A cubic defining algebra for the Links-Gould polynomial

TL;DR

This paper introduces a new cubic algebra quotient of the braid group algebra that supports a Markov trace, enabling the computation of the Links-Gould knot invariant and exploring its properties and conjectures.

Contribution

It defines a novel cubic quotient algebra with a Markov trace that produces the Links-Gould invariant, advancing algebraic understanding of knot invariants.

Findings

Defined a finite-dimensional cubic quotient algebra of the braid group

Established a Markov trace on this algebra for knot invariants

Explored properties and proposed conjectures about the algebra's structure

Abstract

We define a finite-dimensional cubic quotient of the group algebra of the braid group, endowed with a (essentially unique) Markov trace which affords the Links-Grould invariant of knots and links. We investigate several of its properties, and state several conjectures about its structure.