Quadrilateral-octagon coordinates for almost normal surfaces

TL;DR

This paper extends quadrilateral coordinates to octagonal almost normal surfaces, significantly reducing computational complexity and improving the efficiency of 3-sphere recognition algorithms in 3-manifold topology.

Contribution

It develops an analogous coordinate reduction for octagonal surfaces and introduces joint coordinates with favorable geometric properties.

Findings

Reduced dimension from 10n to 6n for octagonal surfaces

Streamlined 3-sphere recognition algorithm with thousands of times faster performance

Introduction of joint coordinates with 3n dimensions

Abstract

Normal and almost normal surfaces are essential tools for algorithmic 3-manifold topology, but to use them requires exponentially slow enumeration algorithms in a high-dimensional vector space. The quadrilateral coordinates of Tollefson alleviate this problem considerably for normal surfaces, by reducing the dimension of this vector space from 7n to 3n (where n is the complexity of the underlying triangulation). Here we develop an analogous theory for octagonal almost normal surfaces, using quadrilateral and octagon coordinates to reduce this dimension from 10n to 6n. As an application, we show that quadrilateral-octagon coordinates can be used exclusively in the streamlined 3-sphere recognition algorithm of Jaco, Rubinstein and Thompson, reducing experimental running times by factors of thousands. We also introduce joint coordinates, a system with only 3n dimensions for octagonal almost normal surfaces that has appealing geometric properties.