Research ArticlesNo Access

On chirality of toroidal embeddings of polyhedral graphs

Senja Barthel

EPFL Valais, Rue de l’Industrie 17, 1951 Sion, Switzerland

https://doi.org/10.1142/S021821651750050XCited by:2 (Source: Crossref)

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 [2, 3]. Building on this and using the chirality of torus knots and links [9, 10], we prove that the 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 et al. [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 generalizing a theorem of Simon [12].

Keywords:

AMSC: 57M25, 05C10, 92E10

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