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The undecakisicosahedral group and a 3-regular carbon network of genus 26. (English) Zbl 1127.92051
Summary: Three projective special linear groups PSL\((2,p)\), those with \(p = 5\), 7 and 11, can be seen as \(p\)-multiples of tetrahedral, octahedral and icosahedral rotational point groups, respectively. The first two have already found applications in carbon chemistry and physics, as PSL\((2,5) \equiv I\) is the rotation group of the fullerene \(C_{60}\) and dodecahedrane \(C_{20}H_{20}\), and PSL\((2,7)\) is the rotation group of the 56-vertex all-heptagon Klein map, an idealisation of the hypothetical genus-3 “plumber’s nightmare” allotrope of carbon. We present an analysis of PSL\((2,11)\) as the rotation group of a 220-vertex, all 11-gon, 3-regular map, which provides the basis for a more exotic hypothetical \(sp ^{2}\) framework of genus 26. The group structure and character table of PSL\((2,11)\) are developed in chemical notation and a three-dimensional (3D) geometrical realisation of the 220-vertex map is derived in terms of a punctured polyhedron model where each of 12 pentagons of the truncated icosahedron is connected by a tunnel to an interior void and the 20 hexagons are connected tetrahedrally in sets of 4.

MSC:
92E10 Molecular structure (graph-theoretic methods, methods of differential topology, etc.)
05C10 Planar graphs; geometric and topological aspects of graph theory
20B25 Finite automorphism groups of algebraic, geometric, or combinatorial structures
20G99 Linear algebraic groups and related topics
57M20 Two-dimensional complexes (manifolds) (MSC2010)
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