The Kauffman skein module of the connected sum of 3-manifolds

Abstract

Let k be an integral domain containing the invertible elements \alpha, s and \frac{1}{s-s^{-1}}. If M is an oriented 3-manifold, let K(M) denote the Kauffman skein module of M over k. Based on the work on Birman-Murakami-Wenzl algebra by Beliakova and Blanchet, we give an ``idempotent-like'' basis for the Kauffman skein module of handlebodies. Gilmer and Zhong have studied the Homflypt skein modules of a connected sum of two 3-manifolds, here we study the case for the Kauffman skein module and show that K(M_1 # M_2) is isomorphic to K(M_1) tensor K(M_2) over a certain localized ring, where M_1 # M_2 is the connected sum of two manifolds M_1 and M_2.