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Surface embeddings of the Klein and the Möbius-Kantor graphs
Martin Cramer Pedersen1, Olaf Delgado-Friedrichs1, Stephen T Hyde1
1Department of Applied Mathematics, Research School of Physics and Engineering, Australian National University, Canberra ACT 2601, Australia.
Abstract:
This paper describes an invariant representation for finite graphs embedded on orientable tori of arbitrary genus, with working examples of embeddings of the Möbius-Kantor graph on the torus, the genus-2 bitorus and the genus-3 tritorus, as well as the two-dimensional, 7-valent Klein graph on the tritorus (and its dual: the 3-valent Klein graph). The genus-2 and -3 embeddings describe quotient graphs of 2- and 3-periodic reticulations of hyperbolic surfaces. This invariant is used to identify infinite nets related to the Möbius-Kantor and 7-valent Klein graphs.
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