A generalized Burau representation for string links

Abstract

A 2-variable matrix B GL n (Z[u 1 , v 1 ]) is defined for any n-string link, generalizing the Burau matrix of an nbraid. The specialization u = 1, v = t -1 recovers the generalized Burau matrix recently defined by X. S. Lin, F. Tian and Z. Wang using probabilistic methods. The specialization u = t -1 , v = 1 results in a matrix with a natural algebraic interpretation, and one that yields homological information about the complement of the closed string link.