A simple algorithm to compute link polynomials defined by using skein relations

Xuezhi Zhao

doi:10.1515/cmb-2020-0102

Abstract

Abstract We give a simple and practical algorithm to compute the link polynomials, which are defined according to the skein relations. Our method is based on a new total order on the set of all braid representatives. As by-product a new complete link invariant are obtained.

Highlights

Link polynomials are important topological invariants to distinguish links and knots
It is known that computing the Jones polynomial is generally ♯P - hard [10], and it is expected to require exponential time in the worst case
HOMFLY polynomial P (·), which contains the information of Alexander polynomial, Conway polynomial, Jones polynomial, and etc., could be obtained inductively as follows: (1.1)

Summary

Introduction

Link polynomials are important topological invariants to distinguish links and knots. It is known that computing the Jones polynomial is generally ♯P - hard [10], and it is expected to require exponential time in the worst case. Many link polynomials can be defined by using the so-called skein relation. P (unknot) = 1, l P (L+) + l−1P (L−) + mP (L0) = 0, (skein relation) where L+, L− and L0 are three link diagrams which are different only on a local region, as indicated in the following figures. We shall provide a simple algorithm to calculate link polynomials, if these polynomials are defined by using skein relations. Our reduction is based on a new total order of the set of all braid representatives.

Braid group and an order of braid representatives

An algorithm

Computing remarks