HOMFLYPT Skein Theory, String Topology and 2-Categories

Abstract

We show that relations in Homflypt type skein theory of an oriented 3manifold M are induced from a 2-groupoid defined from the fundamental 2-groupoid of a space of singular links in M . The module relations are defined by homomorphisms related to string topology. They appear from a representation of the groupoid into free modules on a set of model objects. The construction on the fundamental 2-groupoid is defined by the singularity stratification and relates Vassiliev and skein theory. Several explicit properties are discussed, and some implications for skein modules are derived.