Stated skein modules of 3-manifolds and TQFT

TL;DR

This paper extends the theory of skein modules for 3-manifolds by introducing stated skein modules, analyzing their behavior under gluing, and establishing their algebraic and topological properties, especially at roots of unity.

Contribution

It introduces the notion of stated skein modules for 3-manifolds with boundary markings and explores their non-injectivity and algebraic structures under gluing operations.

Findings

Non-injectivity of natural maps at roots of unity.

Empty skein is zero in connected sums with marked manifolds at roots of unity.

Stated skein modules form a monoidal functor from decorated cobordisms to algebra categories.

Abstract

We study the behaviour of the Kauffman bracket skein modules of 3-manifolds under gluing along surfaces. For this purpose we extend the notion of Kauffman bracket skein modules to $3$-manifolds with marking consisting of open intervals and circles in the boundary. The new module is called the stated skein module. The first main results concern non-injectivity of certain natural maps defined when forming connected sums along a sphere or along a closed disk. These maps are injective for surfaces, or for generic quantum parameter, but we show that in general they are not injective when the quantum parameter is a root of 1. The result applies to the classical skein modules as well. A particular interesting result is that when the quantum parameter is a root of 1, the empty skein is zero in a connected sum where each constituent manifold has non-empty marking. We also prove various non injectivity results for the Chebyshev-Frobenius map and the natural map induced by the deletion of marked balls. We then consider the general case of gluing along a surface, showing that the stated skein module can be interpreted as a monoidal symmetric functor from a category of "decorated cobordisms" to a Morita category of algebras and their bimodules. We apply this result to deduce several properties of stated skein modules as a Van-Kampen like theorem as well as a computation through Heegaard decompositions and a relation to Hochshild homology for trivial circle bundles over surfaces.