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Hyperbolic link

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In mathematics, a hyperbolic link is a link in the 3-sphere with complement that has a complete Riemannian metric of constant negative curvature, i.e. has a hyperbolic geometry. [1]

32 relations: Alternating knot, Borromean rings, Colin Adams (mathematician), Connected space, Curvature, Dehn surgery, Figure-eight knot (mathematics), Hyperbolic 3-manifold, Hyperbolic Dehn surgery, Hyperbolic geometry, Hyperbolic volume, Knot complement, Link (knot theory), Mathematics, Perko pair, Prime knot, Riemannian manifold, Satellite knot, SnapPea, Split link, Stevedore knot (mathematics), The geometry and topology of three-manifolds, Three-twist knot, Topology (journal), Torus knot, William Menasco, William Thurston, (−2,3,7) pretzel knot, 3-sphere, 6₂ knot, 6₃ knot, 7₄ knot.

Alternating knot

In knot theory, a knot or link diagram is alternating if the crossings alternate under, over, under, over, as one travels along each component of the link.

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Borromean rings

In mathematics, the Borromean rings consist of three topological circles which are linked and form a Brunnian link (i.e., removing any ring results in two unlinked rings).

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Colin Adams (mathematician)

Colin Conrad Adams (born October 13, 1956) is a mathematician primarily working in the areas of hyperbolic 3-manifolds and knot theory.

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Connected space

In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets.

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Curvature

In mathematics, curvature is any of a number of loosely related concepts in different areas of geometry.

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Dehn surgery

In topology, a branch of mathematics, a Dehn surgery, named after Max Dehn, is a construction used to modify 3-manifolds.

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Figure-eight knot (mathematics)

In knot theory, a figure-eight knot (also called Listing's knot or a Cavendish knot) is the unique knot with a crossing number of four.

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Hyperbolic 3-manifold

In mathematics, more precisely in topology and differential geometry, a hyperbolic 3–manifold is a manifold of dimension 3 equipped with a hyperbolic metric, that is a Riemannian metric which has all its sectional curvatures equal to -1.

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Hyperbolic Dehn surgery

In mathematics, hyperbolic Dehn surgery is an operation by which one can obtain further hyperbolic 3-manifolds from a given cusped hyperbolic 3-manifold.

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Hyperbolic geometry

In mathematics, hyperbolic geometry (also called Bolyai–Lobachevskian geometry or Lobachevskian geometry) is a non-Euclidean geometry.

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Hyperbolic volume

In the mathematical field of knot theory, the hyperbolic volume of a hyperbolic link is the volume of the link's complement with respect to its complete hyperbolic metric.

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Knot complement

In mathematics, the knot complement of a tame knot K is the three-dimensional space surrounding the knot.

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Link (knot theory)

In mathematical knot theory, a link is a collection of knots which do not intersect, but which may be linked (or knotted) together.

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Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

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Perko pair

In the mathematical theory of knots, the Perko pair, named after Kenneth Perko, is a pair of entries in classical knot tables that actually represent the same knot.

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Prime knot

In knot theory, a prime knot or prime link is a knot that is, in a certain sense, indecomposable.

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Riemannian manifold

In differential geometry, a (smooth) Riemannian manifold or (smooth) Riemannian space (M,g) is a real, smooth manifold M equipped with an inner product g_p on the tangent space T_pM at each point p that varies smoothly from point to point in the sense that if X and Y are differentiable vector fields on M, then p \mapsto g_p(X(p),Y(p)) is a smooth function.

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Satellite knot

In the mathematical theory of knots, a satellite knot is a knot that contains an incompressible, non boundary-parallel torus in its complement.

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SnapPea

SnapPea is free software designed to help mathematicians, in particular low-dimensional topologists, study hyperbolic 3-manifolds.

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Split link

In the mathematical field of knot theory, a split link is a link that has a (topological) 2-sphere in its complement separating one or more link components from the others.

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Stevedore knot (mathematics)

In knot theory, the stevedore knot is one of three prime knots with crossing number six, the others being the 62 knot and the 63 knot.

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The geometry and topology of three-manifolds

The geometry and topology of three-manifolds is a set of widely circulated but unpublished notes by William Thurston from 1978 to 1980 describing his work on 3-manifolds.

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Three-twist knot

In knot theory, the three-twist knot is the twist knot with three-half twists.

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Topology (journal)

Topology was a peer-reviewed mathematical journal covering topology and geometry.

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Torus knot

In knot theory, a torus knot is a special kind of knot that lies on the surface of an unknotted torus in R3.

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William Menasco

William W. Menasco is a topologist and a professor at the University at Buffalo.

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William Thurston

William Paul Thurston (October 30, 1946August 21, 2012) was an American mathematician.

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(−2,3,7) pretzel knot

In geometric topology, a branch of mathematics, the (−2, 3, 7) pretzel knot, sometimes called the Fintushel–Stern knot (after Ron Fintushel and Ronald J. Stern), is an important example of a pretzel knot which exhibits various interesting phenomena under three-dimensional and four-dimensional surgery constructions.

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3-sphere

In mathematics, a 3-sphere, or glome, is a higher-dimensional analogue of a sphere.

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6₂ knot

In knot theory, the 62 knot is one of three prime knots with crossing number six, the others being the stevedore knot and the 63 knot.

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6₃ knot

In knot theory, the 63 knot is one of three prime knots with crossing number six, the others being the stevedore knot and the 62 knot.

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7₄ knot

In mathematical knot theory, 74 is the name of a 7-crossing knot which can be visually depicted in a highly-symmetric form, and so appears in the symbolism and/or artistic ornamentation of various cultures.

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References

[1] https://en.wikipedia.org/wiki/Hyperbolic_link

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