Degree-regular triangulations of the double-torus
-
Basudeb Datta
and
Ashish Kumar Upadhyay
Abstract
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.
© Walter de Gruyter
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- Cohomology of harmonic forms on Riemannian manifolds with boundary
- On the structure and characters of weight modules
- On the vanishing of Ext over formal triangular matrix rings
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Articles in the same Issue
- A summation formula for divisor functions associated to lattices
- On a relative Alexandrov-Fenchel inequality for convex bodies in Euclidean spaces
- Cohomology of harmonic forms on Riemannian manifolds with boundary
- On the structure and characters of weight modules
- On the vanishing of Ext over formal triangular matrix rings
- Homotopy localization of groupoids
- A group-theoretic characterization of the direct product of a ball and a Euclidean space
- Degree-regular triangulations of the double-torus
- Generalized E-algebras over valuation domains
