Cohomology of the tetrahedral complex and quasi-invariants of 2-knots

TL;DR

This paper investigates a cohomological model on 6-valent graphs related to 2-knot diagrams, revealing invariance under Roseman moves and suggesting links to topological quantum field theories in four dimensions.

Contribution

It introduces a novel cohomological approach to 2-knot invariants using tetrahedral complexes, highlighting integrability and potential connections to 4D TQFTs.

Findings

Model invariant under Roseman moves for 2-knot diagrams

Identifies integrability of the model on 3D lattices

Proposes a connection to 4D topological quantum field theories

Abstract

This paper explores a particular statistical model on 6-valent graphs with special properties which turns out to be invariant with respect to certain Roseman moves if the graph is the singular point graph of a diagram of a 2-knot. The approach uses the technic of the tetrahedral complex cohomology. We emphasize that this model considered on regular 3d-lattices appears to be integrable. We also set out some ideas about the possible connection of this construction with the area of topological quantum field theories in dimension 4.