Invariants for links from classical and affine Yokonuma--Hecke algebras

TL;DR

This paper constructs a new family of link invariants using affine Yokonuma--Hecke algebras, relating them to affine Hecke algebras, and simplifies the study of these invariants based on link components.

Contribution

It introduces a novel method to generate link invariants from affine Yokonuma--Hecke algebras via an isomorphism with affine Hecke algebras, reducing the complexity of their analysis.

Findings

Constructed a large class of 3-variable polynomial invariants for links.

Reduced the number of invariants needed to describe links based on their components.

Established a connection between invariants for classical and affine Yokonuma--Hecke algebras.

Abstract

We present a construction of invariants for links using an isomorphism theorem for affine Yokonuma--Hecke algebras. The isomorphism relates affine Yokonuma--Hecke algebras with usual affine Hecke algebras. We use it to construct a large class of Markov traces on affine Yokonuma--Hecke algebras, and in turn, to produce invariants for links in the solid torus. By restriction, this construction contains the construction of invariants for classical links from classical Yokonuma--Hecke algebras. In general, the obtained invariants form an infinite family of 3-variables polynomials. As a consequence of the construction via the isomorphism, we reduce the number of invariants to study, given the number of connected components of a link. In particular, if the link is a classical link with N components, we show that N invariants generate the whole family.