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43
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.
The number of Reidemeister Moves Needed for Unknotting
, 2008
"... There is a positive constant c1 such that for any diagram D representing the unknot, there is a sequence of at most 2 c1n Reidemeister moves that will convert it to a trivial knot diagram, where n is the number of crossings in D. A similar result holds for elementary moves on a polygonal knot K embe ..."
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Cited by 35 (11 self)
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There is a positive constant c1 such that for any diagram D representing the unknot, there is a sequence of at most 2 c1n Reidemeister moves that will convert it to a trivial knot diagram, where n is the number of crossings in D. A similar result holds for elementary moves on a polygonal knot K embedded in the 1skeleton of the interior of a compact, orientable, triangulated PL 3manifold M. There is a positive constant c2 such that for each t ≥ 1, if M consists of t tetrahedra, and K is unknotted, then there is a sequence of at most 2 c2t elementary moves in M which transforms K to a triangle contained inside one tetrahedron of M. We obtain explicit values for c1 and c2.
On AlexanderConway Polynomials for Virtual Knots and Links
, 2001
"... A polynomial invariant of virtual links, arising from an invariant of links in thickened surfaces introduced by Jaeger, Kauffman, and Saleur, is defined and its properties are investigated. Examples are given that the invariant can detect chirality and even noninvertibility of virtual knots and ..."
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Cited by 28 (1 self)
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A polynomial invariant of virtual links, arising from an invariant of links in thickened surfaces introduced by Jaeger, Kauffman, and Saleur, is defined and its properties are investigated. Examples are given that the invariant can detect chirality and even noninvertibility of virtual knots and links. Furthermore, it is shown that the polynomial satisfies a Conwaytype skein relation  in contrast to the Alexander polynomial derived from the virtual link group. Keywords: Virtual Knot Theory, Conway Skein Relation, Alexander Invariants AMS classification: 57M25
Floer homology of surgeries on twobridge knots
 Algebr. Geom. Topol
"... Abstract. We compute the OzsváthSzabó Floer homologies HF ± and ̂ HF for threemanifolds obtained by integer surgery on a twobridge knot. 1. ..."
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Cited by 28 (2 self)
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Abstract. We compute the OzsváthSzabó Floer homologies HF ± and ̂ HF for threemanifolds obtained by integer surgery on a twobridge knot. 1.
Selfadjunctions and matrices
 Journal of Pure and Applied Algebra
"... It is shown that the multiplicative monoids of TemperleyLieb algebras are isomorphic to monoids of endomorphisms in categories where an endofunctor is adjoint to itself. Such a selfadjunction is found in a category whose arrows are matrices, and the functor adjoint to itself is based on the Kronec ..."
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Cited by 10 (4 self)
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It is shown that the multiplicative monoids of TemperleyLieb algebras are isomorphic to monoids of endomorphisms in categories where an endofunctor is adjoint to itself. Such a selfadjunction is found in a category whose arrows are matrices, and the functor adjoint to itself is based on the Kronecker product of matrices. Thereby one obtains a representation of braid groups in matrices, which, though different and presumably new, is related to the standard representation of braid groups in TemperleyLieb algebras. Mathematics Subject Classification (2000): 57M99, 20F36, 18A40 1
Tait's Flyping Conjecture for 4Regular Graphs
, 1998
"... Tait's flyping conjecture, stating that two reduced, alternating, prime link diagrams can be connected by a finite sequence of flypes, is extended to reduced, alternating, prime diagrams of 4regular graphs in S 3 which are vertexseparating, i.e., each vertex can be separated by a nontrivial ..."
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Cited by 4 (1 self)
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Tait's flyping conjecture, stating that two reduced, alternating, prime link diagrams can be connected by a finite sequence of flypes, is extended to reduced, alternating, prime diagrams of 4regular graphs in S 3 which are vertexseparating, i.e., each vertex can be separated by a nontrivial 2string tangle from the rest of the diagram. The proof of this version of the flyping conjecture is based on the fact that the equivalence classes with respect to ambient isotopy and rigid vertex isotopy of graph embeddings are identical on the class of diagrams considered. Keywords: Knotted Graph, Alternating Diagram, Flyping Conjecture AMS classification: 57M25; 57M15
Chebyshev knots
, 2008
"... A Chebyshev knot is a knot which admits a parametrization of the form x(t) = Ta(t); y(t) = Tb(t); z(t) = Tc(t + ϕ), where a, b, c are pairwise coprime, Tn(t) is the Chebyshev polynomial of degree n, and ϕ ∈ R. Chebyshev knots are non compact analogues of the classical Lissajous knots. We show that ..."
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Cited by 4 (4 self)
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A Chebyshev knot is a knot which admits a parametrization of the form x(t) = Ta(t); y(t) = Tb(t); z(t) = Tc(t + ϕ), where a, b, c are pairwise coprime, Tn(t) is the Chebyshev polynomial of degree n, and ϕ ∈ R. Chebyshev knots are non compact analogues of the classical Lissajous knots. We show that there are infinitely many Chebyshev knots with ϕ = 0. We also show that every knot is a Chebyshev knot.
Surgery Untying of Coloured Knots
"... Abstract. For p�3 and for p�5 we prove that there are exactly p equivalence classes of pcoloured knots modulo 1–framed surgeries along unknots in the kernel of a p–colouring. These equivalence classes are represented by connectsums of n lefthand p,2torus knots with a given colouring when n�1, 2, ..."
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Cited by 3 (2 self)
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Abstract. For p�3 and for p�5 we prove that there are exactly p equivalence classes of pcoloured knots modulo 1–framed surgeries along unknots in the kernel of a p–colouring. These equivalence classes are represented by connectsums of n lefthand p,2torus knots with a given colouring when n�1, 2,..., p. This gives us a 3–colour and a 5–colour analogue of the surgery presentation of a knot in the complement of an unknot. 1.
”New” Veneziano amplitudes from ”old” Fermat (hyper)surfaces
, 2003
"... The history of the discovery of bosonic string theory is well documented. This theory evolved as an attempt to find a multidimensional analogue of Euler’s beta function to describe the multiparticle Veneziano amplitudes. Such an analogue had in fact been known in mathematics at least in 1922. Its ma ..."
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Cited by 3 (2 self)
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The history of the discovery of bosonic string theory is well documented. This theory evolved as an attempt to find a multidimensional analogue of Euler’s beta function to describe the multiparticle Veneziano amplitudes. Such an analogue had in fact been known in mathematics at least in 1922. Its mathematical meaning was studied subsequently from different angles by mathematicians such as Selberg, Weil and Deligne among others. The mathematical interpretation of this multidimensional beta function that was developed subsequently is markedly different from that described in physics literature. This work aims to bridge the gap between the mathematical and physical treatments. Using some results of recent publications (e.g. J.Geom.Phys.38 (2001) 81; ibid 43 (2002) 45) new topological, algebrogeometric, numbertheoretic and combinatorial treatment of the multiparticle Veneziano amplitudes is developed. As a result, an entirely new physical meaning of these amplitudes is emerging: they are periods of differential forms associated with homology cycles on Fermat (hyper)surfaces. Such (hyper)surfaces are considered as complex projective varieties of Hodge type. Although the computational formalism developed in this work resembles that used in mirror symmetry calculations, many additional results from mathematics are used along with their suitable physical interpretation. For instance, the Hodge spectrum of the Fermat (hyper)surfaces is in onetoone correspondence with the possible spectrum of particle masses. The formalism also allows us to obtain correlation functions of both conformal field theory and particle physics using the same type of the PicardFuchs equations whose solutions are being interpreted in terms of periods.
Chebyshev diagrams for rational knots
"... We show that every rational knot K of crossing number N admits a polynomial parametrization x = Ta(t), y = Tb(t), z = C(t) where Tk(t) are the Chebyshev polynomials, a = 3 and b + deg C = 3N. We show that every rational knot also admits a polynomial parametrization with a = 4. If C(t) = Tc(t) is a ..."
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Cited by 2 (2 self)
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We show that every rational knot K of crossing number N admits a polynomial parametrization x = Ta(t), y = Tb(t), z = C(t) where Tk(t) are the Chebyshev polynomials, a = 3 and b + deg C = 3N. We show that every rational knot also admits a polynomial parametrization with a = 4. If C(t) = Tc(t) is a Chebyshev polynomial, we call such a knot a harmonic knot. We give the classification of harmonic knots for