HOMFLY Polynomials of Torus Links as Generalized Fibonacci Polynomials

Kemal Taşköprü,İsmet Altıntaş

doi:10.37236/5324

Kemal Taşköprü, İsmet Altıntaş

Open Access

https://doi.org/10.37236/5324

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Abstract

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.

Highlights

An important part of works of the knot theory is about discovering the knot invariants and determining the type of knot by means of the knot invariants.A knot polynomial is a knot invariant whose coefficients encode some of the properties of a given knot
The focus of this paper is to study the HOMFLY polynomial of (2, n)-torus link as a generalized Fibonacci polynomial
We show that the HOMFLY polynomial of (2, n)-torus link can be obtained from its Alexander-Conway polynomial or the classical Fibonacci polynomial

Summary

Introduction

An important part of works of the knot theory is about discovering the knot invariants and determining the type of knot by means of the knot invariants. The Alexander polynomial, ∆L(t), is a Laurent polynomial in the variable t with integer coefficients associated with the link diagram L in an invariant way. The Alexander-Conway polynomial, ∇L(z), is a Laurent polynomial in the variable z with integer coefficients associated with the oriented link diagram L. The Jones polynomial, VL(t), is a Laurent polynomial in the variable t1/2 associated with the oriented link diagram L. The HOMFLY polynomial, PL(a, z), is the electronic journal of combinatorics 22(4) (2015), #P4.8 two variables Laurent polynomial for the oriented link diagram L. PL(a, z) is an ambient isotopy invariant of the link L determined by the following axioms: a−1PL+(a, z) − aPL−(a, z) = zPL0(a, z). We prove identities similar to the identities of Catalan, Cassini and d’Ocagne which are important Fibonacci identities

A generalization of Fibonacci polynomials

Matrix representations