Enumerating fundamental normal surfaces: Algorithms, experiments and invariants

TL;DR

This paper advances normal surface theory algorithms to compute fundamental normal surfaces, enabling the solution of large, complex problems in knot theory and 3-manifold topology through experimental comparison of primal and dual methods.

Contribution

It introduces and implements new algorithms for computing fundamental normal surfaces, extending beyond vertex normal surfaces, with experimental validation on large problems.

Findings

Successfully computed 398 previously-unknown crosscap numbers of knots.

Demonstrated that large, complex problems are now tractable with the new algorithms.

Experimental comparison shows the effectiveness of primal and dual approaches.

Abstract

Computational knot theory and 3-manifold topology have seen significant breakthroughs in recent years, despite the fact that many key algorithms have complexity bounds that are exponential or greater. In this setting, experimentation is essential for understanding the limits of practicality, as well as for gauging the relative merits of competing algorithms. In this paper we focus on normal surface theory, a key tool that appears throughout low-dimensional topology. Stepping beyond the well-studied problem of computing vertex normal surfaces (essentially extreme rays of a polyhedral cone), we turn our attention to the more complex task of computing fundamental normal surfaces (essentially an integral basis for such a cone). We develop, implement and experimentally compare a primal and a dual algorithm, both of which combine domain-specific techniques with classical Hilbert basis algorithms. Our experiments indicate that we can solve extremely large problems that were once though intractable. As a practical application of our techniques, we fill gaps from the KnotInfo database by computing 398 previously-unknown crosscap numbers of knots.