Normal surfaces as combinatorial slicings

TL;DR

This paper studies the properties of normal surfaces within combinatorial 3-manifolds, providing bounds on quadrilaterals based on genus and classifying slicings with maximum edges.

Contribution

It introduces bounds on quadrilaterals in normal surfaces for 2-neighborly 3-manifolds and classifies slicings with maximum edges, advancing understanding of combinatorial manifold structures.

Findings

Lower bounds on quadrilaterals depending on genus g

Sharp bounds achieved for infinitely many g

Classification of slicings with maximum edges

Abstract

We investigate slicings of combinatorial manifolds as properly embedded co-dimension 1 submanifolds. A focus is given to dimension 3 where slicings are normal surfaces. In the case of 2-neighborly 3-manifolds and quadrangulated slicings, a lower bound on the number of quadrilaterals of normal surfaces depending on the genus g is presented. It is shown to be sharp for infinitely many values of g. Furthermore we classify slicings of combinatorial 3-manifolds with a maximum number of edges in the slicing.