Surface realization with the intersection edge functional

Abstract

Deciding realizability of a given polyhedral map on a (compact, connected) surface belongs to the hard problems in discrete geometry, from the theoretical, the algorithmic, and the practical point of view. In this paper, we present a heuristic algorithm for the realization of simplicial maps, based on the intersection edge functional. The heuristic was used to find geometric realizations in R^3 for all vertex-minimal triangulations of the orientable surfaces of genus g=3 and g=4. Moreover, for the first time, examples of simplicial polyhedra in R^3 of genus 5 with 12 vertices were obtained.

22 pages, 11 figures, various minor revisions, to appear in Experimental Mathematics

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• Equivelar and d-Covered Triangulations of Surfaces. II. Cyclic Triangulations and Tessellations
• Minimal flag triangulations of lower-dimensional manifolds
• Unimodular lattice triangulations as small-world and scale-free random graphs
• Equivelar and d-Covered Triangulations of Surfaces. I
• SAT-Based Techniques for Lexicographically Smallest Finite Models
• Polyhedral tori with minimal coordinates
• Spectral properties of unimodular lattice triangulations
• The characterization of $(n-1)$-spheres with $n+4$ vertices having maximal Buchstaber number
• Settling the genus of the $n$-prism
• A new family of triangulations of $\mathbb{R}{P}^d$