THE MULTI-VARIABLE ALEXANDER POLYNOMIAL OF LENS BRAIDS

Let n ≥ 2 be an integer and ℬ_n the n-braid group. A braid β ∈ ℬ_n is said to be a (p,s)-lens braid if there exists α ∈ ℬ_n such that β = α^p(σ₁ σ₂ … σ_n-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.