Kauffman Bracket Skein Module of the Connected Sum of Handlebodies and Non-injectivity

Abstract

For the handlebody Hg of genus g, Przytycki studied the (Kauffman bracket) skein module ∫q(Hn#Hm) of the connected sum Hn#Hm at q. One of his results is that, in the case when 1 - qk is invertible for any k ≠ 0, a homomorphismφ :∫ q(Hn ⊔ Hm) →∫ q(Hn#Hm) is an isomorphism, which is induced by a natural way. In this paper, in the case when n = m = 1, the ground ring is C, and q ∈ C is a 4k-th root of unity (k ≥ 2), we show that φis not injective.