The Computational Complexity of Knot and Link Problems
Document Type
Article
Department
Mathematics (HMC)
Publication Date
3-1999
Abstract
We consider the problem of deciding whether a polygonal knot in 3-dimensional Euclidean space is unknotted, ie., capable of being continuously deformed without self-intersection so that it lies in a plane. We show that this problem, UNKNOTTING PROBLEM is in NP. We also consider the problem, SPLITTING 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 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.
Rights Information
© 1999 ACM
Terms of Use & License Information
DOI
10.1145/301970.301971
Recommended Citation
Joel Hass, Jeffrey C. Lagarias, and Nicholas Pippenger. 1999. The computational complexity of knot and link problems. J. ACM 46, 2 (March 1999), 185-211.
