For an oriented connected \(3\)-manifold \(M\), we will define a skein
module generated by isotopy classes of links in connected sums of
\(M\) with \(j\) copies of \(S^1\times S^2\) for \(j\geq 0\), with skein
relations using Dehn surgery. This construction is motivated by
searching for new viewpoints on HOMFLYPT skein modules and is
strongly related with the Hoste-Przytycki homotopy skein modules.
In this setting we will discuss in detail the homotopy skein modules
of \(S^1\times S^2\) and of the connected sum operation. There are
also generalizations possible based on different Dehn surgeries.
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