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)?



Skein Modules on closed 3-manifolds
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](http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=TI&pg6=PC&pg7=ALLF&pg8=ET&review_format=html&s4=Przytycki&s5=skein%20module&s6=&s7=&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq&r=17&mx-pid=1194712) 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](http://www.math.buffalo.edu/~asikora/Papers/skein.ps).


