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21
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.
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 ..."
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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.
Monte Carlo Explorations Of Polygonal Knot Spaces
, 1998
"... Polygonal knots are embeddings of polygons in three space. For each n, the collection of embedded ngons determines a subset of Euclidean space whose structure is the subject of this paper. Which knots can be constructed with a specified number of edges? What is the likelihood that a randomly cho ..."
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Cited by 5 (5 self)
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Polygonal knots are embeddings of polygons in three space. For each n, the collection of embedded ngons determines a subset of Euclidean space whose structure is the subject of this paper. Which knots can be constructed with a specified number of edges? What is the likelihood that a randomly chosen polygon of nedges will be a knot of a specific topological type? At what point is a given topological type most likely as a function of the number of edges? Are the various orderings of knot types by means of "physical properties" comparable? These and related questions are discussed and supporting evidence, in many cases derived from Monte Carlo explorations, is provided.
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.
Predicting Optimal Lengths of Random Knots
, 2008
"... In thermally fluctuating long linear polymeric chain in solution, the ends come from time to time into a direct contact or a close vicinity of each other. At such an instance, the chain can be regarded as a closed one and thus will form a knot or rather a virtual knot. Several earlier studies of ran ..."
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Cited by 2 (0 self)
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In thermally fluctuating long linear polymeric chain in solution, the ends come from time to time into a direct contact or a close vicinity of each other. At such an instance, the chain can be regarded as a closed one and thus will form a knot or rather a virtual knot. Several earlier studies of random knotting demonstrated that simpler knots show their highest occurrence for shorter random walks than more complex knots. However up to now there were no rules that could be used to predict the optimal length of a random walk, i.e. the length for which a given knot reaches its highest occurrence. Using numerical simulations, we show here that a power law accurately describes the relation between the optimal lengths of random walks leading to the formation of different knots and the previously characterized lengths of ideal knots of the corresponding type.
doi:10.1093/nar/gkn467 DNA supercoiling inhibits DNA knotting
, 2008
"... Despite the fact that in living cells DNA molecules are long and highly crowded, they are rarely knotted. DNA knotting interferes with the normal functioning of the DNA and, therefore, molecular mechanisms evolved that maintain the knotting and catenation level below that which would be achieved if ..."
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Cited by 1 (1 self)
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Despite the fact that in living cells DNA molecules are long and highly crowded, they are rarely knotted. DNA knotting interferes with the normal functioning of the DNA and, therefore, molecular mechanisms evolved that maintain the knotting and catenation level below that which would be achieved if the DNA segments could pass randomly through each other. Biochemical experiments with torsionally relaxed DNA demonstrated earlier that type II DNA topoisomerases that permit inter and intramolecular passages between segments of DNA molecules use the energy of ATP hydrolysis to select passages that lead to unknotting rather than to the formation of knots. Using numerical simulations, we identify here another mechanism by which topoisomerases can keep the knotting level low. We observe that DNA supercoiling, such as found in bacterial cells, creates a situation where intramolecular passages leading to knotting are opposed by the freeenergy change connected to transitions from unknotted to knotted circular DNA molecules.
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.
unknown title
, 2008
"... doi:10.1093/nar/gkn192 3D visualization software to analyze topological outcomes of topoisomerase reactions ..."
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doi:10.1093/nar/gkn192 3D visualization software to analyze topological outcomes of topoisomerase reactions