HOMFLYPT Skein Theory, String Topology and 2-Categories

Abstract: We show that relations in Homflypt type skein theory of an oriented $3$-manifold $M$ 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.