Remarks about the Moebius-Kantor graph
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866917547115282432 |
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| author | Knill, Oliver |
| author_facet | Knill, Oliver |
| contents | The Moebius-Kantor graph MK=G(8,3) is a Cayley graph of three non-abelian groups, the Pauli group P(1), the semi-dihedral group SD(16), as well as the dihedral group D(16) of order 16. In topological graph theory, it illustrates the Heawood number 7 of the torus and leads to the Tucker group Aut(MK), the unique group of genus 2. We compute the Lefschetz numbers to illustrate the Brouwer-Lefschetz fixed point theorem. MK is also the dual of the 2-skeleton complex of the 3-sphere G. The graph represents one of flat Clifford tori of a Hopf fibration in the 3-sphere G=K(2,2,2,2) reflecting that Coxeter saw that MK is a subgraph of the tesseract G*. It carries a metric d so that (MK,d) has only one algebraic group structure (P(1),*) that preserves the metric. It makes the Pauli group natural, similarly as the Moebius ladder M(16) makes the dihedral group D(16) natural, forcing the algebraic structure from the metric structure. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_30799 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Remarks about the Moebius-Kantor graph Knill, Oliver Geometric Topology Discrete Mathematics Combinatorics 68R10 05Cxx The Moebius-Kantor graph MK=G(8,3) is a Cayley graph of three non-abelian groups, the Pauli group P(1), the semi-dihedral group SD(16), as well as the dihedral group D(16) of order 16. In topological graph theory, it illustrates the Heawood number 7 of the torus and leads to the Tucker group Aut(MK), the unique group of genus 2. We compute the Lefschetz numbers to illustrate the Brouwer-Lefschetz fixed point theorem. MK is also the dual of the 2-skeleton complex of the 3-sphere G. The graph represents one of flat Clifford tori of a Hopf fibration in the 3-sphere G=K(2,2,2,2) reflecting that Coxeter saw that MK is a subgraph of the tesseract G*. It carries a metric d so that (MK,d) has only one algebraic group structure (P(1),*) that preserves the metric. It makes the Pauli group natural, similarly as the Moebius ladder M(16) makes the dihedral group D(16) natural, forcing the algebraic structure from the metric structure. |
| title | Remarks about the Moebius-Kantor graph |
| topic | Geometric Topology Discrete Mathematics Combinatorics 68R10 05Cxx |
| url | https://arxiv.org/abs/2605.30799 |