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
Degree-regular triangulations of the double-torus
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.
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