Kauffman bracket skein module of the connected sum of handlebodies and non-injectivity

Abstract

For the handlebody [Formula: see text] of genus [Formula: see text], Przytycki studied the (Kauffman bracket) skein module [Formula: see text] of the connected sum [Formula: see text] at [Formula: see text]. One of his results is that, in the case when [Formula: see text] is invertible for any [Formula: see text], a homomorphism [Formula: see text] is an isomorphism, which is induced by a natural way. In this paper, in the case when [Formula: see text], the ground ring is [Formula: see text], and [Formula: see text] is a [Formula: see text]-th root of unity ([Formula: see text]), we show that [Formula: see text] is not injective.