Hi all! Why are skein modules 1dimensional on closed 3manifolds? The result seems clear on closed manifolds with vanishing first Betti number (e.g. $S^3$), but I don't see how to prove it for, say, $\mathbb{T}^3$. Can anyone point me to an old reference (I'm rather new to this field)?
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The skein module of a closed 3manifold is not 1dimensional. That of the 3torus is infinitedimensional, a basis is given by multicurves on the torus as proved by Przytycki in 1991. Maybe you mean the reduced skein module, defined by Roberts when $A$ is a primitive $4r^{\rm th}$ root of unity: in that case it depends only on $\partial M$ and is hence 1dimensional when $\partial M = \emptyset$. The original paper of Roberts is unfortunately not available (as far as I can see) but you can find a proof in this paper of Sikora. 

