Affine Birman-wenzl-murakami Algebras and Tangles in the Solid Torus

This paper is the first of a series investigating the affine Birman-Wenzl-Murakami (BMW) algebras and their cyclotomic quotients. The purpose of this paper is to establish an isomorphism between the affine BMW algebras (defined by generators and relations) and the algebras of (n, n)–tangles in the solid torus, modulo Kauffman skein relations. The Birman-Wenzl-Murakami algebras were conceived in [1] and [10] as an algebraic framework for the Kauffman link invariant of [7]. The definition of these algebras by generators and relations was motivated by certain canonical elements, and relations satisfied by these elements, in Kauffman tangle algebras in D2 × I – that is, algebras of framed (n, n)–tangles in D2 × I, modulo Kauffman skein relations. It is therefore not surprising, and is sometimes taken as evident, that the BMW algebras are isomorphic to the Kauffman tangle algebras. A proof of the isomorphism was given by Morton and Wassermann [9] in a paper from 1989 that unfortunately was never published. (A related result is given by Kauffman [7], Theorem 4.4. See Remark 3.2.) Some aspects of the BMW algebras can be understood more clearly in the Kauffman tangle picture; for example, the existence of the Markov trace, and conditional expectations is evident from