A basis for the Birman-Wenzl algebra

TL;DR

This paper establishes an explicit algebraic isomorphism between the Birman-Wenzl algebra and the Kauffman algebra by constructing a basis and translating geometric arguments into algebraic form.

Contribution

It provides the first explicit basis for the Birman-Wenzl algebra and bridges the algebraic and geometric perspectives through an explicit isomorphism.

Findings

Constructed an explicit basis for the Birman-Wenzl algebra.

Established an algebraic isomorphism with the Kauffman algebra.

Replaced geometric isotopy arguments with algebraic relations.

Abstract

An explicit isomorphism is constructed between the Birman-Wenzl algebra, defined algebraically by J. Birman and H. Wenzl using generators and relations, and the Kauffman algebra, constructed geometrically by H. R. Morton and P. Traczyk in terms of tangles. The isomorphism is obtained by constructing an explicit basis in the Birman-Wenzl algebra, analogous to a basis previously constructed for the Kauffman algebra using 'Brauer connectors'. The geometric isotopy arguments for the Kauffman algebra are systematically replaced by algebraic versions using the Birman-Wenzl relations.