Spin Structure and the Framing Skein Module of Links in 3-Manifolds
摘要
In this lecture, we show that the only way of changing the framing of a knot or a link by ambient isotopy in an oriented 3-manifold is when the manifold has a properly embedded non-separating sphere. This change of framing is given by the Dirac trick, also known as the light bulb trick. We begin with background information on non-separating spheres, Dehn homeomorphisms, and mapping class groups of 3-manifolds. The main tools used in the proof are based on Darryl McCullough’s work on the mapping class groups of 3-manifolds and spin structures. We then relate these results to the theory of skein modules.