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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.
Euclidean Mahler measure and Twisted Links
, 2005
"... If the twist numbers of a collection of oriented alternating link diagrams are bounded, then the Alexander polynomials of the corresponding links have bounded euclidean Mahler measure (see Definition 1.2). The converse assertion does not hold. Similarly, if a collection of oriented link diagrams, ..."
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Cited by 8 (2 self)
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If the twist numbers of a collection of oriented alternating link diagrams are bounded, then the Alexander polynomials of the corresponding links have bounded euclidean Mahler measure (see Definition 1.2). The converse assertion does not hold. Similarly, if a collection of oriented link diagrams, not necessarily alternating, have bounded twist numbers, then both the Jones polynomials and a parametrization of the 2variable Homflypt polynomials of the corresponding links have bounded Mahler measure.
Computing Linking Numbers of a Filtration
 In Algorithms in Bioinformatics (LNCS 2149
, 2001
"... We develop fast algorithms for computing the linking number of a simplicial complex within a filtration. We give experimental results in applying our work toward the detection of nontrivial tangling in biomolecules, modeled as alpha complexes. ..."
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Cited by 7 (5 self)
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We develop fast algorithms for computing the linking number of a simplicial complex within a filtration. We give experimental results in applying our work toward the detection of nontrivial tangling in biomolecules, modeled as alpha complexes.
The computational complexity of knot genus and spanning area
 electronic), arXiv: math.GT/0205057. MR MR2219001
"... Abstract. We show that the problem of deciding whether a polygonal knot in a closed threedimensional manifold bounds a surface of genus at most g is NPcomplete. 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 NPha ..."
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Cited by 6 (0 self)
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Abstract. We show that the problem of deciding whether a polygonal knot in a closed threedimensional manifold bounds a surface of genus at most g is NPcomplete. 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 NPhard. 1.
Visualization of Seifert Surfaces
 IEEE Transactions on Visualizations and Computer Graphics
, 2006
"... Abstract—The genus of a knot or link can be defined via Seifert surfaces. A Seifert surface of a knot or link is an oriented surface whose boundary coincides with that knot or link. Schematic images of these surfaces are shown in every text book on knot theory, but from these it is hard to understan ..."
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Cited by 5 (1 self)
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Abstract—The genus of a knot or link can be defined via Seifert surfaces. A Seifert surface of a knot or link is an oriented surface whose boundary coincides with that knot or link. Schematic images of these surfaces are shown in every text book on knot theory, but from these it is hard to understand their shape and structure. In this paper, the visualization of such surfaces is discussed. A method is presented to produce different styles of surface for knots and links, starting from the socalled braid representation. Application of Seifert’s algorithm leads to depictions that show the structure of the knot and the surface, while successive relaxation via a physically based model gives shapes that are natural and resemble the familiar representations of knots. Also, we present how to generate closed oriented surfaces in which the knot is embedded, such that the knot subdivides the surface into two parts. These closed surfaces provide a direct visualization of the genus of a knot. All methods have been integrated in a freely available tool, called SeifertView, which can be used for educational and presentation purposes. 1
Topics in Heegaard Floer homology
, 906
"... Abstract. Heegaard Floer homology is an extremely powerful invariant for closed oriented threemanifolds, introduced by Peter Ozsváth and Zoltán Szabó. This invariant was later generalized by them and independently by Jacob Rasmussen to an invariant for knots inside threemanifolds called knot Floer ..."
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Abstract. Heegaard Floer homology is an extremely powerful invariant for closed oriented threemanifolds, introduced by Peter Ozsváth and Zoltán Szabó. This invariant was later generalized by them and independently by Jacob Rasmussen to an invariant for knots inside threemanifolds called knot Floer homology, which was later even further generalized to include the case of links. However the boundary maps in the Heegaard Floer chain complexes were defined by counting the number of points in certain moduli spaces, and there was no algorithm to compute the invariants in general. The primary aim of this thesis is to address this concern. We begin by surveying various areas of this theory and providing the background material to familiarize the reader with the Heegaard Floer homology world. We then describe the algorithm which was discovered by Jiajun Wang and me, that computes the hat version of the threemanifold invariant with coefficients in F2. For the remainder of the thesis, we concentrate on the case of knots and links inside the threesphere. Based on a grid diagram for a knot and following a paper by Ciprian Manolescu, Peter Ozsváth and me, we give a another algorithm for computing the knot Floer homology. We conclude by generalizing the construction to a theory of knot Floer homotopy. ACKNOWLEDGEMENT 3 Acknowledgement My adviser Zoltán Szabó for introducing me to the fascinating world of Heegaard Floer homology and for guiding me throughout the entire course of my graduate studies. My collaborators Matthew Hedden, András Juhász, Ciprian Manolescu, Peter Ozsváth and Jiajun Wang for all the discoveries that we made together, which constitute a significant portion of this thesis.
A New Euler’s Formula for DNA Polyhedra
, 2011
"... DNA polyhedra are cagelike architectures based on interlocked and interlinked DNA strands. We propose a formula which unites the basic features of these entangled structures. It is based on the transformation of the DNA polyhedral links into Seifert surfaces, which removes all knots. The numbers of ..."
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DNA polyhedra are cagelike architectures based on interlocked and interlinked DNA strands. We propose a formula which unites the basic features of these entangled structures. It is based on the transformation of the DNA polyhedral links into Seifert surfaces, which removes all knots. The numbers of components m, of crossings c, and of Seifert circles s are related by a simple and elegant formula: szm~cz2. This formula connects the topological aspects of the DNA cage to the Euler characteristic of the underlying polyhedron. It implies that Seifert circles can be used as effective topological indices to describe polyhedral links. Our study demonstrates that, the new Euler’s formula provides a theoretical framework for the stereochemistry of DNA polyhedra, which can characterize enzymatic transformations of DNA and be used to characterize and design novel cages with higher genus.
LINKING INTEGRAL PROJECTION
, 907
"... Abstract. The linking integral is an invariant of the linktype of two manifolds immersed in a Euclidean space. It is shown that the ordinary Gauss integral in three dimensions may be simplified to a winding number integral in two dimensions. This result is then generalized to show that in certain c ..."
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Abstract. The linking integral is an invariant of the linktype of two manifolds immersed in a Euclidean space. It is shown that the ordinary Gauss integral in three dimensions may be simplified to a winding number integral in two dimensions. This result is then generalized to show that in certain circumstances the linking integral between arbitrary manifolds may be similarly reduced to a lower dimensional integral. 1. Reduction of the Gauss Integral to the Winding Number Integral The linking number of two disjoint oriented closed curves in R 3 is an integer invariant that in some sense measures the extent of linking between the curves. While there are many equivalent ways to compute this number[3], the most wellknown is the linking integral of Gauss. In this section we show that this integral in 3space may always be simplified to an integral in 2space which is equivalent to a winding number integral. Proposition 1. Given two disjoint immersed closed curves s ↦ → γ1(s) and t ↦→ γ2(t) in R 3, the Gauss linking integral of the pair reduces to a sum of winding numbers of one curve about a sequence of points determined by the other, contained in some 2dimensional hyperplane. Proof. The link of γ1 and γ2, lk(γ1, γ2), is given by the Gauss integral, lk(γ1, γ2) = 1 det r,