A chain complex and Quadrilaterals for normal surfaces

TL;DR

This paper introduces a homological chain complex framework to analyze normal surfaces in three-manifolds, clarifying how quadrilateral coordinates uniquely determine such surfaces up to vertex linking spheres.

Contribution

It provides a homological interpretation of normal surfaces, proves quadrilaterals determine surfaces up to vertex linking, and characterizes quadrilateral coordinates in triangulations.

Findings

Quadrilaterals determine a normal surface up to vertex linking spheres.

Homology of a chain complex models normal surfaces.

Characterization of quadrilateral coordinates in triangulations.

Abstract

We interpret a normal surface in a (singular) three-manifold in terms of the homology of a chain complex. This allows us to study the relation between normal surfaces and their quadrilateral co-ordinates. Specifically, we give a proof of an (unpublished) observation independently given by Casson and Rubinstein saying that quadrilaterals determine a normal surface up to vertex linking spheres. We also characterise the quadrilateral coordinates that correspond to a normal surface in a (possibly ideal) triangulation.