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Normalized hyperbolic volume of the complement of a hyperbolic 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
Hyperbolic_volume
Conjecture in knot theory relating quantum invariants and hyperbolic geometry
mathematics called knot theory, the volume conjecture is an open problem that relates quantum invariants of knots to the hyperbolic geometry of their complements
Volume_conjecture
Topological complexity in mathematics
that the simplicial volume of a finite volume hyperbolic manifold is proportional to the hyperbolic volume. The simplicial volume is equal to twice the
Simplicial_volume
Type of mathematical link
knot 74 knot 10 161 knot (the "Perko pair" knot) 12n242 knot SnapPea Hyperbolic volume (knot) Colin Adams (1994, 2004) The Knot Book, American Mathematical
Hyperbolic_link
alternating link is hyperbolic, i.e. the link complement has a hyperbolic geometry, unless the link is a torus link. Thus hyperbolic volume is an invariant
Alternating_knot
Space where every point locally resembles a hyperbolic space
In mathematics, a hyperbolic manifold is a space where every point looks locally like hyperbolic space of some dimension. They are especially studied in
Hyperbolic_manifold
Type of non-Euclidean geometry
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate
Hyperbolic_geometry
Unique knot with a crossing number of four
cusped hyperbolic 3-manifolds of minimum volume, Inventiones Mathematicae, 146 (2001), no. 3, 451–478. MR 1869847 Marc Lackenby, Word hyperbolic Dehn surgery
Figure-eight knot (mathematics)
Figure-eight_knot_(mathematics)
Two interlinked loops with five structural crossings
manifold, respectively one of the minimum-volume hyperbolic manifolds with one cusp and the minimum-volume hyperbolic manifold with no cusps. The Whitehead
Whitehead_link
Simplest non-trivial closed knot with three crossings
3 Braid no. 2 Bridge no. 2 Crosscap no. 1 Crossing no. 3 Genus 1 Hyperbolic volume 0 Stick no. 6 Tunnel no. 1 Unknotting no. 1 Conway notation [3] A–B
Trefoil_knot
Manifold of dimension 3 equipped with a hyperbolic metric
of the 3-dimensional hyperbolic space by a discrete group of isometries (a Kleinian group). Hyperbolic 3-manifolds of finite volume have a particular importance
Hyperbolic_3-manifold
sufficient conditions for Dehn surgery to produce a hyperbolic manifold,[L00] a bound on the hyperbolic volume of a knot complement of an alternating knot,[L04]
Marc_Lackenby
Special case of the polylogarithm
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
Dilogarithm
Non-Euclidean geometry
In mathematics, hyperbolic space of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant negative sectional curvature
Hyperbolic_space
Function of a knot that takes the same value for equivalent knots
Mostow–Prasad rigidity, the hyperbolic structure on the complement of a hyperbolic link is unique, which means the hyperbolic volume is an invariant for these
Knot_invariant
Three linked but pairwise separated rings
are a hyperbolic link: the space surrounding the Borromean rings (their link complement) admits a complete hyperbolic metric of finite volume. Although
Borromean_rings
Mathematical invariant of a knot or link
grows to infinity, the limit value would give the hyperbolic volume of the knot complement. (See Volume conjecture.) In 2000 Mikhail Khovanov constructed
Jones_polynomial
Type of mathematical knot
Dehn surgery slopes which give non-hyperbolic 3-manifolds. Among the enumerated knots, the only other hyperbolic knot with 7 or more is the figure-eight
(−2,3,7)_pretzel_knot
Mathematical knot with crossing number 7
9 Braid no. 4 Bridge no. 2 Crosscap no. 2 Crossing no. 7 Genus 1 Hyperbolic volume 2.82812 Stick no. 9 Unknotting no. 1 Conway notation [52] A–B notation
7_2_knot
Kind of operation in knot theory
as they have a number of the same invariants. They have the same hyperbolic volume (by a result of Ruberman), and have the same HOMFLY polynomials. Conway
Mutation_(knot_theory)
Mathematical knot with crossing number 7
9 Braid no. 4 Bridge no. 2 Crosscap no. 3 Crossing no. 7 Genus 1 Hyperbolic volume 5.13794 Stick no. 9 Unknotting no. 2 Conway notation [313] A–B notation
74_knot
Study of mathematical knots
invariant. Other hyperbolic invariants include the shape of the fundamental parallelogram, length of shortest geodesic, and volume. Modern knot and link
Knot_theory
Simplest nontrivial knot link
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)
Hopf_link
Link formed from a finite number of twisted sections
from Dehn surgery on the (−2,3,7) pretzel knot in particular. The hyperbolic volume of the complement of the (−2,3,8) pretzel link is 4 times Catalan's
Pretzel_link
Type of knot in knot theory
Zeitschrift. 65: 133–170. doi:10.1007/bf01473875. Purcell, Jessica (2020). Hyperbolic knot theory. American Mathematical Society. ISBN 978-1-4704-5499-9. Table
2-bridge_knot
Prime knot named for John Horton Conway
Conway knot Braid no. 3 Hyperbolic volume 11.2191 Conway notation .−(3,2).2 Thistlethwaite 11n34 Other hyperbolic, prime, slice (topological only), chiral
Conway_knot
Mathematical knot with crossing number 5
three-twist knot is not fibered. The three-twist knot is a hyperbolic knot, with its complement having a volume of approximately 2.82812. If the fibre of the knot
Three-twist_knot
Growth function exhibiting a singularity at a finite time
finite variation (a "finite-time singularity") it is said to undergo hyperbolic growth. More precisely, the reciprocal function 1 / x {\displaystyle 1/x}
Hyperbolic_growth
Family of mathematical knots
Chirality Invertible Crosscap no. Crossing no. Finite type invariant Hyperbolic volume Khovanov homology Genus Knot group Link group Linking no. Polynomial
Twist_knot
Geometric figure which has infinite surface area but finite volume
hyperbolico acuto, written in 1643, a truncated acute hyperbolic solid, cut by a plane. Volume 1, part 1 of his Opera geometrica published the following
Gabriel's_horn
Polynomials arising in knot theory
Chirality Invertible Crosscap no. Crossing no. Finite type invariant Hyperbolic volume Khovanov homology Genus Knot group Link group Linking no. Polynomial
HOMFLY_polynomial
Three-dimensional solid
hyperbola) then the solid cylinder is said to be parabolic, elliptic and hyperbolic, respectively. For a right circular cylinder, there are several ways in
Cylinder
Isometric automorphisms of a hyperbolic space
In geometry, hyperbolic motions are isometric automorphisms of a hyperbolic space. Under composition of mappings, the hyperbolic motions form a continuous
Hyperbolic_motion
Quadric surface with one axis of symmetry and no center of symmetry
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
Paraboloid
Motif with two doubly-interlinked loops
Basic Solomon's knot Braid length 7 Braid no. 4 Crossing no. 4 Hyperbolic volume 0 Linking no. 2 Stick no. 5 Unknotting no. 2 Conway notation [4] Thistlethwaite
Solomon's_knot
Mathematical knot with crossing number 7
7 Braid no. 2 Bridge no. 2 Crosscap no. 1 Crossing no. 7 Genus 3 Hyperbolic volume 0 Stick no. 9 Unknotting no. 3 Conway notation [7] A–B notation 71
71_knot
hyperbolic Dehn surgery is an operation by which one can obtain further hyperbolic 3-manifolds from a given cusped hyperbolic 3-manifold. Hyperbolic Dehn
Hyperbolic_Dehn_surgery
Non-trivial knot which cannot be written as the knot sum of two non-trivial knots
Chirality Invertible Crosscap no. Crossing no. Finite type invariant Hyperbolic volume Khovanov homology Genus Knot group Link group Linking no. Polynomial
Prime_knot
Theorem in hyperbolic geometry
theorem, essentially states that the geometry of a complete, finite-volume hyperbolic manifold of dimension greater than two is determined by the fundamental
Mostow_rigidity_theorem
Mathematical concept
precisely in geometric group theory, a hyperbolic group, also known as a word hyperbolic group or Gromov hyperbolic group, is a finitely generated group
Hyperbolic_group
Mathematical knot with crossing number 5
5 Braid no. 2 Bridge no. 2 Crosscap no. 1 Crossing no. 5 Genus 2 Hyperbolic volume 0 Stick no. 8 Unknotting no. 2 Conway notation [5] A–B notation 51
Cinquefoil_knot
Link that consists of finitely many unlinked unknots
two-component unlink. Taizo Kanenobu has shown that for all n > 1 there exists a hyperbolic link of n components such that any proper sublink is an unlink (a Brunnian
Unlink
Two-variable polynomial knot invariant
Chirality Invertible Crosscap no. Crossing no. Finite type invariant Hyperbolic volume Khovanov homology Genus Knot group Link group Linking no. Polynomial
Kauffman_polynomial
American mathematician (1946–2012)
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
William_Thurston
Chirality Invertible Crosscap no. Crossing no. Finite type invariant Hyperbolic volume Khovanov homology Genus Knot group Link group Linking no. Polynomial
Bridge_number
groups of complete noncompact hyperbolic manifolds of finite volume. Further generalizations such as acylindrical hyperbolicity are also explored by current
Relatively_hyperbolic_group
Specific knot in knot theory with 11 crossings
Kinoshita–Terasaka knot Crossing no. 11 Genus 2 Hyperbolic volume 11.2191 Thistlethwaite 11n42 Other hyperbolic, prime, slice
Kinoshita–Terasaka_knot
Polynomial invariant of framed links
Chirality Invertible Crosscap no. Crossing no. Finite type invariant Hyperbolic volume Khovanov homology Genus Knot group Link group Linking no. Polynomial
Bracket_polynomial
Knot invariant
Chirality Invertible Crosscap no. Crossing no. Finite type invariant Hyperbolic volume Khovanov homology Genus Knot group Link group Linking no. Polynomial
Alexander_polynomial
Knot which lies on the surface of a torus in 3-dimensional space
also inconsistent with the pictures that appear in: Alternating knot Hyperbolic knot Irrational winding of a torus Satellite knot Torus Knot on Wolfram
Torus_knot
Argument of the hyperbolic functions
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
Hyperbolic_angle
Prime knot with crossing number 10
10 Braid no. 3 Bridge no. 3 Crosscap no. 2 Crossing no. 10 Genus 3 Hyperbolic volume 5.63877 Unknotting no. 3 Conway notation [3:-20:-20] A–B notation
Perko_pair
Group whose operation is a composition of braids
Chirality Invertible Crosscap no. Crossing no. Finite type invariant Hyperbolic volume Khovanov homology Genus Knot group Link group Linking no. Polynomial
Braid_group
Type of mathematical knot
non boundary-parallel torus in its complement. Every knot is either hyperbolic, a torus, or a satellite knot. The class of satellite knots include composite
Satellite_knot
Integer-valued knot invariant; least number of crossings in a knot diagram
Chirality Invertible Crosscap no. Crossing no. Finite type invariant Hyperbolic volume Khovanov homology Genus Knot group Link group Linking no. Polynomial
Crossing_number_(knot_theory)
Mathematical space
cusped hyperbolic 3-manifold of finite volume. It is non-orientable and has the smallest volume among non-compact hyperbolic manifolds, having volume approximately
3-manifold
Number, approximately 0.916
volume of an ideal hyperbolic octahedron, and therefore 1/4 of the hyperbolic volume of the complement of the Whitehead link. It is 1/8 of the volume
Catalan's_constant
Geometric space with four dimensions
three-dimensional space, he specified an alternative perpendicularity, hyperbolic orthogonality. This notion provides his four-dimensional space with a
Four-dimensional_space
Mathematical knot with crossing number 6
therefore also a slice knot. The stevedore knot is a hyperbolic knot, with its complement having a volume of approximately 3.16396. Figure-eight knot (mathematics)
Stevedore_knot_(mathematics)
Property in knot theory
Chirality Invertible Crosscap no. Crossing no. Finite type invariant Hyperbolic volume Khovanov homology Genus Knot group Link group Linking no. Polynomial
Tricolorability
Every knot or link can be represented as a closed braid
Chirality Invertible Crosscap no. Crossing no. Finite type invariant Hyperbolic volume Khovanov homology Genus Knot group Link group Linking no. Polynomial
Alexander's_theorem
Fundamental group of a knot complement
Chirality Invertible Crosscap no. Crossing no. Finite type invariant Hyperbolic volume Khovanov homology Genus Knot group Link group Linking no. Polynomial
Knot_group
Method for representing and evaluating partial differential equations
variation diminishing Finite volume method for unsteady flow LeVeque, Randall (2002). Finite Volume Methods for Hyperbolic Problems. ISBN 9780511791253
Finite_volume_method
Smallest number of edges of an equivalent polygonal path for a knot
Chirality Invertible Crosscap no. Crossing no. Finite type invariant Hyperbolic volume Khovanov homology Genus Knot group Link group Linking no. Polynomial
Stick_number
Spiral asymptotic to a line
A hyperbolic spiral is a type of spiral with a pitch angle that increases with distance from its center, unlike the constant angles of logarithmic spirals
Hyperbolic_spiral
Flat woven decorative knot
8 Braid no. 3 Bridge no. 3 Crosscap no. 4 Crossing no. 8 Genus 3 Hyperbolic volume 12.35090621 Unknotting no. 2 Conway notation [8*] A–B notation 818
Carrick_mat
Invariant of mathematical knots
Chirality Invertible Crosscap no. Crossing no. Finite type invariant Hyperbolic volume Khovanov homology Genus Knot group Link group Linking no. Polynomial
Khovanov_homology
Loop seen as a trivial knot
Chirality Invertible Crosscap no. Crossing no. Finite type invariant Hyperbolic volume Khovanov homology Genus Knot group Link group Linking no. Polynomial
Unknot
Type of mathematical group
{PSL} _{2}(\mathbb {Z} )} . They, and the hyperbolic surface associated to their action on the hyperbolic plane, often exhibit particularly regular behaviour
Arithmetic_Fuchsian_group
Mathematical knot with crossing number 6
{-2}+2q^{-3}-2q^{-4}+q^{-5}.\,} The 62 knot is a hyperbolic knot, with its complement having a volume of approximately 4.40083. Surface of knot 6.2 Ways
62_knot
Geometric mean and hyperbolic angle as coordinates in quadrant I
In mathematics, hyperbolic coordinates are a method of locating points in quadrant I of the Cartesian plane { ( x , y ) : x > 0 , y > 0 } = Q {\displaystyle
Hyperbolic_coordinates
Interlinked multi-loop construction where cutting one loop frees all the others
four-dimensional hyperbolic space, and considers the hyperbolic convex hulls of the circles. These are two-dimensional subspaces of the hyperbolic space, and
Brunnian_link
Tiling of hyperbolic 3-space by uniform polyhedra
complete set of hyperbolic uniform honeycombs. More unsolved problems in mathematics In hyperbolic geometry, a uniform honeycomb in hyperbolic space is a uniform
Uniform honeycombs in hyperbolic space
Uniform_honeycombs_in_hyperbolic_space
Mathematical notation for describing the structure of knots
Chirality Invertible Crosscap no. Crossing no. Finite type invariant Hyperbolic volume Khovanov homology Genus Knot group Link group Linking no. Polynomial
Dowker–Thistlethwaite notation
Dowker–Thistlethwaite_notation
Notation used to describe knots based on operations on tangles
Chirality Invertible Crosscap no. Crossing no. Finite type invariant Hyperbolic volume Khovanov homology Genus Knot group Link group Linking no. Polynomial
Conway_notation_(knot_theory)
Complement of a knot in three-sphere
Chirality Invertible Crosscap no. Crossing no. Finite type invariant Hyperbolic volume Khovanov homology Genus Knot group Link group Linking no. Polynomial
Knot_complement
Chirality Invertible Crosscap no. Crossing no. Finite type invariant Hyperbolic volume Khovanov homology Genus Knot group Link group Linking no. Polynomial
Knot_polynomial
Model of hyperbolic geometry
model, also called the conformal disk model, is a model of 2-dimensional hyperbolic geometry in which all points are inside the unit disk, and straight lines
Poincaré_disk_model
Type of unbounded quadratic surface-shaped building or work
A hyperbolic paraboloid structure (also called a hypar) is a load-bearing structure with the shape of a truncated region of an infinite hyperbolic paraboloid
Hyperboloid_structure
Type of invariant in Knot theory
Chirality Invertible Crosscap no. Crossing no. Finite type invariant Hyperbolic volume Khovanov homology Genus Knot group Link group Linking no. Polynomial
Finite_type_invariant
Collection of knots that do not intersect, but may be linked
case for Milnor's invariants, for instance. Compare with closed braids. Hyperbolic link Knot (mathematics) Link group Unlink Habegger, Nathan; Lin, X.S.
Link_(knot_theory)
) = y − 2 d x d y {\displaystyle d\nu (\tau )=y^{-2}dxdy} is the hyperbolic volume form. The integral is absolutely convergent and the Petersson inner
Petersson_inner_product
Knot that can't be tied in a string of constant diameter
Chirality Invertible Crosscap no. Crossing no. Finite type invariant Hyperbolic volume Khovanov homology Genus Knot group Link group Linking no. Polynomial
Wild_knot
In mathematics, more precisely in group theory and hyperbolic geometry, Arithmetic Kleinian groups are a special class of Kleinian groups constructed using
Arithmetic hyperbolic 3-manifold
Arithmetic_hyperbolic_3-manifold
Attempt to classify and tabulate all possible knots
2010-07-29. Burton, Benjamin A. (2020). "The Next 350 Million Knots". LIPIcs, Volume 164, SoCG 2020. 164: 25:1–25:17. doi:10.4230/LIPICS.SOCG.2020.25. ISSN 1868-8969
Knot_tabulation
Linear map that preserves areas
1) ⊂ SL(2) – of the subgroup of hyperbolic rotations in the special linear group of transforms preserving area and orientation (a volume form). In the language
Squeeze_mapping
Iranian mathematician (1977–2017)
problem of counting simple closed geodesics on hyperbolic Riemann surfaces by finding a relationship to volume calculations on moduli space. Geodesics are
Maryam_Mirzakhani
Orientable surface whose boundary is a knot or link
Brittenham, Mark (24 September 1998). "Bounding canonical genus bounds volume". arXiv:math/9809142. Agol, Ian; Hass, Joel; Thurston, William (2002-05-19)
Seifert_surface
Chirality Invertible Crosscap no. Crossing no. Finite type invariant Hyperbolic volume Khovanov homology Genus Knot group Link group Linking no. Polynomial
Crosscap_number
Invariant of a knot diagram
Chirality Invertible Crosscap no. Crossing no. Finite type invariant Hyperbolic volume Khovanov homology Genus Knot group Link group Linking no. Polynomial
Writhe
Chirality Invertible Crosscap no. Crossing no. Finite type invariant Hyperbolic volume Khovanov homology Genus Knot group Link group Linking no. Polynomial
List_of_prime_knots
Category of coordinate systems
In the hyperbolic plane, as in the Euclidean plane, each point can be uniquely identified by two real numbers. Several qualitatively different ways of
Coordinate systems for the hyperbolic plane
Coordinate_systems_for_the_hyperbolic_plane
Branch of mathematics
between points in the Euclidean plane, while the hyperbolic metric measures the distance in the hyperbolic plane. Other important examples of metrics include
Geometry
Type of mathematical knot
Chirality Invertible Crosscap no. Crossing no. Finite type invariant Hyperbolic volume Khovanov homology Genus Knot group Link group Linking no. Polynomial
Ribbon_knot
Three dimensional analogue of uniformization conjecture
product of a hyperbolic surface with a circle, or more generally the mapping torus of an isometry of a hyperbolic surface. Finite volume manifolds with
Geometrization_conjecture
Mathematical tool for studying knots
Chirality Invertible Crosscap no. Crossing no. Finite type invariant Hyperbolic volume Khovanov homology Genus Knot group Link group Linking no. Polynomial
Skein_relation
One of three types of isotopy-preserving local changes to a knot diagram
Chirality Invertible Crosscap no. Crossing no. Finite type invariant Hyperbolic volume Khovanov homology Genus Knot group Link group Linking no. Polynomial
Reidemeister_move
Invariant of framed knots
Chirality Invertible Crosscap no. Crossing no. Finite type invariant Hyperbolic volume Khovanov homology Genus Knot group Link group Linking no. Polynomial
Self-linking_number
Embedding of the circle in three dimensional Euclidean space
Chirality Invertible Crosscap no. Crossing no. Finite type invariant Hyperbolic volume Khovanov homology Genus Knot group Link group Linking no. Polynomial
Knot_(mathematics)
How many times curves wind around each other
Chirality Invertible Crosscap no. Crossing no. Finite type invariant Hyperbolic volume Khovanov homology Genus Knot group Link group Linking no. Polynomial
Linking_number
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