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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)
##### Keywords:
PSL(2,11); topology; carbon allotrope
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##### References:
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