1998/07/03 by Joel Hass, Jeffrey C. Lagarias, Nicholas Pippenger · 2 citations
Mathematics · #math.GT #msc:57M25 #msc:68Q15
published as J. A. C. M. 46 (1999) 185--211. · 32 pages, 1 figure
arxiv created 1998/07/03 · arxiv updated 2009/11/30
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.