A polynomial invariant for knots via von Neumann algebras

Abstract

Thus, the trivial link with n components is represented by the pair (l ,n), and the unknot is represented by (si$2 * * • s n i , n) for any n, where si, $2, • • • > sn_i are the usual generators for Bn. The second example shows that the correspondence of (b, n) with b is many-to-one, and a theorem of A. Markov [15] answers, in theory, the question of when two braids represent the same link. A Markov move of type 1 is the replacement of (6, n) by (gbg~, n) for any element g in Bn, and a Markov move of type 2 is the replacement of (6, n) by (6s J 1 , n-hl). Markov's theorem asserts that (6, n) and (c, ra) represent the same closed braid (up to link isotopy) if and only if they are equivalent for the equivalence relation generated by Markov moves of types 1 and 2 on the disjoint union of the braid groups. Unforunately, although the conjugacy problem has been solved by F. Garside [8] within each braid group, there is no known algorithm to decide when (6, n) and (c, m) are equivalent. For a proof of Markov's theorem see J. Birman's book [4]. The difficulty of applying Markov's theorem has made it difficult to use braids to study links. The main evidence that they might be useful was the existence of a representation of dimension n — 1 of Bn discovered by W. Burau in [5]. The representation has a parameter t, and it turns out that the determinant of 1-(Burau matrix) gives the Alexander polynomial of the closed braid. Even so, the Alexander polynomial occurs with a normalization which seemed difficult to predict.