Excision for Spaces of Admissible Skeins

TL;DR

This paper proves that admissible skein modules in any dimension satisfy excision, allowing their computation via boundary data, and explores their relations to cylinder categories and modified traces.

Contribution

It establishes the excision property for admissible skein modules in all dimensions and connects these modules to boundary values, cylinder categories, and modified traces.

Findings

Proved excision property for admissible skein modules in any dimension.

Expressed skein modules as a coend over boundary values.

Related skein modules for different subcategories S and to modified traces.

Abstract

The skein module for a d-dimensional manifold is a vector space spanned by embedded framed graphs decorated by a category A with suitable extra structure depending on the dimension d, modulo local relations which hold inside d-balls. For a full subcategory S of A, an S-admissible skein module is defined analogously, except that local relations for a given ball may only be applied if outside the ball at least one edge is coloured in S. In this paper we prove that admissible skein modules in any dimension satisfy excision, namely that the skein module of a glued manifold is expressed as a coend over boundary values on the boundary components glued together. We furthermore relate skein modules for different choices of S, apply our result to cylinder categories, and recover the relation to modified traces.