Bounds for the genus of a normal surface

TL;DR

This paper establishes precise linear bounds on the genus of normal surfaces in 3-manifolds based on quadrilaterals, with applications to minimal triangulations and the realization problem.

Contribution

It introduces sharp linear bounds on the genus of normal surfaces in 3-manifolds and applies these bounds to specific problems in triangulation and realization theory.

Findings

Determined minimal triangulations of the product of a surface and an interval.

Established sharp linear bounds on the genus based on quadrilaterals.

Compared the effectiveness of normal surface theory with dual polytope methods.

Abstract

This paper gives sharp linear bounds on the genus of a normal surface in a triangulated compact, orientable 3--manifold in terms of the quadrilaterals in its cell decomposition---different bounds arise from varying hypotheses on the surface or triangulation. Two applications of these bounds are given. First, the minimal triangulations of the product of a closed surface and the closed interval are determined. Second, an alternative approach to the realisation problem using normal surface theory is shown to be less powerful than its dual method using subcomplexes of polytopes.