Braid equivalences in 3-manifolds with rational surgery description (original) (raw)
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Braid equivalence in 3-manifolds with rational surgery description
arXiv (Cornell University), 2013
In this paper we describe braid equivalence for knots and links in a 3-manifold M obtained by rational surgery along a framed link in S 3. We first prove a sharpened version of the Reidemeister theorem for links in M. We then give geometric formulations of the braid equivalence via mixed braids in S 3 using the L-moves and the braid band moves. We finally give algebraic formulations in terms of the mixed braid groups Bm,n using cabling and the techniques of parting and combing for mixed braids. We also provide concrete formuli of the braid equivalence in the case where M is a lens space, a Seifert manifold or a homology sphere obtained from the trefoil. The algebraic classification of knots and links in a 3-manifold via mixed braids is a useful tool for studying skein modules of 3-manifolds.
Studying links via closed braids. VI. A nonfiniteness theorem
Pacific Journal of Mathematics, 1992
Exchange moves were introduced in an earlier paper by the same authors. They take one closed n-braid representative of a link to another, and can lead to examples where there are infinitely many conjugacy classes of n-braids representing a single link type. THEOREM I. If a link type has infinitely many conjugacy classes of closed n-braid representatives, then n > 4 and the infinitely many classes divide into finitely many equivalence classes under the equivalence relation generated by exchange moves. This theorem is the last of the preliminary steps in the authors' program for the development of a calculus on links in S 3. THEOREM 2. Choose integers n, g > 1. Then there are at most finitely many link types with braid index n and genus g.
On framings of knots in 3-manifolds
2020
We show that the only way of changing the framing of a knot or a link by ambient isotopy in an oriented 333-manifold is when the manifold has a properly embedded non-separating S2S^2S2. This change of framing is given by the Dirac trick, also known as the light bulb trick. The main tool we use is based on McCullough's work on the mapping class groups of 333-manifolds. We also relate our results to the theory of skein modules.
An Algebra Involving Braids and Ties
In this note we study a family of algebras E n (u) with one parameter defined by generators and relations. The set of generators contains the generators of the usual braids algebra, and another set of generators which is interpreted as ties between consecutive strings. We also study the representations theory of the algebra when the parameter is specialized to 1. Foreword This article was written in the 2000 year and published as ICTP preprint [1]. The authors submitted it to a journal, but it was rejected since according to the referee report the considered algebra was the Hecke algebra with another vesture; moreover, this algebra should be not related to knot theory. However, in the last years the algebras E n (u) was revisited by Ryom-Hansen [5], who found explicit bases and classified the irreducible representations. Moreover, in 2013 Banjo [6] determined the complex generic representation theory, showing that a certain specialization of this algebra is isomorphic to the small ramified partition algebra. The authors of the article have successively proved that E n (u)E n (u) admits a trace [2] and introduced the tied links, defining different invariants for them [3]. More recently, they have considered Kauffman type invariants for tied links [4] and have introduced a new algebra (a generalization of the BMW algebra) which is related to E n (u). A Referee of [4] suggested to put in ArXiv the present article.
2014
We construct graph-valued analogues of the Kuperberg sl(3) and G2 invariants for virtual knots. The restriction of the sl(3) or G2 invariants for classical knots coincides with the usual Homflypt sl(3) invariant and G2 invariants. For virtual knots and graphs these invariants provide new graphical information that allows one to prove minimality theorems and to construct new invariants for free knots (unoriented and unlabeled Gauss codes taken up to abstract Reidemeister moves). A novel feature of this approach is that some knots are of sufficient complexity that they evaluate themselves in the sense that the invariant is the knot itself seen as a combinatorial structure. The paper generalizes these structures to virtual braids and discusses the relationship with the original Penrose bracket for graph colorings.
arXiv: Geometric Topology, 2020
We define the property (D) for nontrivial knots. We show that the fundamental group of the manifold obtained by Dehn surgery on a knot KKK with property (D) with slope fracpqge2g(K)−1\frac{p}{q}\ge 2g(K)-1fracpqge2g(K)−1 is not left orderable. By making full use of the fixed point method, we prove that (1) nontrivial knots which are closures of positive 111-bridge braids have property (D); (2) L-space satellite knots, with positive 111-bridge braid patterns, and companion with property (D), have property (D); (3) the fundamental group of the manifold obtained by Dehn filling on v2503v2503v2503 is not left orderable. Additionally, we prove that L-space twisted torus knots of form Tp,kppm1l,mT_{p,kp\pm 1}^{l,m}Tp,kppm1l,m are closures of positive 111-bridge braids.
Journal of Knot Theory and Its Ramifications, 2015
We construct graph-valued analogues of the Kuperberg sl(3) and G2 invariants for virtual knots. The restriction of the sl(3) and G2 invariants for classical knots coincides with the usual Homflypt sl(3) invariant and G2 invariants. For virtual knots and graphs these invariants provide new graphical information that allows one to prove minimality theorems and to construct new invariants for free knots (unoriented and unlabeled Gauss codes taken up to abstract Reidemeister moves). A novel feature of this approach is that some knots are of sufficient complexity that they evaluate themselves in the sense that the invariant is the knot itself seen as a combinatorial structure. The paper generalizes these structures to virtual braids and discusses the relationship with the original Penrose bracket for graph colorings.
5-MOVE Equivalence Classes of Links and Their Algebraic Invariants
Journal of Knot Theory and Its Ramifications, 2007
We start a systematic analysis of links up to 5-move equivalence. Our motivation is to develop tools which later can be used to study skein modules based on the skein relation being deformation of a 5-move (in an analogous way as the Kauffman skein module is a deformation of a 2-move, i.e. a crossing change). Our main tools are Jones and Kauffman polynomials and the fundamental group of the 2-fold branch cover of S3 along a link. We use also the fact that a 5-move is a composition of two rational ±(2, 2)-moves (i.e. [Formula: see text]-moves) and rational moves can be analyzed using the group of Fox colorings and its non-abelian version, the Burnside group of a link. One curious observation is that links related by one (2, 2)-move are not 5-move equivalent. In particular, we partially classify (up to 5-moves) 3-braids, pretzel and Montesinos links, and links up to 9 crossings.
Ju n 20 06 Virtual Braids and the L – Move
2008
In this paper we prove a Markov Theorem for the virtual braid group and for some analogs of this structure. The virtual braid group is the natural companion to the category of virtual knots, just as the Artin braid group is to classical knots and links. In classical knot theory the braid group gives a fundamental algebraic structure associated with knots. The Alexander Theorem tells us that every knot or link can be isotoped to braid form. The capstone of this relationship is the Markov Theorem, giving necessary and sufficient conditions for two braids to close to the same link (where sameness of two links means that they are ambient isotopic).