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17
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 78 (9 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.
The arithmetic and geometry of Salem numbers
 Bull. Amer. Math. Soc
, 1991
"... Abstract. A Salem number is a real algebraic integer, greater than 1, with the property that all of its conjugates lie on or within the unit circle, and at least one conjugate lies on the unit circle. In this paper we survey some of the recent appearances of Salem numbers in parts of geometry and ar ..."
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Cited by 18 (3 self)
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Abstract. A Salem number is a real algebraic integer, greater than 1, with the property that all of its conjugates lie on or within the unit circle, and at least one conjugate lies on the unit circle. In this paper we survey some of the recent appearances of Salem numbers in parts of geometry and arithmetic, and discuss the possible implications for the ‘minimization problem’. This is an old question in number theory which asks whether the set of Salem numbers is bounded away from 1. Contents
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 17 (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 Visualization and Computer Graphics
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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 10 (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.
a, over Vector bundles Computational topology
"... cal, th bee ncr d L pu g th giving a more general theoretical and computational account of the underlying ideas and their relationships. Building on this we describe how the modified Laplacians and the corresponding computations can be extended to threedimensional Riemannian manifolds, yielding a m ..."
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cal, th bee ncr d L pu g th giving a more general theoretical and computational account of the underlying ideas and their relationships. Building on this we describe how the modified Laplacians and the corresponding computations can be extended to threedimensional Riemannian manifolds, yielding a method that is persp e defin n phe n M. es from tion p g wel ensions is more eral theory. ting power, the rest in computanumerically computed Laplacian invariants are shape and image Contents lists available at SciVerse ScienceDirect journal homepage: www.e
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.
Topics in Heegaard Floer homology
, 2009
"... 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 ..."
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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.