Excision of Skein Categories and 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. † The Kauffman bracket skein algebra is the case G = SL 2 .

Relating stated skein algebras and internal skein algebras

“…The above functor RT describes a way to obtain invariants of coloured tangles, and actually coloured ribbon graphs, from a ribbon category V. We want to study this invariant, and say that two ribbon graphs are identified if they give the same invariant, namely the same morphism in V after evaluation under RT. Skein categories, studied in [Coo19], generalize this construction for coloured ribbon graphs on a surface. The idea is to take the relations between ribbon graph that are true locally, on an embedded cube.…”

“…However, such an object does not always exist in Sk V (R 2 ), and actually lives in its cocompletion. The internal skein algebra A Σ of the surface is the internal endomorphism algebra of the empty set Hom(∅, ∅), see [GJS19] or [BBJ18] together with [Coo19]. This means one can understand ribbon graphs in Sk V (Σ) with boundary points on the bottom and near the boundary edge as morphisms in (the Free cocompletion of) Sk V (R 2 ) V with target A Σ .…”

“…The goal of this paper is to link stated skein algebras of [CL19] and internal skein algebras of [GJS19] and [BBJ18] in a way that would be understandable by researchers from both fields. For this purpose we adopt the point of view of [GJS19] using the results of [Coo19] to compute factorisation homology used in [BBJ18] by skein categories, so we do not use factorisation homology.…”

“…Skein categories, introduced in [Wal06] and [J-F19] and studied in [Coo19], are a generalisation of the Reshetekhin-Turaev construction to surfaces, as well as a categorical extension of skein algebras. Given a k-linear ribbon category V, in [RT90] (see also [Tur10]) one constructs a functor RT from the category Tan f r…”

The $\mathfrak{gl}_2$-Skein Module of Lens Spaces via the Torus and Solid Torus

“…It may also be possible to view the gl 2 -skein module in the framework of factorization homology following [BBJ18]. This was explicitly done for the Kauffman bracket skein module in [Coo19].…”