Computing closed essential surfaces in 3-manifolds

TL;DR

This paper introduces a practical algorithm for detecting closed essential surfaces in 3-manifolds, combining enumeration and optimization techniques from normal surface theory, with successful application to numerous examples.

Contribution

The paper presents a new, practical algorithm based on the Jaco-Oertel framework for identifying closed essential surfaces in 3-manifolds, improving computational methods in the field.

Findings

Algorithm successfully tested on many 3-manifolds and knot exteriors.

Yields new results in the understanding of 3-manifold structures.

Demonstrates the algorithm's effectiveness and practicality.

Abstract

We present a practical algorithm to test whether a 3-manifold given by a triangulation or an ideal triangulation contains a closed essential surface. This property has important theoretical and algorithmic consequences. As a testament to its practicality, we run the algorithm over a comprehensive body of closed 3-manifolds and knot exteriors, yielding results that were not previously known. The algorithm derives from the original Jaco-Oertel framework, involves both enumeration and optimisation procedures, and combines several techniques from normal surface theory. Our methods are relevant for other difficult computational problems in 3-manifold theory, such as the recognition problem for knots, links and 3-manifolds.