an:05230002
Zbl 1127.92051
Lijnen, Erwin; Ceulemans, Arnout; Fowler, Patrick W.; Deza, Michel
The undecakisicosahedral group and a 3-regular carbon network of genus 26
EN
J. Math. Chem. 42, No. 3, 617-644 (2007).
00212421
2007
j
92E10 05C10 20B25 20G99 57M20
PSL(2,11); topology; carbon allotrope
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.