Skein Modules

Given the question if two knots inside the 3-dimensional Euclidean space are isotopic, a number of polynomial knot invariants have been constructed in order to distinguish them. We will take a closer look at the Kauffman bracket, the Alexander-Conway polynomial, and the HOMFLY polynomial. To generalize these polynomials to knots in other 3-dimensional manifolds, Skein modules were introduced. To each 3-manifold we can associate infinitely many different Skein modules, the vast majority of which have not yet been computed. We will overview current results and show in what sense Skein modules relate to the classical polynomials.