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

Abstract

Let k be an integral domain containing the invertible elements α, s and 1 s−s −1 . Let M be a compact 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 [Math. Ann. 321 (2001) 347], we give an “idempotent-like” basis for the Kauffman skein module of handlebodies. We study the Kauffman skein module of a connected sum of two 3-manifolds M 1 and M 2 and prove that K(M 1 # M 2) is isomorphic to K( M 1)⊗ 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.