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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 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.
COFINITELY HOPFIAN GROUPS, OPEN MAPPINGS AND KNOT COMPLEMENTS
"... Abstract. A group Γ is defined to be cofinitely Hopfian if every homomorphism Γ → Γ whose image is of finite index is an automorphism. Geometrically significant groups enjoying this property include certain relatively hyperbolic groups and many lattices. A knot group is cofinitely Hopfian if and onl ..."
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Abstract. A group Γ is defined to be cofinitely Hopfian if every homomorphism Γ → Γ whose image is of finite index is an automorphism. Geometrically significant groups enjoying this property include certain relatively hyperbolic groups and many lattices. A knot group is cofinitely Hopfian if and only if the knot is not a torus knot. A freebycyclic group is cofinitely Hopfian if and only if it has trivial centre. Applications to the theory of open mappings between manifolds are presented. 1.
RELATIVE KNOT INVARIANTS: PROPERTIES AND APPLICATIONS
, 909
"... Abstract. We state Bennequin inequalities in the relative case, and show that the relative invariants are additive under relative connected sums. We show they exhibit similar limitations as their classical analogues. We study relatively Legendrian simple knots and give some classification results. 1 ..."
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Abstract. We state Bennequin inequalities in the relative case, and show that the relative invariants are additive under relative connected sums. We show they exhibit similar limitations as their classical analogues. We study relatively Legendrian simple knots and give some classification results. 1.
A HOMOLOGICAL APPROACH TO RELATIVE KNOT INVARIANTS
, 909
"... Abstract. We define relative versions of the classical invariants of Legendrian and transverse knots in contact 3manifolds for knots that are homologous to a fixed reference knot. We show these invariants are welldefined and give some basic properties. 1. ..."
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Abstract. We define relative versions of the classical invariants of Legendrian and transverse knots in contact 3manifolds for knots that are homologous to a fixed reference knot. We show these invariants are welldefined and give some basic properties. 1.