Identifying the invariants for classical knots and links from the Yokonuma-Hecke algebras

TL;DR

This paper introduces a new family of 2-variable polynomial invariants for classical links derived from Yokonuma-Hecke algebras, extending the Homflypt polynomial and providing stronger link invariants.

Contribution

It establishes the existence of new invariants from Yokonuma-Hecke algebras, relates them to the Homflypt polynomial, and generalizes to a 3-variable invariant with a closed formula.

Findings

Invariants are topologically equivalent to Homflypt on knots.

New invariants distinguish certain links that Homflypt cannot.

A closed formula for the 3-variable invariant is provided.

Abstract

In this paper we announce the existence of a family of new $2$-variable polynomial invariants for oriented classical links defined via a Markov trace on the Yokonuma-Hecke algebra of type $A$. Yokonuma-Hecke algebras are generalizations of Iwahori-Hecke algebras, and this family contains the Homflypt polynomial, the famous $2$-variable invariant for classical links arising from the Iwahori-Hecke algebra of type $A$. We show that these invariants are topologically equivalent to the Homflypt polynomial on knots, but not on links, by providing pairs of Homflypt-equivalent links that are distinguished by our invariants. In order to do this, we prove that our invariants can be defined diagrammatically via a special skein relation involving only crossings between different components. We further generalize this family of invariants to a new $3$-variable skein link invariant which is stronger than the Homflypt polynomial. Finally, we present a closed formula for this invariant, by W.B.R. Lickorish, which uses Homflypt polynomials of sublinks and linking numbers of a given oriented link.