Drinfeld-Manin solutions of the Yang-Baxter equation coming from cube complexes

TL;DR

This paper introduces new solutions to the Yang-Baxter equation derived from higher-dimensional cube complexes, expanding the geometric interpretations beyond traditional braid and knot frameworks.

Contribution

It presents explicit constructions of cube complexes covered by products of trees that yield solutions to the D-state Yang-Baxter equation for arbitrary D.

Findings

Cube complexes can solve higher-dimensional Yang-Baxter equations.

Explicit constructions involve products of trees.

New solutions extend geometric interpretations of the Yang-Baxter equation.

Abstract

The most common geometric interpretation of the Yang-Baxter equation is by braids, knots and relevant Reidemeister moves. So far, cubes were used for connections with the third Reidemeister move only. We will show that there are higher-dimensional cube complexes solving the $D$-state Yang-Baxter equation for arbitrarily large $D$. More precisely, we introduce explicit constructions of cube complexes covered by products of $n$ trees and show that these cube complexes lead to new solutions of the Yang-Baxter equations.