Abstract

The Jones polynomial and its cousins are invariants of knots and links in the 3-sphere, which are determined by local so-called skein relations. This allows a simple definition of an invariant of oriented 3-manifolds M: the space of all framed links in M modulo the skein relations. Of particular interest are these invariants for thickened surfaces, in which case they carry an algebra structure and act on the invariants of 3-manifolds co-bounding the surface. They are also related to character varieties, quantum Teichmueller spaces and feature in several important conjectures in quantum topology. After surveying this area, I will talk about positive bases for skein algebras that were found by D. Thurston, and how they might be related to Khovanov’s categorification of the Jones polynomial and its desired extension to a 4-dimensional TQFT .