Affiliation:
1. EPFL Valais, Rue de l’Industrie 17, 1951 Sion, Switzerland
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].
Funder
Imperial College London
Deutscher Akademischer Austauschdienst
Evangelisches Studienwerk Villigst
Publisher
World Scientific Pub Co Pte Lt
Subject
Algebra and Number Theory
Cited by
2 articles.
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1. Symmetric Tangling of Honeycomb Networks;Symmetry;2022-08-31
2. Symmetric tangled Platonic polyhedra;Proceedings of the National Academy of Sciences;2022-01-04