The Computational Complexity of Knot and Link Problems

TL;DR

This paper analyzes the computational complexity of knot and link problems, establishing their placement in NP and PSPACE classes, and provides exponential bounds for algorithms solving these problems.

Contribution

It proves that the unknotting and splitting problems are in NP and that the genus problem is in PSPACE, extending the understanding of their computational difficulty.

Findings

Unknotting problem is in NP.

Splitting problem is in NP.

Genus determination is in PSPACE.

Abstract

We consider the problem of deciding whether a polygonal knot in 3-dimensional Euclidean space is unknotted, capable of being continuously deformed without self-intersection so that it lies in a plane. We show that this problem, {\sc unknotting problem} is in {\bf NP}. We also consider the problem, {\sc unknotting problem} of determining whether two or more such polygons can be split, or continuously deformed without self-intersection so that they occupy both sides of a plane without intersecting it. We show that it also is in NP. Finally, we show that the problem of determining the genus of a polygonal knot (a generalization of the problem of determining whether it is unknotted) is in {\bf PSPACE}. We also give exponential worst-case running time bounds for deterministic algorithms to solve each of these problems. These algorithms are based on the use of normal surfaces and decision procedures due to W. Haken, with recent extensions by W. Jaco and J. L. Tollefson.