THE MULTI-VARIABLE ALEXANDER POLYNOMIAL OF LENS BRAIDS

Abstract

Let n ≥ 2 be an integer and ℬn the n-braid group. A braid β ∈ ℬn is said to be a (p,s)-lens braid if there exists α ∈ ℬn such that β = αp(σ1 σ2 … σn-1)ns. In this paper we use the multi-variable Alexander polynomial to find a necessary condition for a braid to be a (p,s)-lens braid, for p prime. Our main tool here is the multi-variable Burau representation of the n-braid group.