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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.