Polyhedral 2-manifolds inE 3 with unusually large genus

Abstract

An equivelar polyhedral 2-manifold in the class ℳp,q is one embedded inE 3 in which every face is a convexp-gon and every vertex isq-valent. In this paper, examples are constructed, to show that each of the classes ℳ3,q (q≧7), ℳ4,q (q≧5) and ℳp,4 (p≧5) contains infinitely many distinct combinatorial types. As particular examples, there are polyhedral 2-manifolds with 576 vertices and genus 577, and with 4096 faces and genus 4097. A modification of one construction shows that there is a constantk, such that for eachg≧2, there exists a closed polyhedral 2-manifold inE 3 of genusg with at mostkg/logg vertices.