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The computational Complexity of Knot and Link Problems
 J. ACM
, 1999
"... We consider the problem of deciding whether a polygonal knot in 3dimensional Euclidean space is unknotted, capable of being continuously deformed without selfintersection so that it lies in a plane. We show that this problem, unknotting problem is in NP. We also consider the problem, unknotting pr ..."
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Cited by 55 (6 self)
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We consider the problem of deciding whether a polygonal knot in 3dimensional Euclidean space is unknotted, capable of being continuously deformed without selfintersection 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 selfintersection 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.
Interactive Topological Drawing
, 1998
"... The research presented here examines topological drawing, a new mode of constructing and interacting with mathematical objects in threedimensional space. In topological drawing, issues such as adjacency and connectedness, which are topological in nature, take precedence over purely geometric issues ..."
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Cited by 18 (1 self)
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The research presented here examines topological drawing, a new mode of constructing and interacting with mathematical objects in threedimensional 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 threedimensional 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 selfintersections. 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
Energy Functions for Knots: Beginning to Predict Physical Behavior
 in Proc. of 1994 IMA Conference on Geometry and Topology of DNA
, 1994
"... 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 contrib ..."
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Cited by 7 (5 self)
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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 knottype invariants, e.g by considering the minimum of u(K) as K ranges over all the knots of a given knottype. We also are motivated by the desire to understand and predict how knottype 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...
Motion Planning for Knot Untangling
 Int. J. of Robotics Research
, 2002
"... 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 a ..."
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Cited by 6 (3 self)
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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.
Universal characteristics of polygonal knot probabilities, in Physical and Numerical Models in Knot Theory
 Knots Everything
, 2005
"... There is a striking qualitative similarity among the graphs of the relative probabilities of corresponding knot types across a wide range of random polygon models. In many cases one has theoretical results describing the asymptotic decay of these knot probabilities but, in the finite range, there is ..."
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Cited by 3 (3 self)
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There is a striking qualitative similarity among the graphs of the relative probabilities of corresponding knot types across a wide range of random polygon models. In many cases one has theoretical results describing the asymptotic decay of these knot probabilities but, in the finite range, there is little theoretical knowledge and a variety of functional models have been used to fit the observed structures. In this paper we compare a selection of these models and study the extent to which each provides a successful fit for five distinct random knot models. One consequence of this study is that while such models are quite successful in this finite range, they do not provide the theoretically predicted asymptotic structure. A second result is the observed similarity between the global knot probabilities and those arising from small perturbations of three ideal knots. 1.
Linking of Random pSpheres in Z^d
"... We consider the number of embeddings of k pspheres in Z , with p+2 d 2p+1, stratified by the pdimensional volumes of the spheres. We show for p + 2 ! d that the number of embeddings of a fixed link type of k equivolume pspheres grows with increasing pdimensional volume at an exponential ..."
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We consider the number of embeddings of k pspheres in Z , with p+2 d 2p+1, stratified by the pdimensional volumes of the spheres. We show for p + 2 ! d that the number of embeddings of a fixed link type of k equivolume pspheres grows with increasing pdimensional 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 pspheres and for "augmented" links where each component psphere can have any knot type, and similar but weaker results when the spheres are of specified knot type.
Knots In Graphs In Subsets Of Z³
"... 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 ..."
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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 ...
Monte Carlo Results for Projected SelfAvoiding Polygons: A Twodimensional Model for Knotted Polymers
, 2008
"... We introduce a twodimensional 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 configurat ..."
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We introduce a twodimensional 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 twodimensional 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 selfavoiding 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.
DOI: 10.1142/S0218216510008078 KNOTS, SLIPKNOTS, AND EPHEMERAL KNOTS IN RANDOM WALKS AND EQUILATERAL POLYGONS
, 2009
"... The probability that a random walk or polygon in the 3space or in the simple cubic lattice contains a knot goes to one at the length goes to infinity. Here, we prove that this is also true for slipknots consisting of unknotted portions, called the slipknot, that contain a smaller knotted portion, c ..."
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The probability that a random walk or polygon in the 3space or in the simple cubic lattice contains a knot goes to one at the length goes to infinity. Here, we prove that this is also true for slipknots consisting of unknotted portions, called the slipknot, that contain a smaller knotted portion, called the ephemeral knot. As is the case with knots, we prove that any topological knot type occurs as the ephemeral knotted portion of a slipknot.