scholarly journals Multivariable link invariants arising from sl(2|1) and the Alexander polynomial

2007 ◽  
Vol 210 (1) ◽  
pp. 283-298 ◽  
Author(s):  
Nathan Geer ◽  
Bertrand Patureau-Mirand
Keyword(s):  
Alexander Polynomial ◽  
Link Invariants

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.

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.

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].

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 ON THE COLORED HOMFLY-PT, MULTIVARIABLE AND KASHAEV LINK INVARIANTS

2008 ◽  
Vol 10 (supp01) ◽  
pp. 993-1011 ◽  
Author(s):  
NATHAN GEER ◽  
BERTRAND PATUREAU-MIRAND
Keyword(s):  
Alexander Polynomial ◽  
Link Invariants ◽  
Alexander Invariants

We study various specializations of the colored HOMFLY-PT polynomial. These specializations are used to show that the multivariable link invariants arising from a complex family of 𝔰𝔩(m|n) super-modules previously defined by the authors contain both the multivariable Alexander polynomial and Kashaev's invariants. We conjecture these multivariable link invariants also specialize to the generalized multivariable Alexander invariants defined by Y. Akutsu, T. Deguchi, and T. Ohtsuki.

On the Arf invariant of links

1986 ◽  
Vol 100 (2) ◽  
pp. 355-359 ◽  
Author(s):  
Ying-Qing Wu
Keyword(s):  
Partial Solution ◽  
Alexander Polynomial ◽  
Link Invariants ◽  
Arf Invariant ◽  
Image Position

In ([2], p. 310) the problem of relating the Arf invariant of a link to other link invariants, and finding practical ways of computing it, was raised. Murasugi gave a partial solution to this in [9]: if L has two components, then Arf(L), when defined, is given bywhere l1, l2 are the components of L, and ΔL(t1, t2) denotes the Alexander polynomial of L.

scholarly journals POLYNOMIAL KNOT AND LINK INVARIANTS FROM THE VIRTUAL BIQUANDLE

2013 ◽  
Vol 22 (04) ◽  
pp. 1340004 ◽  
Author(s):  
ALISSA S. CRANS ◽  
ALLISON HENRICH ◽  
SAM NELSON
Keyword(s):  
Polynomial Ring ◽  
Laurent Polynomial ◽  
Alexander Polynomial ◽  
Virtual Knot ◽  
Link Invariants ◽  
Virtual Knots ◽  
Monomial Ordering ◽  
Laurent Polynomial Ring ◽  
Future Work ◽  
Knots And Links

The Alexander biquandle of a virtual knot or link is a module over a 2-variable Laurent polynomial ring which is an invariant of virtual knots and links. The elementary ideals of this module are then invariants of virtual isotopy which determine both the generalized Alexander polynomial (also known as the Sawollek polynomial) for virtual knots and the classical Alexander polynomial for classical knots. For a fixed monomial ordering <, the Gröbner bases for these ideals are computable, comparable invariants which fully determine the elementary ideals and which generalize and unify the classical and generalized Alexander polynomials. We provide examples to illustrate the usefulness of these invariants and propose questions for future work.

VASSILIEV INVARIANTS AND DOUBLE DATING TANGLES

2002 ◽  
Vol 11 (04) ◽  
pp. 527-544 ◽  
Author(s):  
MYEONG-JU JEONG ◽  
CHAN-YOUNG PARK
Keyword(s):  
Finite Type ◽  
Alexander Polynomial ◽  
Necessary Condition ◽  
Link Invariant ◽  
Vassiliev Invariants ◽  
Link Invariants ◽  
Knot Invariant ◽  
Vassiliev Invariant ◽  
Framed Link

In [1], E. Appleboim introduced the notion of double dating linking class-P invariants of finite type for framed links with a fixed linking matrix P and showed that all Vassiliev link invariants are of finite type for any linking matrix and in [13], R. Trapp provided a necessary condition for a knot invariant to be a Vassiliev invariant by using twist sequences. In this paper we provide a necessary condition for a framed link invariant to be a DD-linking class-P invariant of finite type by considering sequence of links induced from a double dating tangle. As applications we give a generalization of R. Trapp's result to see whether a link invariant is a Vassiliev invariant or not and apply the criterion for all non-zero coefficients of the Jones, HOMFLY, Q-, and Alexander polynomial.

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.

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