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

Let k be an integral domain containing the invertible elements α, s and 1s−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 M1 and M2 and prove that K(M1 # M2) is isomorphic to K(M1)⊗K(M2) over a certain localized ring, where M1 # M2 is the connected sum of two manifolds M1 and M2.