. Whether there exist others has been an open question for over two decades [9, Question 1]. Its answer has important consequences for commensurability classes of knot complements, see [11] and [2]. The partial answers that we know are all negative. Aside from the figure-eight, there are no knots with hidden symmetries with at most fifteen crossings [6] and no two-bridge knots with hidden symmetries [11]. Macasieb{Mattman showed that no hyperbolic (−2; 3; n) pretzel knot, n 2 Z, has hidden symmetries [8]. Hoffman showed the dodecahedral knots are commensurable with no others [7].
    [Show full text]
  • Introduction to the Berge Conjecture
    Introduction to the Berge Conjecture Gemma Halliwell School of Mathematics and Statistics, University of Sheffield 8th June 2015 Outline Introduction Dehn Surgery Definition Example Lens Spaces and the Berge conjecture Lens Spaces Berge Knots Martelli and Petronio Baker Families of Berge Knots Outline Introduction Dehn Surgery Definition Example Lens Spaces and the Berge conjecture Lens Spaces Berge Knots Martelli and Petronio Baker Families of Berge Knots It is not yet known whether [the partial filling on the 3-chain link]... gives rise to Berge knots. In this talk I will aim to answer this question and discuss how this relates to the Berge conjecture and future work. Introduction In their 2008 paper, “Dehn Surgery and the magic 3-manifold”, Martelli and Pertronio ended with the following statement: In this talk I will aim to answer this question and discuss how this relates to the Berge conjecture and future work. Introduction It is not yet known whether [the partial filling on the 3-chain link]... gives rise to Berge knots. Introduction It is not yet known whether [the partial filling on the 3-chain link]... gives rise to Berge knots. In this talk I will aim to answer this question and discuss how this relates to the Berge conjecture and future work. Outline Introduction Dehn Surgery Definition Example Lens Spaces and the Berge conjecture Lens Spaces Berge Knots Martelli and Petronio Baker Families of Berge Knots I A closed tubular neighbourhood N of L. I a specifed simple closed curve J in @N. Then we can construct the 3-manifold: ◦ [ M = (S3 − N) N h ◦ where N denotes the interior of N, and h is a homeomorphism which takes the meridian, µ, of N to the specifed J.
    [Show full text]
  • Dehn Surgery on Knots of Wrapping Number 2
    Dehn surgery on knots of wrapping number 2 Ying-Qing Wu Abstract Suppose K is a hyperbolic knot in a solid torus V intersecting a meridian disk D twice. We will show that if K is not the Whitehead knot and the frontier of a regular neighborhood of K ∪ D is incom- pressible in the knot exterior, then K admits at most one exceptional surgery, which must be toroidal. Embedding V in S3 gives infinitely many knots Kn with a slope rn corresponding to a slope r of K in V . If r surgery on K in V is toroidal then either Kn(rn) are toroidal for all but at most three n, or they are all atoroidal and nonhyperbolic. These will be used to classify exceptional surgeries on wrapped Mon- tesinos knots in solid torus, obtained by connecting the top endpoints of a Montesinos tangle to the bottom endpoints by two arcs wrapping around the solid torus. 1 Introduction A Dehn surgery on a hyperbolic knot K in a compact 3-manifold is excep- tional if the surgered manifold is non-hyperbolic. When the manifold is a solid torus, the surgery is exceptional if and only if the surgered manifold is either a solid torus, reducible, toroidal, or a small Seifert fibered manifold whose orbifold is a disk with two cone points. Solid torus surgeries have been classified by Berge [Be] and Gabai [Ga1, Ga2], and by Scharlemann [Sch] there is no reducible surgery. For toroidal surgery, Gordon and Luecke [GL2] showed that the surgery slope must be either an integral or a half integral slope.
    [Show full text]
  • Dehn Filling of the “Magic” 3-Manifold
    communications in analysis and geometry Volume 14, Number 5, 969–1026, 2006 Dehn filling of the “magic” 3-manifold Bruno Martelli and Carlo Petronio We classify all the non-hyperbolic Dehn fillings of the complement of the chain link with three components, conjectured to be the smallest hyperbolic 3-manifold with three cusps. We deduce the classification of all non-hyperbolic Dehn fillings of infinitely many one-cusped and two-cusped hyperbolic manifolds, including most of those with smallest known volume. Among other consequences of this classification, we mention the following: • for every integer n, we can prove that there are infinitely many hyperbolic knots in S3 having exceptional surgeries {n, n +1, n +2,n+3}, with n +1,n+ 2 giving small Seifert manifolds and n, n + 3 giving toroidal manifolds. • we exhibit a two-cusped hyperbolic manifold that contains a pair of inequivalent knots having homeomorphic complements. • we exhibit a chiral 3-manifold containing a pair of inequivalent hyperbolic knots with orientation-preservingly homeomorphic complements. • we give explicit lower bounds for the maximal distance between small Seifert fillings and any other kind of exceptional filling. 0. Introduction We study in this paper the Dehn fillings of the complement N of the chain link with three components in S3, shown in figure 1. The hyperbolic structure of N was first constructed by Thurston in his notes [28], and it was also noted there that the volume of N is particularly small. The relevance of N to three-dimensional topology comes from the fact that by filling N, one gets most of the hyperbolic manifolds known and most of the interesting non-hyperbolic fillings of cusped hyperbolic manifolds.
    [Show full text]
  • Berge–Gabai Knots and L–Space Satellite Operations
    BERGE-GABAI KNOTS AND L-SPACE SATELLITE OPERATIONS JENNIFER HOM, TYE LIDMAN, AND FARAMARZ VAFAEE Abstract. Let P (K) be a satellite knot where the pattern, P , is a Berge-Gabai knot (i.e., a knot in the solid torus with a non-trivial solid torus Dehn surgery), and the companion, K, is a non- trivial knot in S3. We prove that P (K) is an L-space knot if and only if K is an L-space knot and P is sufficiently positively twisted relative to the genus of K. This generalizes the result for cables due to Hedden [Hed09] and the first author [Hom11]. 1. Introduction In [OS04d], Oszv´ath and Szab´ointroduced Heegaard Floer theory, which produces a set of invariants of three- and four-dimensional manifolds. One example of such invariants is HF (Y ), which associates a graded abelian group to a closed 3-manifold Y . When Y is a rational homologyd three-sphere, rk HF (Y ) ≥ |H1(Y ; Z)| [OS04c]. If equality is achieved, then Y is called an L-space. Examples included lens spaces, and more generally, all connected sums of manifolds with elliptic geometry [OS05]. L-spaces are of interest for various reasons. For instance, such manifolds do not admit co-orientable taut foliations [OS04a, Theorem 1.4]. A knot K ⊂ S3 is called an L-space knot if it admits a positive L-space surgery. Any knot with a positive lens space surgery is then an L-space knot. In [Ber], Berge gave a conjecturally complete list of knots that admit lens space surgeries, which includes all torus knots [Mos71].
    [Show full text]
  • 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.
    [Show full text]
  • Knots and Links in Lens Spaces
    Alma Mater Studiorum Università di Bologna Dottorato di Ricerca in MATEMATICA Ciclo XXVI Settore Concorsuale di afferenza: 01/A2 Settore Scientifico disciplinare: MAT/03 Knots and links in lens spaces Tesi di Dottorato presentata da: Enrico Manfredi Coordinatore Dottorato: Relatore: Prof.ssa Prof. Giovanna Citti Michele Mulazzani Esame Finale anno 2014 Contents Introduction 1 1 Representation of lens spaces 9 1.1 Basic definitions . 10 1.2 A lens model for lens spaces . 11 1.3 Quotient of S3 model . 12 1.4 Genus one Heegaard splitting model . 14 1.5 Dehn surgery model . 15 1.6 Results about lens spaces . 17 2 Links in lens spaces 19 2.1 General definitions . 19 2.2 Mixed link diagrams . 22 2.3 Band diagrams . 23 2.4 Grid diagrams . 25 3 Disk diagram and Reidemeister-type moves 29 3.1 Disk diagram . 30 3.2 Generalized Reidemeister moves . 32 3.3 Standard form of the disk diagram . 36 3.4 Connection with band diagram . 38 3.5 Connection with grid diagram . 42 4 Group of links in lens spaces via Wirtinger presentation 47 4.1 Group of the link . 48 i ii CONTENTS 4.2 First homology group . 52 4.3 Relevant examples . 54 5 Twisted Alexander polynomials for links in lens spaces 57 5.1 The computation of the twisted Alexander polynomials . 57 5.2 Properties of the twisted Alexander polynomials . 59 5.3 Connection with Reidemeister torsion . 61 6 Lifting links from lens spaces to the 3-sphere 65 6.1 Diagram for the lift via disk diagrams . 66 6.2 Diagram for the lift via band and grid diagrams .
    [Show full text]
  • Introduction to Vassiliev Knot Invariants First Draft. Comments
    Introduction to Vassiliev Knot Invariants First draft. Comments welcome. July 20, 2010 S. Chmutov S. Duzhin J. Mostovoy The Ohio State University, Mansfield Campus, 1680 Univer- sity Drive, Mansfield, OH 44906, USA E-mail address: [email protected] Steklov Institute of Mathematics, St. Petersburg Division, Fontanka 27, St. Petersburg, 191011, Russia E-mail address: [email protected] Departamento de Matematicas,´ CINVESTAV, Apartado Postal 14-740, C.P. 07000 Mexico,´ D.F. Mexico E-mail address: [email protected] Contents Preface 8 Part 1. Fundamentals Chapter 1. Knots and their relatives 15 1.1. Definitions and examples 15 § 1.2. Isotopy 16 § 1.3. Plane knot diagrams 19 § 1.4. Inverses and mirror images 21 § 1.5. Knot tables 23 § 1.6. Algebra of knots 25 § 1.7. Tangles, string links and braids 25 § 1.8. Variations 30 § Exercises 34 Chapter 2. Knot invariants 39 2.1. Definition and first examples 39 § 2.2. Linking number 40 § 2.3. Conway polynomial 43 § 2.4. Jones polynomial 45 § 2.5. Algebra of knot invariants 47 § 2.6. Quantum invariants 47 § 2.7. Two-variable link polynomials 55 § Exercises 62 3 4 Contents Chapter 3. Finite type invariants 69 3.1. Definition of Vassiliev invariants 69 § 3.2. Algebra of Vassiliev invariants 72 § 3.3. Vassiliev invariants of degrees 0, 1 and 2 76 § 3.4. Chord diagrams 78 § 3.5. Invariants of framed knots 80 § 3.6. Classical knot polynomials as Vassiliev invariants 82 § 3.7. Actuality tables 88 § 3.8. Vassiliev invariants of tangles 91 § Exercises 93 Chapter 4.
    [Show full text]