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, unknotting problem is in NP. We also consider the problem, 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 PSPACE. We also give exponential worstcase 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.

The research presented here examines topological drawing, a new mode of constructing and interacting with mathematical objects in three-dimensional space. In topological drawing, issues such as adjacency and connectedness, which are topological in nature, take precedence over purely geometric issues. Because the domain of application is mathematics, topological drawing is also concerned with the correct representation and display of these objects on a computer. By correctness we mean that the essential topological features of objects are maintained during interaction. We have chosen to limit the scope of topological drawing to knot theory, a domain that consists essentially of one class of object (embedded circles in three-dimensional space) yet is rich enough to contain a wide variety of difficult problems of research interest. In knot theory, two embedded circles (knots) are considered equivalent if one may be smoothly deformed into the other without any cuts or self-intersections. This notion of equivalence may be thought of as the heart of knot theory. We present methods for the computer construction and interactive manipulation of a

Several definitions have been proposed for the "energy" of a knot. The intuitive goal is to define a number u(K) that somehow measures how "tangled" or "crumpled" a knot K is. Typically, one starts with the idea that a small piece of the knot somehow repels other pieces, and then adds up the contributions from all the pieces. From a purely mathematical standpoint, one may hope to define new knot-type invariants, e.g by considering the minimum of u(K) as K ranges over all the knots of a given knot-type. We also are motivated by the desire to understand and predict how knot-type affects the behavior of physically real knots, in particular DNA loops in gel electrophoresis or random knotting experiments. Despite the physical naivet'e of recently studied knot energies, there now is enough laboratory data on relative gel velocity, along with computer calculations of idealized knot energies, to justify the assertion that knot energies can predict relative knot behavior in physical systems. Th...

When given a very tangled but unknotted circular piece of string it is usually quite easy to move it around and tug on parts of it until it untangles. However solving this problem by computer, either exactly or heuristically, is challenging. Various approaches have been tried, employing ideas from algebra, geometry, topology and optimization. This paper investigates the application of motion planning techniques to the untangling of mathematical knots. Such an approach brings together robotics and knotting at the intersection of these fields: rational manipulation of a physical model. In the past, simulated annealing and other energy minimization methods have been used to find knot untangling paths for physical models. Using a probabilistic planner, we have untangled some standard benchmarks described by over four hundred variables much more quickly than has been achieved with minimization. We also show how to produce candidates with minimal number of segments for a given knot. We discuss novel motion planning techniques that were used in our algorithm and some possible applications of our untangling planner in computational topology and in the study of DNA rings.

We consider the number of embeddings of k p-spheres in Z , with p+2 d 2p+1, stratified by the p-dimensional volumes of the spheres. We show for p + 2 ! d that the number of embeddings of a fixed link type of k equivolume p-spheres grows with increasing p-dimensional volume at an exponential rate which is independent of the link type. For d = p+2 we derive similar results both for links of unknotted p-spheres and for "augmented" links where each component p-sphere can have any knot type, and similar but weaker results when the spheres are of specified knot type.

The probability that an embedding of a graph in Z³ is knotted is investigated. For any given graph (embeddable in Z³) without cut edges, it is shown that this probability approaches 1 at an exponential rate as the number of edges in the embedding goes to infinity. Furthermore, at least for a subset of these graphs, the rate at which the probability approaches 1 does not depend on the particular graph being embedded. Results analogous to these are proved to be true for embeddings of graphs in a subset of Z³ bounded by two parallel planes (a slab). In order to investigate the knotting probability of embeddings of graphs in a rectangular prism (an infinitely long rectangular tube in Z³), a pattern theorem for selfavoiding polygons in a prism is proved. From this it is possible to prove that for any given eulerian graph, the probability that an embedding of the graph in a prism is knotted goes to 1 as the number of edges in the embedding goes to infinity. Then, just as for ...

We introduce a two-dimensional lattice model for the description of knotted polymer rings. A polymer configuration is modeled by a closed polygon drawn on the square diagonal lattice, with possible crossings describing pairs of strands of polymer passing on top of each other. Each polygon configuration can be viewed as the two-dimensional projection of a particular knot. We study numerically the statistics of large polygons with a fixed knot type, using a generalization of the BFACF algorithm for self-avoiding walks. This new algorithm incorporates both the displacement of crossings and the three types of Reidemeister transformations preserving the knot topology. Its ergodicity within a fixed knot type is not proven here rigorously but strong arguments in favor of this ergodicity are given together with a tentative sketch of proof. Assuming this ergodicity, we obtain numerically the following results for the statistics of knotted polygons: In the limit of a low crossing fugacity, we find a localization along the polygon of all the primary factors forming the knot.