Joel Hass
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.
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Index Terms
- The computational complexity of knot and link problems
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Reviews
Patrick J. Ryan
For the purposes of this paper, a “knot” is a simple polygon in 3-space. A finite collection of disjoint knots is a “link.” Two knots are “isotopic” if there is a (piecewise-linear) isotopy transforming one to the other. A knot is “trivial” if it is isotopic to a (simple) plane polygon. A link is represented by a link diagram. This is a planar graph obtained by projection of the link. Its vertices are the crossings of the projection and are specially labeled to indicate which edge of the projection passes over the other. An edge of the link diagram corresponds to the portion of a link between two crossings. The projection is chosen to be in “general position” so that all vertices of the link diagram have degree 4. The first question is to determine whether a given link diagram represents a trivial knot. This is shown to be an NP problem. A link is “splittable” if it is isotopic to the union of two links that can be separated by a plane. The splitting problem (determining if a link is splittable) is also shown to be in NP.<__?__Pub Fmt eos-space>Exponential worst-case running time bounds for deterministic algorithms to solve each of these problems are given. These algorithms are based on the use of normal surfaces and decision procedures due to Haken and others [1,2], which are applied to the piecewise-linear category. It is convenient to work in the 3-sphere, which can be triangulated in a way that is adapted to the link. An appropriate triangulation includes a tubular neighborhood of each knot, and the link itself lies on its 1-skeleton. The tetrahedra of the triangulation not belonging to the tubular neighborhoods form the “link complement” M K . To show that the link is a trivial knot, one needs to find a disk in M K (an “essential disk”) whose boundary has nontrivial Z 2 homology in the homology of the boundary of M K . For a given link diagram with crossing measure n (the number of vertices plus the number of components minus 1), the authors construct a triangulated link complement having t?420n tetrahedra. Normal surfaces are parameterized by vectors v in Z 7t , which code the way the surface intersects each tetrahedron. Then one has to search the admissible values of v to find the desired essential disk. Related problems are solved in a similar way. Although the topological background required to fully understand the paper is formidable, the authors do a good job of presenting their results in an intuitive way. The estimates obtained are not shown to be sharp,<__?__Pub Caret> and the authors state that no lower bounds or hardness results are known. This is not surprising, since prior to the appearance of the paper under review, only decidability had been established (in Haken [1]). It appears that there are many opportunities for further analysis of the problems discussed.
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