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On the structure of skein modules

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Rhea Palak Bakshi (George Washington)

Abstract: Skein modules were introduced in 1987 by Józef H. Przytycki as generalisations of the various polynomial link invariants in the 3-sphere
to arbitrary 3-manifolds. Over time they have evolved into one of the most important objects in knot theory and quantum topology having strong ties with algebraic and hyperbolic geometry, quantum cluster algebras, and the Witten-Reshetikhin-Turaev 3-manifold invariants and Topological Quantum Field Theories, to name a few. There are many different skein modules since we can take links in the 3-manifolds to be either oriented or unoriented, framed or unframed, or up to homotopy or isotopy. In this talk we will focus on the structure of two different skein modules: the framing skein module and the Kauffman bracket skein module of 3-manifolds:
1. We show that the only way of changing the framing of a link by ambient isotopy in an oriented 3-manifold is when the manifold admits a properly embedded non-separating 2-sphere. This change of framing is given by the Dirac trick. We use this result to completely determine the structure of the framing skein module and show that it detects the presence of non-separating 2-spheres by way of torsion.
2. We disprove a twenty-two-year-old theorem about the structure of the Kauffman bracket skein module of the connected sum of two handlebodies. We achieve this by analysing handle slidings on compressing discs in a handlebody.