The Yokonuma-Hecke algebras and the HOMFLYPT polynomial

TL;DR

This paper compares the Yokonuma-Hecke algebra-based invariant with the HOMFLYPT polynomial, showing they generally differ except in trivial cases, highlighting differences in their ability to distinguish links.

Contribution

It provides a detailed comparison between the Yokonuma-Hecke algebra invariant and the HOMFLYPT polynomial, clarifying their relationship and differences in link invariants.

Findings

The invariants do not coincide for most links.

They only match in a few trivial cases.

The comparison clarifies the distinct capabilities of each invariant.

Abstract

We compare the invariant for classical knots and links defined using the Juyumaya trace on the Yokonuma-Hecke algebras with the HOMFLYPT polynomial. We show that the two invariants, as maps on the set ${\mathcal L}$ of oriented link types in $S^3$, do not coincide except in a few trivial cases.