On the link invariants from the Yokonuma-Hecke algebras

TL;DR

This paper investigates properties of traces on Yokonuma-Hecke algebras and introduces link invariants, finding they match the Homflypt polynomial for knots but differ for links.

Contribution

It defines new link invariants from Yokonuma-Hecke algebras and compares them to the Homflypt polynomial, revealing their equivalence on knots but divergence on links.

Findings

Invariants are topologically equivalent to the Homflypt polynomial on knots.

Invariants behave differently from Homflypt polynomial on links.

Developed computational tools to evaluate invariants on knot pairs.

Abstract

In this paper we study properties of the Markov trace ${\rm tr}_d$ and the specialized trace ${\rm tr}_{d,D}$ on the Yokonuma-Hecke algebras, such as behaviour under inversion of a word, connected sums and mirror imaging. We then define invariants for framed, classical and singular links through the trace ${\rm tr}_{d,D}$ and also invariants for transverse links through the trace ${\rm tr}_d$. In order to compare the invariants for classical links with the Homflypt polynomial we develop computer programs and we evaluate them on several Homflypt-equivalent pairs of knots and links. Our computations lead to the result that these invariants are topologically equivalent to the Homflypt polynomial on knots. However, they do not demonstrate the same behaviour on links.