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

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.

14 pages, 2 figures. arXiv admin note: substantial text overlap with arXiv:1710.10306, arXiv:1304.5549