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

Stephen C Power

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

Combinatorics 2018-02-06 v2 Mathematical Physics math.MP

Authors: Stephen C Power

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.

Keywords

mechanical modes triangulated surface hidden symmetry sparse determinant topological stability

Cite

@article{arxiv.1609.01275,
  title  = {The mechanical modes of a $2$-periodic triangulated surface},
  author = {Stephen C Power},
  journal= {arXiv preprint arXiv:1609.01275},
  year   = {2018}
}

Comments

There is a serious error 3 lines below "Step (II)". It is not true that "It follows that the set C' of all oriented colourings for ... is equal to the set of extension colourings.. ." (In fact they only account for half of the extension colourings.)

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

Abstract

Keywords

Cite

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