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. The volume is necessarily a finite real number, and is a topological invariant of the link. [1] As a link invariant, it was first studied by William Thurston in connection with his geometrization conjecture. [2]
A hyperbolic link is a link in the 3-sphere whose complement (the space formed by removing the link from the 3-sphere) can be given a complete Riemannian metric of constant negative curvature, giving it the structure of a hyperbolic 3-manifold, a quotient of hyperbolic space by a group acting freely and discontinuously on it. The components of the link will become cusps of the 3-manifold, and the manifold itself will have finite volume. By Mostow rigidity, when a link complement has a hyperbolic structure, this structure is uniquely determined, and any geometric invariants of the structure are also topological invariants of the link. In particular, the hyperbolic volume of the complement is a knot invariant. In order to make it well-defined for all knots or links, the hyperbolic volume of a non-hyperbolic knot or link is often defined to be zero.
There are only finitely many hyperbolic knots for any given volume. [2] A mutation of a hyperbolic knot will have the same volume, [3] so it is possible to concoct examples with equal volumes; indeed, there are arbitrarily large finite sets of distinct knots with equal volumes. [2] In practice, hyperbolic volume has proven very effective in distinguishing knots, utilized in some of the extensive efforts at knot tabulation. Jeffrey Weeks's computer program SnapPea is the ubiquitous tool used to compute hyperbolic volume of a link. [1]
| Knot/link | Volume | Reference |
|---|---|---|
| Figure-eight knot | [4] | |
| Three-twist knot | 2.82812 | [ citation needed ] |
| Stevedore knot | 3.16396 | [ citation needed ] |
| 62 knot | 4.40083 | [ citation needed ] |
| Endless knot | 5.13794 | [ citation needed ] |
| Perko pair | 5.63877 | [ citation needed ] |
| 63 knot | 5.69302 | [ citation needed ] |
| Borromean rings | [4] |
More generally, the hyperbolic volume may be defined for any hyperbolic 3-manifold. The Weeks manifold has the smallest possible volume of any closed manifold (a manifold that, unlike link complements, has no cusps); its volume is approximately 0.9427. [5]
Thurston and Jørgensen proved that the set of real numbers that are hyperbolic volumes of 3-manifolds is well-ordered, with order type ωω. [6] The smallest limit point in this set of volumes is given by the knot complement of the figure-eight knot, [7] and the smallest limit point of limit points is given by the complement of the Whitehead link. [8]
William Paul Thurston was an American mathematician. He was a pioneer in the field of low-dimensional topology and was awarded the Fields Medal in 1982 for his contributions to the study of 3-manifolds.
In knot theory, a figure-eight knot is the unique knot with a crossing number of four. This makes it the knot with the third-smallest possible crossing number, after the unknot and the trefoil knot. The figure-eight knot is a prime knot.
In mathematics, Thurston's geometrization conjecture states that each of certain three-dimensional topological spaces has a unique geometric structure that can be associated with it. It is an analogue of the uniformization theorem for two-dimensional surfaces, which states that every simply connected Riemann surface can be given one of three geometries.
In mathematics, low-dimensional topology is the branch of topology that studies manifolds, or more generally topological spaces, of four or fewer dimensions. Representative topics are the structure theory of 3-manifolds and 4-manifolds, knot theory, and braid groups. This can be regarded as a part of geometric topology. It may also be used to refer to the study of topological spaces of dimension 1, though this is more typically considered part of continuum theory.
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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. It is generally required that this metric be also complete: in this case the manifold can be realised as a quotient of the 3-dimensional hyperbolic space by a discrete group of isometries.
Algorithmic topology, or computational topology, is a subfield of topology with an overlap with areas of computer science, in particular, computational geometry and computational complexity theory.
In mathematics, a Kleinian group is a discrete subgroup of the group of orientation-preserving isometries of hyperbolic 3-space H3. The latter, identifiable with PSL(2, C), is the quotient group of the 2 by 2 complex matrices of determinant 1 by their center, which consists of the identity matrix and its product by −1. PSL(2, C) has a natural representation as orientation-preserving conformal transformations of the Riemann sphere, and as orientation-preserving conformal transformations of the open unit ball B3 in R3. The group of Möbius transformations is also related as the non-orientation-preserving isometry group of H3, PGL(2, C). So, a Kleinian group can be regarded as a discrete subgroup acting on one of these spaces.
In mathematics, a hyperbolic manifold is a space where every point looks locally like hyperbolic space of some dimension. They are especially studied in dimensions 2 and 3, where they are called hyperbolic surfaces and hyperbolic 3-manifolds, respectively. In these dimensions, they are important because most manifolds can be made into a hyperbolic manifold by a homeomorphism. This is a consequence of the uniformization theorem for surfaces and the geometrization theorem for 3-manifolds proved by Perelman.
In knot theory, the Whitehead link, named for J. H. C. Whitehead, is one of the most basic links. It can be drawn as an alternating link with five crossings, from the overlay of a circle and a figure-eight shaped loop.
Colin Conrad Adams is an American mathematician primarily working in the areas of hyperbolic 3-manifolds and knot theory. His book, The Knot Book, has been praised for its accessible approach to advanced topics in knot theory. He is currently Francis Christopher Oakley Third Century Professor of Mathematics at Williams College, where he has been since 1985. He writes "Mathematically Bent", a column of math for the Mathematical Intelligencer. His nephew is popular American singer Still Woozy.
In mathematics, the Weeks manifold, sometimes called the Fomenko–Matveev–Weeks manifold, is a closed hyperbolic 3-manifold obtained by (5, 2) and (5, 1) Dehn surgeries on the Whitehead link. It has volume approximately equal to 0.942707… and David Gabai, Robert Meyerhoff, and Peter Milley showed that it has the smallest volume of any closed orientable hyperbolic 3-manifold. The manifold was independently discovered by Jeffrey Weeks as well as Sergei V. Matveev and Anatoly T. Fomenko.
In mathematics, hyperbolic Dehn surgery is an operation by which one can obtain further hyperbolic 3-manifolds from a given cusped hyperbolic 3-manifold. Hyperbolic Dehn surgery exists only in dimension three and is one which distinguishes hyperbolic geometry in three dimensions from other dimensions.
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In mathematics, more precisely in group theory and hyperbolic geometry, Arithmetic Kleinian groups are a special class of Kleinian groups constructed using orders in quaternion algebras. They are particular instances of arithmetic groups. An arithmetic hyperbolic three-manifold is the quotient of hyperbolic space by an arithmetic Kleinian group.
James W. Cannon is an American mathematician working in the areas of low-dimensional topology and geometric group theory. He was an Orson Pratt Professor of Mathematics at Brigham Young University.
In hyperbolic geometry, the ending lamination theorem, originally conjectured by William Thurston, states that hyperbolic 3-manifolds with finitely generated fundamental groups are determined by their topology together with certain "end invariants", which are geodesic laminations on some surfaces in the boundary of the manifold.
In three-dimensional hyperbolic geometry, an ideal polyhedron is a convex polyhedron all of whose vertices are ideal points, points "at infinity" rather than interior to three-dimensional hyperbolic space. It can be defined as the convex hull of a finite set of ideal points. An ideal polyhedron has ideal polygons as its faces, meeting along lines of the hyperbolic space.
Robert F. Riley was an American mathematician. He is known for his work in low-dimensional topology using computational tools and hyperbolic geometry, being one of the inspirations for William Thurston's later breakthroughs in 3-dimensional topology.
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