Reflection equation and link polynomials for arbitrary genus solid tori

TL;DR

This paper establishes a connection between braid groups on solid tori of any genus and algebraic operators satisfying the Yang-Baxter and reflection equations, proposing a new approach to defining link invariants in three-manifolds.

Contribution

It introduces a representation of the braid group for arbitrary genus solid tori using R-matrices and explores the reflection equation's characteristic as a skein relation for link invariants.

Findings

Representation of braid groups via R-matrices for arbitrary genus

Reflection equation characteristic as skein relation

Potential for defining invariants on three-manifolds

Abstract

The correspondence between the braid group on a solid torus of arbitrary genus and the algebra of Yang-Baxter and reflection equation operators is shown. A representation of this braid group in terms of $R$-matrices is given. The characteristic equation of the reflection equation matrix is considered as an additional skein relation. This could lead to an intrinsic definition of invariant link polynomials on solid tori and, via Heegaard splitting, to invariant link polynomials on arbitrary three-manifolds without boundary.