Strongly spin-preserving solutions of the Yang-Baxter equation and their link invariants

1993 ◽  
Vol 113 (2) ◽  
pp. 401-411
Author(s):  
M. A. Hennings
Keyword(s):  
Jones Polynomial ◽  
Alexander Polynomial ◽  
The Other ◽  
Baxter Equation ◽  
Link Invariants

Recently Kauffman[2, 3] has classified the (strongly) spin-preserving solutions of the Yang—Baxter equation, and in particular discussed two solutions. One of these can be used to obtain the Jones polynomial, while the other (with care) leads to the Alexander polynomial. In this paper we shall complete this analysis, and shall describe all strongly spin-preserving invertible solutions of the Yang-Baxter equation which lead to link invariants. Having done this, we investigate what sort of link invariants can be obtained from these solutions. It turns out that these invariants are precisely those which have been described in Hennings [l].

scholarly journals ALL LINK INVARIANTS FOR TWO DIMENSIONAL SOLUTIONS OF YANG–BAXTER EQUATION AND DRESSINGS

2006 ◽  
Vol 15 (10) ◽  
pp. 1279-1301
Author(s):  
N. AIZAWA ◽  
M. HARADA ◽  
M. KAWAGUCHI ◽  
E. OTSUKI
Keyword(s):  
Jones Polynomial ◽  
Alexander Polynomial ◽  
Baxter Equation ◽  
Two Dimensional ◽  
Link Invariant ◽  
Three Solutions ◽  
Polynomial Invariants ◽  
Link Invariants ◽  
Higher Dimensional

All polynomial invariants of links for two dimensional solutions of Yang–Baxter equation is constructed by employing Turaev's method. As a consequence, it is proved that the best invariant so constructed is the Jones polynomial and there exist three solutions connecting to the Alexander polynomial. Invariants for higher dimensional solutions, obtained by the so-called dressings, are also investigated. It is observed that the dressings do not improve link invariant unless some restrictions are put on dressed solutions.

FERMIONS AND LINK INVARIANTS

1992 ◽  
Vol 07 (supp01a) ◽  
pp. 493-532 ◽  
Author(s):  
L. Kauffman ◽  
H. Saleur
Keyword(s):  
Braid Group ◽  
Degrees Of Freedom ◽  
Knot Theory ◽  
Alexander Polynomial ◽  
Baxter Equation ◽  
Group Representations ◽  
R Matrix ◽  
Link Invariants ◽  
Linear Representations ◽  
State Models

This paper deals with various aspects of knot theory when fermionic degrees of freedom are taken into account in the braid group representations and in the state models. We discuss how the Ř matrix for the Alexander polynomial arises from the Fox differential calculus, and how it is related to the quantum group Uqgl(1,1). We investigate new families of solutions of the Yang Baxter equation obtained from "linear" representations of the braid group and exterior algebra. We study state models associated with Uqsl(n,m), and in the case n=m=1 a state model for the multivariable Alexander polynomial. We consider invariants of links in solid handlebodies and show how the non trivial topology lifts the boson fermion degeneracy that is present in S3. We use "gauge like" changes of basis to obtain invariants in thickened surfaces Σ×[0,1].

scholarly journals FROM RIBBON CATEGORIES TO GENERALIZED YANG–BAXTER OPERATORS AND LINK INVARIANTS (AFTER KITAEV AND WANG)

2013 ◽  
Vol 24 (01) ◽  
pp. 1250126 ◽  
Author(s):  
SEUNG-MOON HONG
Keyword(s):  
The Other ◽  
Polynomial Invariant ◽  
Link Invariants ◽  
Fusion Categories ◽  
New Family ◽  
Kauffman Polynomial ◽  
Twist Structure

We consider two approaches to isotopy invariants of oriented links: one from ribbon categories and the other from generalized Yang–Baxter (gYB) operators with appropriate enhancements. The gYB-operators we consider are obtained from so-called gYBE objects following a procedure of Kitaev and Wang. We show that the enhancement of these gYB-operators is canonically related to the twist structure in ribbon categories from which the operators are produced. If a gYB-operator is obtained from a ribbon category, it is reasonable to expect that two approaches would result in the same invariant. We prove that indeed the two link invariants are the same after normalizations. As examples, we study a new family of gYB-operators which is obtained from the ribbon fusion categories SO (N)2, where N is an odd integer. These operators are given by 8 × 8 matrices with the parameter N and the link invariants are specializations of the two-variable Kauffman polynomial invariant F.

scholarly journals BIRACK SHADOW MODULES AND THEIR LINK INVARIANTS

2013 ◽  
Vol 22 (10) ◽  
pp. 1350056 ◽  
Author(s):  
SAM NELSON ◽  
KATIE PELLAND
Keyword(s):  
Associative Algebra ◽  
Alexander Polynomial ◽  
Link Invariants ◽  
Knots And Links

We introduce an associative algebra ℤ[X, S] associated to a birack shadow and define enhancements of the birack counting invariant for classical knots and links via representations of ℤ[X, S] known as shadow modules. We provide examples which demonstrate that the shadow module enhanced invariants are not determined by the Alexander polynomial or the unenhanced birack counting invariants.

scholarly journals Khovanov Homology of Three-Strand Braid Links

Symmetry ◽  
2018 ◽  
Vol 10 (12) ◽  
pp. 720
Author(s):  
Young Kwun ◽  
Abdul Nizami ◽  
Mobeen Munir ◽  
Zaffar Iqbal ◽  
Dishya Arshad ◽  
...  
Keyword(s):  
Euler Characteristic ◽  
Jones Polynomial ◽  
Khovanov Homology ◽  
Homology Groups ◽  
Link Invariants ◽  
Chain Complexes ◽  
Chain Homotopy

Khovanov homology is a categorication of the Jones polynomial. It consists of graded chain complexes which, up to chain homotopy, are link invariants, and whose graded Euler characteristic is equal to the Jones polynomial of the link. In this article we give some Khovanov homology groups of 3-strand braid links Δ 2 k + 1 = x 1 2 k + 2 x 2 x 1 2 x 2 2 x 1 2 ⋯ x 2 2 x 1 2 x 1 2 , Δ 2 k + 1 x 2 , and Δ 2 k + 1 x 1 , where Δ is the Garside element x 1 x 2 x 1 , and which are three out of all six classes of the general braid x 1 x 2 x 1 x 2 ⋯ with n factors.

scholarly journals Estimating Jones and HOMFLY polynomials with one clean qubit

2009 ◽  
Vol 9 (3&4) ◽  
pp. 264-289
Author(s):  
S.P. Jordan ◽  
P. Wocjan
Keyword(s):  
Quantum Computing ◽  
Quantum Computer ◽  
Jones Polynomial ◽  
Single Variable ◽  
Initial State ◽  
Link Invariants ◽  
Root Of Unity ◽  
Complete Problem ◽  
The One ◽  
Homfly Polynomials

The Jones and HOMFLY polynomials are link invariants with close connections to quantum computing. It was recently shown that finding a certain approximation to the Jones polynomial of the trace closure of a braid at the fifth root of unity is a complete problem for the one clean qubit complexity class\cite{Shor_Jordan}. This is the class of problems solvable in polynomial time on a quantum computer acting on an initial state in which one qubit is pure and the rest are maximally mixed. Here we generalize this result by showing that one clean qubit computers can efficiently approximate the Jones and single-variable HOMFLY polynomials of the trace closure of a braid at \emph{any} root of unity.

scholarly journals Writhe-like invariants of alternating links

2021 ◽  
Vol 30 (01) ◽  
pp. 2150004
Author(s):  
Yuanan Diao ◽  
Van Pham
Keyword(s):  
Jones Polynomial ◽  
Link Diagram ◽  
Link Invariant ◽  
Link Invariants ◽  
Link Diagrams ◽  
Alternating Links ◽  
Alternating Link

It is known that the writhe calculated from any reduced alternating link diagram of the same (alternating) link has the same value. That is, it is a link invariant if we restrict ourselves to reduced alternating link diagrams. This is due to the fact that reduced alternating link diagrams of the same link are obtainable from each other via flypes and flypes do not change writhe. In this paper, we introduce several quantities that are derived from Seifert graphs of reduced alternating link diagrams. We prove that they are “writhe-like” invariants, namely they are not general link invariants, but are invariants when restricted to reduced alternating link diagrams. The determination of these invariants are elementary and non-recursive so they are easy to calculate. We demonstrate that many different alternating links can be easily distinguished by these new invariants, even for large, complicated knots for which other invariants such as the Jones polynomial are hard to compute. As an application, we also derive an if and only if condition for a strongly invertible rational link.

Equivalence of Cables Of Mutants of Knots

1989 ◽  
Vol 41 (2) ◽  
pp. 250-273 ◽  
Author(s):  
Józef H. Przytycki
Keyword(s):  
Mirror Image ◽  
Alexander Polynomial ◽  
The Other ◽  
Homfly Polynomial ◽  
Similar Formula ◽  
Conway Polynomial ◽  
Other Hand ◽  
Limited Possibility

There is the nice formula which links the Alexander polynomial of (m, k)-cable of a link with the Alexander polynomial of the link [5] [36] [38]. H. Morton and H. Short investigated whether a similar formula holds for the Jones-Conway (Homfly) polynomial and they found that it is very unlikely. Morton and Short made many calculations of the Jones-Conway polynomial of (2, q)-cables along knots (2 was chosen because of limited possibility of computers) and they get very interesting experimental material [24], [25]. In particular they found that using their method they were able to distinguish some Birman [4] and Lozano-Morton [22] examples (all which they tried) and the 942 knot (in the Rolfsen [37] notation) from its mirror image. On the other hand they were unable to distinguish the Conway knot and the Kinoshita-Terasaka knot.

The Tait conjecture in S1 × S2

2016 ◽  
Vol 25 (11) ◽  
pp. 1650063 ◽  
Author(s):  
Alessio Carrega
Keyword(s):  
Analogous Result ◽  
Jones Polynomial ◽  
The Other ◽  
Complete Answer ◽  
Other Hand ◽  
Alternating Links

The Tait conjecture states that reduced alternating diagrams of links in [Formula: see text] have the minimal number of crossings. It has been proved in 1987 by M. Thistlethwaite, L. H. Kauffman and K. Murasugi studying the Jones polynomial. In this paper, we prove an analogous result for alternating links in [Formula: see text] giving a complete answer to this problem. In [Formula: see text] we find a dichotomy: the appropriate version of the statement is true for [Formula: see text]-homologically trivial links, and our proof also uses the Jones polynomial. On the other hand, the statement is false for [Formula: see text]-homologically non-trivial links, for which the Jones polynomial vanishes.

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