A construction of surface skein TQFTs and their extension to 4-dimensional 2-handlebodies

TL;DR

This paper constructs a new surface skein TQFT based on Frobenius algebras, extending it partially to 4-dimensional 2-handlebodies, and provides computational methods for these invariants.

Contribution

It introduces a novel surface skein TQFT for 3-manifolds and extends it to 4-dimensional 2-handlebodies using an inductive state-sum approach.

Findings

Surface skein modules carry boundary actions enabling gluing.

Partial extension of the TQFT to 4D 2-handlebodies achieved.

Examples demonstrate computation of 4D 2-handlebody invariants.

Abstract

For a commutative Frobenius algebra $A$, we construct a $(2,3,3+\varepsilon)$-dimensional TQFT $\mathsf{AFK}_A$ that assigns to a 3-manifold a skein module of embedded $A$-decorated surfaces. These surface skein modules have been first defined by Asaeda--Frohman and Kaiser using skein relations that generalize the combinatorics of Bar-Natan's dotted cobordisms. For 3-manifolds with boundary, we show that surface skein modules carry an action by a certain surface skein category associated with the boundary, which yields a gluing formalism. Our main result concerns a partial extension of $\mathsf{AFK}_A$ to dimension 4, which uses an inductive state-sum construction following Walker. As an example, the equivariant version of Lee's deformation of dotted cobordisms yields a TQFT that extends to 4-dimensional 2-handlebodies but not 3-handlebodies. Finally, we characterize the attachment of 4-dimensional 2-handles by means of a certain Kirby color and use it to compute the invariants of 4-dimensional 2-handlebodies in examples.