Non-trivial links and plats with trivial Gassner matrices

Abstract

LetBndenote the Artin braid group on ‘n-strings[ and PBnits normal subgroup consisting of all the pure braids [Bi, Mo]. These groups have been considerably scrutinized by both topologists and algebraists [BL]. One question whose answer has so far eluded us is whether or not the Gassner representationG: PBn→Mn×n(λ), into the group ofn-by-nmatrices over, is faithful (see Section 1) [Bi; ·3] [Ga]. Recently the less discriminating Burau representation B: PBn→Mn×n(Z[t±1] ) was shown to have a non-trivial kernel for each n ≥ 6 [M, LP] but these techniques have not yet yielded an element of kernel(G). This paper is a partial step in that direction.