Combinatorial Sutured TQFT as Exterior Algebra

TL;DR

This paper constructs an elementary sutured TQFT using exterior algebras of homology groups, demonstrating its equivalence to Honda et al.'s theory and applying it to contact structures on solid tori.

Contribution

It introduces a new, simple sutured TQFT model based on exterior algebra, showing its equivalence to existing theories and enabling new insights into contact structures.

Findings

The exterior algebra-based sutured TQFT matches Honda et al.'s theory with Z2 coefficients.

The ring structure of the exterior algebra provides tools for analyzing tight contact structures.

Application to solid tori illustrates the practical utility of the new TQFT model.

Abstract

The idea of a sutured topological quantum field theory was introduced by Honda, Kazez and Mati\'c (2008). A sutured TQFT associates a group to each sutured surface and an element of this group to each dividing set on this surface. The notion was originally introduced to talk about contact invariants in Sutured Floer Homology. We provide an elementary example of a sutured TQFT, which comes from taking exterior algebras of certain singular homology groups. We show that this sutured TQFT coincides with that of Honda et al. using $\Z_2$-coefficients. The groups in our theory, being exterior algebras, naturally come with the structure of a ring with unit. We give an application of this ring structure to understanding tight contact structures on solid tori.