Extended reflection equation algebras, the braid group on a handlebody and associated link polynomials

TL;DR

This paper explores the algebraic structures related to braid groups on handlebodies, deriving representations and invariants that generalize link polynomials to 3-manifolds using extended reflection equations and skein relations.

Contribution

It introduces a new algebraic framework connecting braid groups on handlebodies with extended reflection equations, leading to novel link invariants on 3-manifolds.

Findings

Representation of braid groups via R-matrices

Derivation of skein relations from reflection equations

Extension of Jones polynomial to handlebodies

Abstract

The correspondence of the braid group on a handlebody of arbitrary genus to the algebra of Yang-Baxter and extended reflection equation operators is shown. Representations of the infinite dimensional extended reflection equation algebra in terms of direct products of quantum algebra generators are derived, they lead to a representation of this braid group in terms of $R$-matrices. Restriction to the reflection equation operators only gives the coloured braid group. The reflection equation operators, describing the effect of handles attached to a 3-ball, satisfy characteristic equations which give rise to additional skein relations and thereby invariants of links on handlebodies. The origin of the skein relations is explained and they are derived from an adequately adapted handlebody version of the Jones polynomial. Relevance of these results to the construction of link polynomials on closed 3-manifolds via Heegard splitting and surgery is indicated.