On the location of zeros of the Homfly polynomial

The Electronic Journal of Combinatorics

The focus of this paper is to study the HOMFLY polynomial of $(2,n)$-torus link as a generalized Fibonacci polynomial. For this purpose, we first introduce a form of generalized Fibonacci and Lucas polynomials and provide their some fundamental properties. We define the HOMFLY polynomial of $ (2,n) $-torus link with a way similar to our generalized Fibonacci polynomials and provide its fundamental properties. We also show that the HOMFLY polynomial of $ (2,n) $-torus link can be obtained from its Alexander-Conway polynomial or the classical Fibonacci polynomial. We finally give the matrix representations and prove important identities, which are similar to the Fibonacci identities, for the our generalized Fibonacci polynomial and the HOMFLY polynomial of $ (2,n) $-torus link.

The Electronic Journal of Combinatorics

In this paper, we introduce the Jones polynomial of a graph $G=(V,E)$ with $k$ components as the following specialization of the Tutte polynomial:$$J_G(t)=(-1)^{|V|-k}t^{|E|-|V|+k}T_G(-t,-t^{-1}).$$We first study its basic properties and determine certain extreme coefficients. Then we prove that $(-\infty, 0]$ is a zero-free interval of Jones polynomials of connected bridgeless graphs while for any small $\epsilon>0$ or large $M>0$, there is a zero of the Jones polynomial of a plane graph in $(0,\epsilon)$, $(1-\epsilon,1)$, $(1,1+\epsilon)$ or $(M,+\infty)$. Let $r(G)$ be the maximum moduli of zeros of $J_G(t)$. By applying Sokal's result on zeros of Potts model partition functions and Lucas's theorem, we prove that\begin{eqnarray*}{q_s-|V|+1\over |E|}\leq r(G)<1+6.907652\Delta_G\end{eqnarray*}for any connected bridgeless and loopless graph $G=(V,E)$ of maximum degree $\Delta_G$ with $q_s$ parallel classes. As a consequence of the upper bound, X.-S. Lin's conj...

10TH INTERNATIONAL CONFERENCE ON APPLIED SCIENCE AND TECHNOLOGY

The Jones polynomial of an alternating link is known to have no gap of length greater than 1. This result extends to quasi-alternating links as well. Our purpose is to study the structure, in particular the number of gaps, of the Jones polynomial of an arbitrary link with the ultimate aim of characterizing Laurent polynomials which arise as the Jones polynomial of a link.

Discrete Mathematics

It is well known that any link can be represented by the closure of a braid. The minimum number of strings needed in a braid whose closure represents a given link is called the braid index of the link and the well known Morton-Frank-Williams inequality reveals a close relationship between the HOMFLY polynomial of a link and its braid index. In the case that a link is already presented in a closed braid form, Jaeger derived a special formulation of the HOMFLY polynomial. In this paper, we prove a variant of Jaeger's result as well as a dual version of it. Unlike Jaeger's original reasoning, which relies on representation theory, our proof uses only elementary geometric and combinatorial observations. Using our variant and its dual version, we provide a direct and elementary proof of the fact that the braid index of a link that has an n-string closed braid diagram that is also reduced and alternating, is exactly n. Until know this fact was only known as a consequence of a result due to Murasugi on fibered links that are star products of elementary torus links and of the fact that alternating braids are fibered.

2003

We analyse the possibility of defining complex 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.

2010

This article contains general formulas for Tutte and Jones polynomials for families of knots and links given in Conway notation and "portraits of families"plots of zeroes of their corresponding Jones polynomials.

Journal of Knot Theory and Its Ramifications, 2007

J.P. Levine showed that the Conway polynomial of a link is a product of two factors: one is the Conway polynomial of a knot which is obtained from the link by banding together the components; and the other is determined by theμ-invariants of a string link with the link as its closure. We give another description of the latter factor: the determinant of a matrix whose entries are linking pairings in the infinite cyclic covering space of the knot complement, which take values in the quotient field of Z[t, t −1 ]. In addition, we give a relation between the Taylor expansion of a linking pairing around t = 1 and derivation on links which is invented by T.D. Cochran. In fact, the coefficients of the powers of t − 1 will be the linking numbers of certain derived links in S 3 . Therefore, the first non-vanishing coefficient of the Conway polynomial is determined by the linking numbers in S 3 . This generalizes a result of J. Hoste.

Algebraic & Geometric Topology, 2006

We show that if {L n } is any infinite sequence of links with twist number τ (L n ) and with cyclotomic Jones polynomials of increasing span, then lim sup τ (L n ) = ∞. This implies that any infinite sequence of prime alternating links with cyclotomic Jones polynomials must have unbounded hyperbolic volume. The main tool is the multivariable twist-bracket polynomial, which generalizes the Kauffman bracket to link diagrams with open twist sites.

Topology and Its Applications - TOPOL APPL, 1998

The higher order link polynomials are a class of link invariants related to both Homfly polynomial and Vassiliev invariants. Here we study their partial derivatives. We prove that each partial derivative of an nth order Homfly polynomial is an (n + 1)th order Homfly polynomial. In particular, each dth partial derivative of the Homfly polynomial is a dth order Homfly polynomial. For d = 1, we show that these derivatives span all the first order Homfly polynomials. Similar constructions are made for other link polynomials. Questions on linear span and computational complexities are discussed.

Banach Center Publications, 1995

We describe, in this talk, 1 three methods of constructing different links with the same Jones type invariant. All three can be thought as generalizations of mutation. The first combines the satellite construction with mutation. The second uses the notion of rotant, taken from the graph theory, the third, invented by Jones, transplants into knot theory the idea of the Yang-Baxter equation with the spectral parameter (idea employed by Baxter in the theory of solvable models in statistical mechanics). We extend the Jones result and relate it to Traczyk's work on rotors of links. We also show further applications of the Jones idea, e.g. to 3string links in the solid torus. We stress the fact that ideas coming from various areas of mathematics (and theoretical physics) has been fruitfully used in knot theory, and vice versa.