Remarks about the Moebius-Kantor graph

2026-05-28

AI Summary

This study systematically investigates the multifaceted roles of the Möbius–Kantor graph in group theory, topology, and geometry. Employing tools from algebraic graph theory, topological graph theory, group actions, Lefschetz fixed-point theory, and metric geometry, the work establishes the graph’s uniqueness as the Cayley graph of three non-abelian groups—most notably the Pauli group—and demonstrates how its algebraic properties arise naturally from its metric structure. Key contributions include confirming its Heawood embedding on the torus, realizing it as the dual 2-skeleton of a triangulation of the 3-sphere, revealing its manifestation of the Clifford torus within the Hopf fibration, and verifying the Brouwer–Lefschetz fixed-point theorem via Lefschetz numbers. The research underscores the graph’s central role in bridging topology, geometry, and group-theoretic structures relevant to quantum information.

Abstract

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.

Problem

• Moebius-Kantor graph
• Cayley graph
• Pauli group
• topological graph theory
• metric structure

Innovation

• Moebius-Kantor graph
• Pauli group
• metric-preserving group structure
• Cayley graph
• Hopf fibration