Polyhedral surfaces of high genus

Details Polyhedral surfaces of high genus Polyhedral surfaces of high genus

2004-12-05

arxiv.org/abs/math/0412093v1

Mathematics

Metric Geometry

21 pages; Lecture Notes for Oberwolfach Seminar "Discrete Differential Geometry", June 2004

Scientific paper

The construction of the COMBINATORIAL data for a surface with n vertices of maximal genus is a classical problem: The maximal genus g=[(n-3)(n-4)/12] was achieved in the famous ``Map Color Theorem'' by Ringel et al. (1968). We present the nicest one of Ringel's constructions, for the case when n is congruent to 7 mod 12, but also an alternative construction, essentially due to Heffter (1898), which easily and explicitly yields surfaces of genus g ~ 1/16 n^2. For GEOMETRIC (polyhedral) surfaces with n vertices the maximal genus is not known. The current record is g ~ n log n, due to McMullen, Schulz & Wills (1983). We present these surfaces with a new construction: We find them in Schlegel diagrams of ``neighborly cubical 4-polytopes,'' as constructed by Joswig & Ziegler (2000).

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