Surface embeddings of the Klein and the Möbius–Kantor graphs

Martin Cramer Pedersen; Olaf Delgado-Friedrichs; Stephen T. Hyde

doi:10.1107/S2053273318002036

Surface embeddings of the Klein and the Möbius–Kantor graphs

Martin Cramer Pedersen^*, Olaf Delgado-Friedrichs, Stephen T. Hyde

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

5 Citations (Scopus)

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.

Original language	English
Pages (from-to)	223-232
Number of pages	10
Journal	Acta Crystallographica Section A: Foundations and Advances
Volume	74
Issue number	3
DOIs	https://doi.org/10.1107/S2053273318002036
Publication status	Published - May 2018

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abstract = "This paper describes an invariant representation for finite graphs embedded on orientable tori of arbitrary genus, with working examples of embeddings of the M{\"o}bius–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 M{\"o}bius–Kantor and 7-valent Klein graphs.",

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AU - Delgado-Friedrichs, Olaf

AU - Hyde, Stephen T.

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N2 - 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.

AB - 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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Surface embeddings of the Klein and the Möbius–Kantor graphs

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