# Skein Modules on closed 3-manifolds

Hi all! Why are skein modules 1-dimensional on closed 3-manifolds? 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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## 1 Answer

The skein module of a closed 3-manifold is not 1-dimensional. That of the 3-torus is infinite-dimensional, 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 1-dimensional 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.

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Thanks! This is precisely what I was looking for. –  CVK Jul 25 '12 at 14:08