Kauffman bracket skein module of the connected sum of handlebodies: a counterexample (Q2114210)

The paper concerns a theorem formulated 22 years ago by one of the authors (Józef H. Przytycki) and predicting the form of Kauffman bracket skein modules for connected sums of handlebodies. The important ingredient for such considerations are the Theorem 1.1 proved in his paper about the Kauffman bracket skein module of a connected sum of compact oriented 3-manifolds [\textit{J. H. Przytycki}, Manuscr. Math. 101, No. 2, 199--207 (2000; Zbl 0947.57018)], which is the tensor product of the corresponding Kauffman bracket skein modules of the component manifolds. Then, since a handlebody of genus \(n\) can be considered as a cylinder over a sphere with \(n+1\) holes, it seems quite natural to predict, as Przytycki did, the form of the Kauffman bracket skein module of the connected sum of handlebodies to be isomorphic to the quotient of the skein module for a handlebody of the genus being a sum of genuses of the component handlebodies by certain relations. However, as this paper proves, in the case of \(n\ge2\), \(m\ge1\) the submodule by which division is made is larger, which is in detail shown in the paper. There is also formulated a conjecture about the form of the Kauffman bracket skein module in the case of the connected sum of two \(H_1\)'s, which, as claim the authors, has been later proved in another paper by the present authors together with \textit{T. T. Q. Lê}.