Normalized hyperbolic volume of the complement of a hyperbolic knot
The hyperbolic volume of the figure-eight knot is 2.0298832.
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]
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theory, the hyperbolicvolume of a hyperbolic link is the volume of the link's complement with respect to its complete hyperbolic metric. The volume is necessarily...
alternating link is hyperbolic, i.e. the link complement has a hyperbolic geometry, unless the link is a torus link. Thus hyperbolicvolume is an invariant...
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate...
In mathematics, a hyperbolic manifold is a space where every point looks locally like hyperbolic space of some dimension. They are especially studied in...
called knot theory, the volume conjecture is the following open problem that relates quantum invariants of knots to the hyperbolic geometry of knot complements...
Mostow–Prasad rigidity, the hyperbolic structure on the complement of a hyperbolic link is unique, which means the hyperbolicvolume is an invariant for these...
In mathematics, hyperbolic space of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant sectional curvature equal...
volume to prove that hyperbolicvolume decreases under hyperbolic Dehn surgery. Benedetti, Riccardo; Petronio, Carlo (1992), Lectures on hyperbolic geometry...
t}{1-t}}dt=\operatorname {Li} _{2}(1-v).} In hyperbolic geometry the dilogarithm can be used to compute the volume of an ideal simplex. Specifically, a simplex...
grows to infinity, the limit value would give the hyperbolicvolume of the knot complement. (See Volume conjecture.) In 2000 Mikhail Khovanov constructed...
manifold, respectively one of the minimum-volumehyperbolic manifolds with one cusp and the minimum-volumehyperbolic manifold with no cusps. The Whitehead...
This space has a locally Euclidean geometry, so the Hopf link is not a hyperbolic link. The knot group of the Hopf link (the fundamental group of its complement)...
from Dehn surgery on the (−2,3,7) pretzel knot in particular. The hyperbolicvolume of the complement of the (−2,3,8) pretzel link is 4 times Catalan's...
plane parallel to the axis of symmetry is a parabola. The paraboloid is hyperbolic if every other plane section is either a hyperbola, or two crossing lines...
precisely in geometric group theory, a hyperbolic group, also known as a word hyperbolic group or Gromov hyperbolic group, is a finitely generated group...
are a hyperbolic link: the space surrounding the Borromean rings (their link complement) admits a complete hyperbolic metric of finite volume. Although...
Thurston, there were only a handful of known examples of hyperbolic 3-manifolds of finite volume, such as the Seifert–Weber space. The independent and distinct...
hyperbola) then the solid cylinder is said to be parabolic, elliptic and hyperbolic, respectively. For a right circular cylinder, there are several ways in...
invariant. Other hyperbolic invariants include the shape of the fundamental parallelogram, length of shortest geodesic, and volume. Modern knot and link...
Chirality Invertible Crosscap no. Crossing no. Finite type invariant Hyperbolicvolume Khovanov homology Genus Knot group Link group Linking no. Polynomial...
In geometry, hyperbolic motions are isometric automorphisms of a hyperbolic space. Under composition of mappings, the hyperbolic motions form a continuous...
finite variation (a "finite-time singularity") it is said to undergo hyperbolic growth. More precisely, the reciprocal function 1 / x {\displaystyle 1/x}...
In geometry, hyperbolic angle is a real number determined by the area of the corresponding hyperbolic sector of xy = 1 in Quadrant I of the Cartesian plane...
Chirality Invertible Crosscap no. Crossing no. Finite type invariant Hyperbolicvolume Khovanov homology Genus Knot group Link group Linking no. Polynomial...
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