Homotopy and q-homotopy skein modules of 3-manifolds: an example in Algebra Situs

Algebra Situs is a branch of mathematics which has its roots in Jones' construction of his polynomial invariant of links and Drinfeld's work on quantum groups. It encompasses the theory of quantum invariants of knots and 3-manifolds, algebraic topology based on knots, operads, planar algebras, q-deformations, quantum groups, and overlaps with algebraic geometry, non-commutative geometry and statistical mechanics. Algebraic topology based on knots may be characterized as a study of properties of manifolds by considering links (submanifolds) in a manifold and their algebraic structure. The main objects of the discipline are skein modules, which are quotients of free modules over ambient isotopy classes of links in a manifold by properly chosen local (skein) relations. We concentrate, in this lecture, on one relatively simple example of a skein module of 3-manifolds -- the q-homotopy skein module. This skein module already has many ingredients of the theory: algebra structure, associated Lie algebra, quantization, state models...