Jones type invariants and the topology of 3-manifolds
by
Uwe Kaiser
Universität Siegen

In 1984 Vaughn Jones discovered a fascinating new polynomial invariant of knots and links in 3-space. The skein module of a compact oriented 3-manifold is a natural generalization of this polynomial. While easily definable from local combinatorial structures, it turns out to be very difficult to determine the structure of the skein module from the global topology of the 3-manifold. For thickened orientable surfaces the skein module has been determined by V. Turaev and J. Przytycki. We will prove the following results: It is possible to consider the skein module as a module over the skein algebra of the 3-ball, which carries those relations to be satisfied in the skein module of each 3-manifold. Now if \pi2(M) = 0 and M is atoroidal then the skein module always contains natural free submodules corresponding to the free homotopy classes of loops in M. On the other hand, if M is homotopically nontrivial in the sense above then very often it is possible to construct torsion. In the case of Lens spaces different from S²×S¹ this allows to prove that the skein module is free. We discuss how the stratified topology of the string space of M, i.e. the space of maps of loops into M with the stratification given by intersections and singularities of loops, encodes the structure of skein modules. Then skein modules appear as a kind of stratified homology of the string space.

Date received: February 28, 2001