Two-color solutions of set-theoretic tetrahedron equation and their cohomologies

TL;DR

This paper classifies all two-color solutions to the set-theoretic tetrahedron equation, analyzes their properties, and computes their 3-cohomologies, with implications for 3D statistical physics models.

Contribution

It provides a complete enumeration of 406 two-color solutions to the set-theoretic tetrahedron equation and analyzes their cohomological properties.

Findings

406 solutions identified, most are degenerate

3-cohomologies computed for the solutions

discusses relevance to 3D statistical physics

Abstract

All solutions of the set-theoretic constant tetrahedron equation with two colors are found, and some of their properties are analyzed. The list includes 406 solutions - we call them R-operators, - most of which are degenerate (non-bijective). Then, we calculate the 3-cohomologies for our R-operators, and discuss the applicability of our results to 3-dimensional statistical physics.