Computing closed essential surfaces in knot complements

TL;DR

This paper introduces a practical algorithm for detecting closed essential surfaces in knot complements, enabling systematic analysis of a large set of knots and advancing computational 3-manifold topology.

Contribution

The paper presents a new, efficient algorithm based on normal surface theory that improves the practicality of testing for closed essential surfaces in knot complements.

Findings

Successfully tested 2979 knots, including complex dodecahedral knots.

Achieved results previously infeasible due to computational complexity.

Demonstrated the algorithm's applicability to other 3-manifold problems.

Abstract

We present a new, practical algorithm to test whether a knot complement contains a closed essential surface. This property has important theoretical and algorithmic consequences; however, systematically testing it has until now been infeasibly slow, and current techniques only apply to specific families of knots. As a testament to its practicality, we run the algorithm over a comprehensive body of 2979 knots, including the two 20-crossing dodecahedral knots, 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. This represents substantial progress in the practical implementation of normal surface theory, in that we can systematically solve a theoretically double exponential-time problem for significant inputs. Our methods are relevant for other difficult computational problems in 3-manifold theory, ranging from testing for Haken-ness to the recognition problem for knots, links and 3-manifolds.