Burau Representation and Random Walks Onstring

Using a probabilistic interpretation of the Burau representation of the braid group ooered by Vaughan Jones, we generalize the Burau representation to a representation of the semigroup of string links. This representation is determined by a linear system, and is dominated by nite type string link invariants. For positive string links, the representation matrix can be interpreted as the transition matrix of a Markov process. For positive non-separable links, we show that all states are persistent. 1. Beginning of the story: Burau representation. The Burau matrices i , 1 i n?1, are nn matrices given as follows: Let t 6 = 0; 1 be a complex number. If we think of i as a linear transformation on C n , then i = (1) (1) | {z } i?1 copies 1 ? t t 1 0 ! (1) (1) i?1 copies 0 1 t 1 ? t ! (1) (1) | {z } n?i?1 copies : Here we use t to denote t ?1 for simplicity.