A matrix solution to any polygon equation

TL;DR

"This work develops explicit matrix realizations of the $n$-gon polygon equation arising from Pachner moves on triangulated $n$-gons. The authors extend the existing odd$-$n construction to even $n$ by forming Vandermonde$-$based matrices and augmenting them with fixed simplex data, then prove that the resulting extended matrices satisfy the $n$-gon equation for all $n$. The approach hinges on attaching vectors to $(n-3)$-simplices, proving their linear independence, and showing that Pachner moves act compatibly on these vectors, which yields equality of the two sides of the equation. Concrete examples for the pentagon, hexagon, and heptagon illustrate the method, and the work connects to Grassmannian$/$Pl"ucker coordinates with potential implications for invariants of triangulated manifolds via simplicial cocycles."

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 $n$.

Figures (4)

• Figure 1: Pachner 2-3 move.
• Figure 2: The $n$-gon equation involves two sequences of Pachner moves from the initial triangulation to the final triangulation of the $n$-gon. Each arrow with label $q$ denotes the Pachner move which does not involve the vertex $q$.
• Figure 3: Triangulation graph for the pentagon.
• Figure 4: Triangulation graph for the shifted octahedron.

Theorems & Definitions (15)

• Theorem 3.1
• Lemma 3.3
• proof
• Lemma 4.1
• proof
• Proposition 4.2
• proof
• Proposition 4.3
• proof
• Proposition 4.4