MODULAR CATEGORIES AND 3-MANIFOLD INVARIANTS

1992 ◽  
Vol 06 (11n12) ◽  
pp. 1807-1824 ◽  
Author(s):  
VLADIMIR G. TURAEV
Keyword(s):  
Quantum Groups ◽  
Tensor Category ◽  
Jones Polynomial ◽  
Knot Invariants ◽  
Modular Tensor Category ◽  
Modular Category

The aim of this paper is to give a concise introduction to the theory of knot invariants and 3-manifold invariants which generalize the Jones polynomial and which may be considered as a mathematical version of the Witten invariants. Such a theory was introduced by N. Reshetikhin and the author on the ground of the theory of quantum groups. Here we use more general algebraic objects, specifically, ribbon and modular categories. Such categories in particular arise as the categories of representations of quantum groups. The notion of modular category, interesting in itself, is closely related to the notion of modular tensor category in the sense of G. Moore and N. Seiberg. For simplicity we restrict ourselves in this paper to the case of closed 3-manifolds.

scholarly journals ON THE ANALYTIC PROPERTIES OF THE z-COLOURED JONES POLYNOMIAL

2005 ◽  
Vol 14 (04) ◽  
pp. 435-466 ◽  
Author(s):  
JOÃO FARIA MARTINS
Keyword(s):  
Jones Polynomial ◽  
Unitary Representations ◽  
Target Space ◽  
Analytic Extension ◽  
Torus Knots ◽  
Infinite Dimensional ◽  
Knot Invariants ◽  
Starting Point ◽  
Zero Radius ◽  
Formal Power

We analyse the possibility of defining ℂ-valued Knot invariants associated with infinite-dimensional unitary representations of SL(2,ℝ) and the Lorentz Group taking as starting point the Kontsevich integral and the notion of infinitesimal character. This yields a family of knot invariants whose target space is the set of formal power series in ℂ, which contained in the Melvin–Morton expansion of the coloured Jones polynomial. We verify that for some knots the series have zero radius of convergence and analyse the construction of functions of which this series are asymptotic expansions by means of Borel re-summation. Explicit calculations are done in the case of torus knots which realise an analytic extension of the values of the coloured Jones polynomial to complex spins. We present a partial answer in the general case.

scholarly journals Two Lectures on the Jones Polynomial and Khovanov Homology

2017 ◽  
Author(s):  
Edward Witten
Keyword(s):  
Gauge Theory ◽  
Three Dimensional ◽  
Jones Polynomial ◽  
Khovanov Homology ◽  
Theory Approach ◽  
Laurent Polynomials ◽  
Four Dimensions ◽  
Knot Invariants ◽  
Fifth Dimension ◽  
Instanton Number

In the first of these two lectures I describe a gauge theory approach to understanding quantum knot invariants as Laurent polynomials in a complex variable q. The two main steps are to reinterpret three-dimensional Chern-Simons gauge theory in four dimensional terms and then to apply electric-magnetic duality. The variable q is associated to instanton number in the dual description in four dimensions. In the second lecture, I describe how Khovanov homology can emerge upon adding a fifth dimension.

scholarly journals Classification of globally colorized categories of partitions

2018 ◽  
Vol 21 (04) ◽  
pp. 1850029 ◽  
Author(s):  
Daniel Gromada
Keyword(s):  
Quantum Group ◽  
Quantum Groups ◽  
Tensor Category ◽  
Set Partitions

Set partitions closed under certain operations form a tensor category. They give rise to certain subgroups of the free orthogonal quantum group [Formula: see text], the so-called easy quantum groups, introduced by Banica and Speicher in 2009. This correspondence was generalized to two-colored set partitions, which, in addition, assign a black or white color to each point of a set. Globally colorized categories of partitions are those categories that are invariant with respect to arbitrary permutations of colors. This paper presents a classification of globally colorized categories. In addition, we show that the corresponding unitary quantum groups can be constructed from the orthogonal ones using tensor complexification.

RESHETIKHIN-TURAEV AND CRANE-KOHNO-KONTSEVICH 3-MANIFOLD INVARIANTS COINCIDE

1993 ◽  
Vol 02 (01) ◽  
pp. 65-95 ◽  
Author(s):  
SERGEY PIUNIKHIN
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Explicit Formula ◽  
Quantum Groups ◽  
Jones Polynomial ◽  
Matrix Elements ◽  
Conformal Blocks ◽  
Conformal Field ◽  
Wznw Model ◽  
The Matrix

The coincidence of two different presentations of Witten 3-manifold invariants is proved. One of them, invented by Reshetikhin and Turaev, is based on the surgery presentation a of 3-manifold and the representation theory of quantum groups; another one, invented by Kohno and Crane and, in slightly different language by Kontsevich, is based on a Heegaard decomposition of a 3-manifold and representations of the Teichmuller group, arising in conformal field theory. The explicit formula for the matrix elements of generators of the Teichmuller group in the space of conformal blocks in the SU(2) k, WZNW-model is given,using the Jones polynomial of certain links.

scholarly journals Quotients of the Gordian and H(2)-Gordian graphs

2021 ◽  
pp. 2150037
Author(s):  
Christopher Flippen ◽  
Allison H. Moore ◽  
Essak Seddiq
Keyword(s):  
Complete Graph ◽  
Jones Polynomial ◽  
Graph Isomorphism ◽  
Isomorphism Type ◽  
Equivalence Relations ◽  
Knot Invariants ◽  
Quotient Graphs ◽  
Gromov Hyperbolic ◽  
Vertex Sets

The Gordian graph and H(2)-Gordian graphs of knots are abstract graphs whose vertex sets represent isotopy classes of unoriented knots, and whose edge sets record whether pairs of knots are related by crossing changes or H(2)-moves, respectively. We investigate quotients of these graphs under equivalence relations defined by several knot invariants including the determinant, the span of the Jones polynomial, and an invariant related to tricolorability. We show, in all cases considered, that the quotient graphs are Gromov hyperbolic. We then prove a collection of results about the graph isomorphism type of the quotient graphs. In particular, we find that the H(2)-Gordian graph of links modulo the relation induced by the span of the Jones polynomial is isomorphic with the complete graph on infinitely many vertices.

scholarly journals Crosscap number of knots and volume bounds

2020 ◽  
Vol 31 (13) ◽  
pp. 2050111
Author(s):  
Noboru Ito ◽  
Yusuke Takimura
Keyword(s):  
Jones Polynomial ◽  
Upper Bounds ◽  
Crossing Number ◽  
Proved Theorem ◽  
Knot Invariants ◽  
Chord Diagrams ◽  
Hyperbolic Volume ◽  
Knot Invariant ◽  
Twist Number

In this paper, we obtain the crosscap number of any alternating knots by using our recently-introduced diagrammatic knot invariant (Theorem 1). The proof is given by properties of chord diagrams (Kindred proved Theorem 1 independently via other techniques). For non-alternating knots, we give Theorem 2 that generalizes Theorem 1. We also improve known formulas to obtain upper bounds of the crosscap number of knots (alternating or non-alternating) (Theorem 3). As a corollary, this paper connects crosscap numbers and our invariant with other knot invariants such as the Jones polynomial, twist number, crossing number, and hyperbolic volume (Corollaries 1–7). In Appendix A, using Theorem 1, we complete giving the crosscap numbers of the alternating knots with up to 11 crossings including those of the previously unknown values for [Formula: see text] knots (Tables A.1).

scholarly journals RIGIDITY AND MODULARITY OF VERTEX TENSOR CATEGORIES

2008 ◽  
Vol 10 (supp01) ◽  
pp. 871-911 ◽  
Author(s):  
YI-ZHI HUANG
Keyword(s):  
Matrix Element ◽  
Vertex Operator ◽  
Vertex Operator Algebra ◽  
Tensor Category ◽  
General Version ◽  
Natural Structure ◽  
Tensor Categories ◽  
Modular Tensor Category ◽  
Verlinde Formula ◽  
Completely Reducible

Let V be a simple vertex operator algebra satisfying the following conditions: (i) V(n)= 0 for n < 0, V(0)= ℂ1 and V′ is isomorphic to V as a V-module. (ii) Every ℕ-gradable weak V-module is completely reducible. (iii) V is C2-cofinite. (In the presence of Condition (i), Conditions (ii) and (iii) are equivalent to a single condition, namely, that every weak V-module is completely reducible.) Using the results obtained by the author in the formulation and proof of the general version of the Verlinde conjecture and in the proof of the Verlinde formula, we prove that the braided tensor category structure on the category of V-modules is rigid, balanced and nondegenerate. In particular, the category of V-modules has a natural structure of modular tensor category. We also prove that the tensor-categorical dimension of an irreducible V-module is the reciprocal of a suitable matrix element of the fusing isomorphism under a suitable basis.

scholarly journals Knot theory realizations in nematic colloids

2015 ◽  
Vol 112 (6) ◽  
pp. 1675-1680 ◽  
Author(s):  
Simon Čopar ◽  
Uroš Tkalec ◽  
Igor Muševič ◽  
Slobodan Žumer
Keyword(s):  
Phase Diagram ◽  
Knot Theory ◽  
Colloidal Particles ◽  
Jones Polynomial ◽  
Laser Tweezers ◽  
Knot Invariants ◽  
Topological Parameters ◽  
Twisted Nematic ◽  
2D Arrays ◽  
Knots And Links

Nematic braids are reconfigurable knots and links formed by the disclination loops that entangle colloidal particles dispersed in a nematic liquid crystal. We focus on entangled nematic disclinations in thin twisted nematic layers stabilized by 2D arrays of colloidal particles that can be controlled with laser tweezers. We take the experimentally assembled structures and demonstrate the correspondence of the knot invariants, constructed graphs, and surfaces associated with the disclination loop to the physically observable features specific to the geometry at hand. The nematic nature of the medium adds additional topological parameters to the conventional results of knot theory, which couple with the knot topology and introduce order into the phase diagram of possible structures. The crystalline order allows the simplified construction of the Jones polynomial and medial graphs, and the steps in the construction algorithm are mirrored in the physics of liquid crystals.

Quantum Groups at a Root of 1 and Tangle Invariants

1993 ◽  
Vol 07 (20n21) ◽  
pp. 3715-3726 ◽  
Author(s):  
MARC ROSSO
Keyword(s):  
Quantum Groups ◽  
Dimensional Representation ◽  
Knot Invariants ◽  
Conway Polynomial

One uses certain representations (in the De Concini-Kac picture) of the quantum groups [Formula: see text] for q a root of 1 to produce for R-matrices depending on rank [Formula: see text] continuous parameters. Using the formalism of Reshetikhin and Turaev, this allows to produce tangle and knot invariants depending on rank [Formula: see text] parameters. The simplest example ([Formula: see text] in the 2-dimensional representation) gives the Alexander-Conway polynomial.

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