On chirality of toroidal embeddings of polyhedral graphs

TL;DR

This paper proves that nontrivial embeddings of simple 3-connected planar graphs on the torus are chiral, using properties of torus knots, links, and minors, extending previous results with new proofs and generalizations.

Contribution

It introduces an alternative proof for chirality of graph embeddings using minors and generalizes the chirality of Hopf ladders with multiple rungs.

Findings

Nontrivial embeddings contain chiral knots or links.

Embeddings with nontrivial knots are proven to be chiral.

Hopf ladders with three or more rungs are chiral.

Abstract

We investigate properties of spatial graphs on the standard torus. It is known that nontrivial embeddings of planar graphs in the torus contain a nontrivial knot or a nonsplit link due to [1],[2]. Building on this and using the chirality of torus knots and links [3],[4], we prove that nontrivial embeddings of simple 3-connected planar graphs in the standard torus are chiral. For the case that the spatial graph contains a nontrivial knot, the statement was shown by Castle [5]. We give an alternative proof using minors instead of the Euler characteristic. To prove the case in which the graph embedding contains a nonsplit link, we show the chirality of Hopf ladders with at least three rungs, thus generalising a theorem of Simon [6].