Excision of Skein Categories and Factorisation Homology

TL;DR

This paper proves that skein categories satisfy excision, establishing their role as $k$-linear factorisation homology, and shows how their free cocompletion relates to locally finitely presentable factorisation homology, with applications to quantum groups.

Contribution

It demonstrates that skein categories satisfy excision and are equivalent to $k$-linear factorisation homology, connecting skein algebras to quantisations of character varieties.

Findings

Skein categories satisfy excision property.

Skein categories are $k$-linear factorisation homology.

Cocompletion of skein categories recovers locally finitely presentable factorisation homology.

Abstract

We prove that the skein categories of Walker--Johnson-Freyd satisfy excision. This allows us to conclude that skein categories are $k$-linear factorisation homology and taking the free cocompletion of skein categories recovers locally finitely presentable factorisation homology. An application of this is that the skein algebra of a punctured surface related to any quantum group with generic parameter gives a quantisation of the associated character variety.