No abstract

We consider the problem of deciding whether a polygonal knot in 3-dimensional Euclidean space is unknotted, i.e., 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.

Abstract. We show that the problem of deciding whether a polygonal knot in a closed three-dimensional manifold bounds a surface of genus at most g is NP-complete. We also show that the problem of deciding whether a curve in a PL manifold bounds a surface of area less than a given constant C is NP-hard.

No abstract

There is a positive constant c 1 c_1 such that for any diagram D \mathcal {D} representing the unknot, there is a sequence of at most 2 c 1 n 2^{c_1 n} Reidemeister moves that will convert it to a trivial knot diagram, where n n is the number of crossings in D \mathcal {D} . A similar result holds for elementary moves on a polygonal knot K K embedded in the 1-skeleton of the interior of a compact, orientable, triangulated P L PL 3-manifold M M . There is a positive constant c 2 c_2 such that for each t ≥ 1 t \geq 1 , if M M consists of t t tetrahedra and K K is unknotted, then there is a sequence of at most 2 c 2 t 2^{c_2 t} elementary moves in M M which transforms K K to a triangle contained inside one tetrahedron of M M . We obtain explicit values for c 1 c_1 and c 2 c_2 .