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14
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.
Finiteness of classifying spaces of relative diffeomorphism groups of 3manifolds, Geometry and Topology 1
, 1997
"... The main theorem shows that if M is an irreducible compact connected orientable 3manifold with nonempty boundary, then the classifying space BDiff (M rel ∂M) of the space of diffeomorphisms of M which restrict to the identity map on ∂M has the homotopy type of a finite aspherical CWcomplex. This a ..."
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Cited by 13 (4 self)
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The main theorem shows that if M is an irreducible compact connected orientable 3manifold with nonempty boundary, then the classifying space BDiff (M rel ∂M) of the space of diffeomorphisms of M which restrict to the identity map on ∂M has the homotopy type of a finite aspherical CWcomplex. This answers, for this class of manifolds, a question posed by M. Kontsevich. The main theorem follows from a more precise result, which asserts that for these manifolds the mapping class group H(M rel ∂M) is built up as a sequence of extensions of free abelian groups and subgroups of finite index in relative mapping class groups of compact connected surfaces.
Algorithms for recognizing knots and 3manifolds
 Chaos, Solitons and Fractals
, 1998
"... Algorithms are of interest to geometric topologists for two reasons. First, they have bearing on the decidability of a problem. Certain topological questions, such as finding a classification of four dimensional manifolds, admit no solution. ..."
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Cited by 6 (3 self)
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Algorithms are of interest to geometric topologists for two reasons. First, they have bearing on the decidability of a problem. Certain topological questions, such as finding a classification of four dimensional manifolds, admit no solution.
Small 3manifolds of large genus
"... Abstract. We prove the existence of pure braids with arbitrarily many strands which are small, i.e. they contain no closed incompressible surface in the complement which is not boundary parallel. This implies the existence of irreducible nonHaken 3manifolds of arbitrarily high Heegaard genus. 1. ..."
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Cited by 4 (0 self)
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Abstract. We prove the existence of pure braids with arbitrarily many strands which are small, i.e. they contain no closed incompressible surface in the complement which is not boundary parallel. This implies the existence of irreducible nonHaken 3manifolds of arbitrarily high Heegaard genus. 1.
Families of knots for which Morton’s inequality is strict
, 2006
"... Abstract. We describe a procedure for creating infinite families of knots, each having the maximum degree of their HOMFLY polynomial strictly less than twice their canonical genus. §0 ..."
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Abstract. We describe a procedure for creating infinite families of knots, each having the maximum degree of their HOMFLY polynomial strictly less than twice their canonical genus. §0
unknown title
, 2006
"... ISSN numbers are printed here 1 Topology of knot spaces in dimension 3 ..."
unknown title
, 2006
"... ISSN numbers are printed here 1 JSJdecompositions of knot and link complements in S 3 ..."
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ISSN numbers are printed here 1 JSJdecompositions of knot and link complements in S 3
unknown title
, 2009
"... ISSN numbers are printed here 1 Topology of knot spaces in dimension 3 ..."
unknown title
, 2007
"... ISSN numbers are printed here 1 JSJdecompositions of knot and link complements in S 3 ..."
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ISSN numbers are printed here 1 JSJdecompositions of knot and link complements in S 3