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Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

Ed. by Cohen, Frederick R. / Droste, Manfred / Darmon, Henri / Duzaar, Frank / Echterhoff, Siegfried / Gordina, Maria / Neeb, Karl-Hermann / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna

IMPACT FACTOR 2018: 0.867

CiteScore 2018: 0.71

SCImago Journal Rank (SJR) 2018: 0.898
Source Normalized Impact per Paper (SNIP) 2018: 0.964

Mathematical Citation Quotient (MCQ) 2017: 0.67

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Volume 18, Issue 6


Degree-regular triangulations of the double-torus

Basudeb Datta / Ashish Kumar Upadhyay
Published Online: 2007-02-27 | DOI: https://doi.org/10.1515/FORUM.2006.051


A connected combinatorial 2-manifold is called degree-regular if each of its vertices have the same degree. A connected combinatorial 2-manifold is called weakly regular if it has a vertex-transitive automorphism group. Clearly, a weakly regular combinatorial 2-manifold is degree-regular and a degree-regular combinatorial 2-manifold of Euler characteristic –2 must contain 12 vertices.

In 1982, McMullen et al. constructed a 12-vertex geometrically realized triangulation of the double-torus in ℝ3. As an abstract simplicial complex, this triangulation is a weakly regular combinatorial 2-manifold. In 1999, Lutz showed that there are exactly three weakly regular orientable combinatorial 2-manifolds of Euler characteristic –2. In this article, we classify all the orientable degree-regular combinatorial 2-manifolds of Euler characteristic –2. There are exactly six such combinatorial 2-manifolds. This classifies all the orientable equivelar polyhedral maps of Euler characteristic –2.

About the article

Received: 2005-05-04

Revised: 2005-07-11

Published Online: 2007-02-27

Published in Print: 2006-11-20

Citation Information: Forum Mathematicum, Volume 18, Issue 6, Pages 1011–1025, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/FORUM.2006.051.

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