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Journal of Mathematical Chemistry

, Volume 42, Issue 3, pp 617–644 | Cite as

The Undecakisicosahedral Group and a 3-regular Carbon Network of Genus 26

  • Erwin Lijnen
  • Arnout Ceulemans
  • Patrick W. Fowler
  • Michel Deza
Article

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) ≡ I is the rotation group of the fullerene C60 and dodecahedrane C20H20, 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. Here, 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.

Keywords

PSL(211) group theory undecakisicosahedral group topology carbon allotrope 

AMS

05C10 20B25 57M20 

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Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • Erwin Lijnen
    • 1
  • Arnout Ceulemans
    • 1
  • Patrick W. Fowler
    • 2
  • Michel Deza
    • 3
  1. 1.Departement ChemieK.U.LeuvenLeuvenBelgium
  2. 2.Department of ChemistryUniversity of SheffieldLeuvenUK
  3. 3.Laboratoire de Géométrie AppliquéLIGA-EuJC, École Normale SupérieureParis Cedex 05France

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