A matrix solution to any polygon equation

TL;DR

This paper introduces matrix constructions linked to Pachner moves for all polygon sizes, proving they satisfy the polygon equation universally, thus advancing algebraic understanding of topological transformations.

Contribution

It provides a novel matrix-based framework for Pachner moves applicable to any polygon size, establishing their universal satisfaction of the polygon equation.

Findings

Matrices satisfy the n-gon equation for all n

Constructs are rational functions of formal variables

Applicable to both odd and even n cases

Abstract

In this article, we construct matrices associated to Pachner $\frac{n-1}{2}$-$\frac{n-1}{2}$ moves for odd $n$ and matrices associated to Pachner $(\frac{n}{2}-1)$-$\frac{n}{2}$ moves for even $n$. The entries of these matrices are rational functions of formal variables in a field. We prove that these matrices satisfy the $n$-gon equation for any $n$.