The mechanical modes of a $2$-periodic triangulated surface

TL;DR

This paper proves a conjecture about the dimension of mechanical modes in 2-periodic triangulated surfaces, revealing a palindromic property of an associated determinant and establishing topological stability results.

Contribution

It introduces an indexing method for determinant terms using 3-colorings and proves the conjecture on the dimension of mechanical modes for 2-periodic triangulated surfaces.

Findings

The determinant p_O(z_1,z_2) is palindromic or antipalindromic.

The conjectured dimension 1 phenomenon is confirmed.

Topological stability of modes in 1-periodic nanotubes is established.

Abstract

A recent "hidden symmetry" conjecture of B. Gin-ge Chen et al is resolved, concerning the dimension of the mechanical modes of a generic $2$-periodic triangulated surface $O$ in $R^3$ whose structure graph corresponds to a triangular tiling of $R^2$. We introduce an indexing of the terms in an associated sparse determinant $p_O(z_1, z_2)$ by means of oriented $3$-colourings of the underlying sparse graph, and use this and vertex splitting arguments to show that $p_O(z_1,z_2)$ is palindromic or antipalindromic, up to a shift index. This implies the conjectured dimension $1$ phenomenon for these surfaces. As a corollary we obtain the topological stability of the generic modes of a $1$-periodic triangulated nanotube.