Title: Homotopy and q-homotopy skein modules of 3-manifolds:
an example in Algebra Situs (to appear in the Proceedings of the Conference in Low-Dimensional Topology in Honor of Joan Birman's 70th Birthday, Columbia University/Barnard College, New York, March 14--15, 1998).

Author:  Jozef H. Przytycki

Preprint: GWUM-1999-13 

Abstract: Algebra Situs is a branch of mathematics which has its roots in Jones' construction of his polynomial invariant of links, Jones polynomial, 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 the properties of manifolds by considering the space of links (submanifolds) in a manifold and its 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.
Of course, this is not a complete definition of the field, which has its purely algebraic component (skein algebras of groups), higher manifold generalization and rich internal structure, but at least it gives the idea of our subject.

We concentrate, here, 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 rich theory: algebra structure, associated Lie algebra, Hopf algebra,
quantization, state models, relation to graph theory. For example, we obtain a Lie algebra being a deformation of
the Heisenberg Lie algebra.