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Knot which lies on the surface of a torus in 3-dimensional space
In knot theory, a torus knot is a special kind of knot that lies on the surface of an unknotted torus in R3. Similarly, a torus link is a link which lies
Torus_knot
Simplest non-trivial closed knot with three crossings
3t\end{aligned}}} The (2,3)-torus knot is also a trefoil knot. The following parametric equations give a (2,3)-torus knot lying on torus ( r − 2 ) 2 + z 2 = 1
Trefoil_knot
Non-trivial knot which cannot be written as the knot sum of two non-trivial knots
The trefoil is actually a (2, 3)-torus knot. The figure-eight knot, with four crossings, is the simplest non-torus knot. For any positive integer n, there
Prime_knot
Type of mathematical knot
isotopic to the central core curve of the solid torus. Then tie up the solid torus into a nontrivial knot. This means there is a non-trivial embedding f
Satellite_knot
Mathematical knot with crossing number 7
In knot theory, the 71 knot, also known as the septoil knot, the septafoil knot, or the (7, 2)-torus knot, is one of seven prime knots with crossing number
71_knot
Doughnut-shaped surface of revolution
is called a torus of revolution, also known as a ring torus. If the axis of revolution is tangent to the circle, the surface is a horn torus. If the axis
Torus
Knot that can't be tied in a string of constant diameter
mathematical theory of knots, a knot is tame if it can be "thickened", that is, if there exists an extension to an embedding of the solid torus S 1 × D 2 {\displaystyle
Wild_knot
Unique knot with a crossing number of four
In knot theory, a figure-eight knot (also called Listing's knot) is the unique knot with a crossing number of four. This makes it the knot with the third-smallest
Figure-eight knot (mathematics)
Figure-eight_knot_(mathematics)
Complement of a knot in three-sphere
solid torus. The knot complement is then the complement of N, X K = M − interior ( N ) . {\displaystyle X_{K}=M-{\mbox{interior}}(N).} The knot complement
Knot_complement
Fundamental group of a knot complement
q)-torus knot has knot group with presentation ⟨ x , y ∣ x p = y q ⟩ . {\displaystyle \langle x,y\mid x^{p}=y^{q}\rangle .} The figure eight knot has
Knot_group
Mathematical knot with crossing number 5
the three-twist knot. It is listed as the 51 knot in the Alexander-Briggs notation, and can also be described as the (5,2)-torus knot. The cinquefoil
Cinquefoil_knot
Family of mathematical knots
knots, and are considered the simplest type of knots after the torus knots. A twist knot is obtained by linking together the two ends of a twisted loop
Twist_knot
knot - (2,3)-torus knot, the two loose ends of a common overhand knot joined together 41 knot/Figure-eight knot (mathematics) - a prime knot with a crossing
List of mathematical knots and links
List_of_mathematical_knots_and_links
Loop seen as a trivial knot
of knots, the unknot, not knot, or trivial knot, is the least knotted of all knots. Intuitively, the unknot is a closed loop of rope without a knot tied
Unknot
known as dynamical systems theory, a linear flow on the torus is a flow on the n-dimensional torus T n = S 1 × S 1 × ⋯ × S 1 ⏟ n , {\displaystyle \mathbb
Linear_flow_on_the_torus
Study of mathematical knots
significance (though in each number of crossings the twist knot comes after the torus knot). Links are written by the crossing number with a superscript to denote
Knot_theory
Integer-valued knot invariant; least number of crossings in a knot diagram
particular knot out of those with this many crossings is meant (this sub-ordering is not based on anything in particular, except that torus knots then twist
Crossing_number_(knot_theory)
Type of mathematical link
torus link is hyperbolic by a result of William Menasco. 41 knot (the figure-eight knot) 52 knot (the three-twist knot) 61 knot (the stevedore knot)
Hyperbolic_link
Orientable surface whose boundary is a knot or link
unknot is the only knot with genus zero. The trefoil knot has genus 1, as does the figure-eight knot. The genus of a (p, q)-torus knot is (p − 1)(q − 1)/2
Seifert_surface
Class of ornamental knots
double overhand knot. A two lead, three bight Turk's head is also a trefoil knot if the ends are joined together. (2,n) alternating torus knots are (2,n) Turk's
Turk's_head_knot
Smooth closed surface with g holes
In mathematics, a genus g surface (also known as a g-torus or g-holed torus) is a surface formed by the connected sum of g distinct tori: the interior
Genus_g_surface
concept of a knot. Two classes of knots: torus knots and pretzel knots Cinquefoil knot also known as a (5, 2) torus knot. Figure-eight knot (mathematics)
List_of_knot_theory_topics
Mathematical knot with crossing number 5
In knot theory, the three-twist knot is the twist knot with three-half twists. It is listed as the 52 knot in the Alexander-Briggs notation, and is one
Three-twist_knot
Property in knot theory
with a tricolorable separable component is also tricolorable. If the torus knot/link denoted by (m,n) is tricolorable, then so are (j*m,i*n) and (i*n
Tricolorability
Simplest nontrivial knot link
2)-torus link with the braid word σ 1 2 {\displaystyle \sigma _{1}^{2}} . The knot complement of the Hopf link is R × S1 × S1, the cylinder over a torus
Hopf_link
For p = q {\displaystyle p=q} the knot is a torus knot. In braid form these knots can be defined in a square solid torus (i.e. the cube [ − 1 , 1 ] 3 {\displaystyle
Lissajous-toric_knot
Knot invariant
sections of the curves passing through them. Torus links are subject to similar considerations as torus knots, discussed above. The ropelength of Borromean
Ropelength
Types of knots (and links) Torus knot Prime knot Alternating knot Hyperbolic link Knot invariants Crossing number Linking number Skein relation Knot polynomials
List of geometric topology topics
List_of_geometric_topology_topics
Conjecture in knot theory relating quantum invariants and hyperbolic geometry
knot (Tobias Ekholm), The three-twist knot (Rinat Kashaev and Yoshiyuki Yokota), The Borromean rings (Stavros Garoufalidis and Thang Le), Torus knots
Volume_conjecture
Mathematical knot with crossing number 7
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
74_knot
Knot that is not equivalent to its mirror image
be chiral by Max Dehn. All nontrivial torus knots are chiral. The Alexander polynomial cannot distinguish a knot from its mirror image, but the Jones polynomial
Chiral_knot
Minimum number of times a specific knot must be passed through itself to become untied
{\displaystyle (p,q)} -torus knot is equal to ( p − 1 ) ( q − 1 ) / 2 {\displaystyle (p-1)(q-1)/2} . The unknotting numbers of prime knots with nine or fewer
Unknotting_number
Mathematical knot with crossing number 6
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. The stevedore knot
Stevedore_knot_(mathematics)
Theorem that the slice genus of the (p, q) torus knot is (p-1)(q-1)/2
In knot theory, the Milnor conjecture says that the slice genus of the ( p , q ) {\displaystyle (p,q)} torus knot is ( p − 1 ) ( q − 1 ) 2 . {\displaystyle
Milnor conjecture (knot theory)
Milnor_conjecture_(knot_theory)
Artistic representation of three circular leaf shapes used in architecture
Hidden Mickey Quatrefoil Shamrock Trefoil arch Trefoil domain Trefoil knot Torus knot The French terms 'quartefeuille' and 'quintefeuille' are translated
Trefoil
Proposed tube-like space habitat
several times around the local star, in a geometric figure known as a torus knot. Topopolises are also called cosmic spaghetti. A topopolis with big enough
Topopolis
Knot defined by parametric equations defining Lissajous curves
domains, for instance in a cylinder or in a (flat) solid torus (Lissajous-toric knot). Because a knot cannot be self-intersecting, the three integers n x
Lissajous_knot
Topics referred to by the same term
overhand knot Double torus knot Double windsor Karash double loop This disambiguation page lists articles associated with the title Double knot. If an internal
Double_knot
Research institute in Kerala, India
Indian Institute of Technology Palakkad Inspired from the (5,3) torus knot, the logo highlights the notion of synergy and dynamic inter-relationship between
IIT_Palakkad
Group of isotopy classes of a topological automorphism group
can be realized as symmetry groups of knots. The symmetry group of a torus knot is known to be of order two Z2. Notice that there is an induced action
Mapping_class_group
Prime knot named for John Horton Conway
In mathematics, specifically in knot theory, the Conway knot (or Conway's knot) is a particular knot with 11 crossings, named after John Horton Conway
Conway_knot
List of concrete topologies and topological spaces
Irrational winding of a torus/Irrational cable on a torus Knot (mathematics) Linear flow on the torus Space-filling curve Torus knot Wild knot The following topologies
List_of_topologies
Mathematical knot with crossing number 6
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. This knot is sometimes
62_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
Alternating_knot
Smallest number of edges of an equivalent polygonal path for a knot
Jin determined the stick number of a ( p , q ) {\displaystyle (p,q)} -torus knot T ( p , q ) {\displaystyle T(p,q)} in case the parameters p {\displaystyle
Stick_number
3D computer graphics program
an array of values called knots specifies the extent of influence of each control vertex (CV) on the curve or surface. Knots are invisible in 3D space
Autodesk_3ds_Max
Type of mathematical knot
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
(−2,3,7)_pretzel_knot
Topics referred to by the same term
sagittal torus, a structure found in crania Torus, a structure of the xylem Solid torus, a solid whose surface is a torus. Torus knot Algebraic torus Umbilic
Torus_(disambiguation)
Flat woven decorative knot
The carrick mat is a flat woven decorative knot which can be used as a mat or pad. Its name is based on the mat's decorative-type carrick bend with the
Carrick_mat
Mathematical knot with crossing number 6
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. It is alternating
63_knot
Formulas in differential geometry
N in green, B in purple) along Viviani's curve. On the example of a torus knot, the tangent vector T, the normal vector N, and the binormal vector B
Frenet–Serret_formulas
Operation combining two oriented knots
omitted. Smooth knots, for example, are always tame. A framed knot is the extension of a tame knot to an embedding of the solid torus D2 × S1 in S3. The
Knot_(mathematics)
British mathematician (born 1963)
Milnor, that the four-ball genus of a ( p , q ) {\displaystyle (p,q)} -torus knot is ( p − 1 ) ( q − 1 ) / 2 {\displaystyle (p-1)(q-1)/2} . They then went
Peter_B._Kronheimer
crosscap number of the trefoil knot is 1, as it bounds a Möbius strip and is not trivial. The crosscap number of a torus knot was determined by M. Teragaito
Crosscap_number
The unknot is the only knot with tunnel number 0. The trefoil knot has tunnel number 1. In general, any nontrivial torus knot has tunnel number 1. Every
Tunnel_number
Chaotic model of atmospheric convection
values of ρ, the system displays knotted periodic orbits. For example, with ρ = 99.96 it becomes a T(3,2) torus knot. In Figure 4 of his paper, Lorenz
Lorenz_system
Motif with two doubly-interlinked loops
classified as a link, and is not a true knot according to the definitions of mathematical knot theory. The Solomon's knot consists of two closed loops, which
Solomon's_knot
Mathematical knot with crossing number 7
In knot theory, the Pentatwist knot, also known as the five-twist knot, or the 72, is one of seven prime knots with crossing number seven. It is the fifth
7_2_knot
American mathematician
factor. He showed that on the contrary, the stretch factor of certain torus knots could be arbitrarily large. His proof was published in the Annals of
John_Pardon
Manifold of dimension 3 equipped with a hyperbolic metric
example, any knot which is not either a satellite knot or a torus knot is hyperbolic. Moreover, almost all Dehn surgeries on a hyperbolic knot yield a hyperbolic
Hyperbolic_3-manifold
Number of "holes" of a surface
genus is the number of "holes" of a surface. A sphere has genus 0, while a torus has genus 1. The genus of a connected, orientable surface is an integer
Genus_(mathematics)
diagrammatic representation of a G-coloring reduces to a Fox n-coloring. The torus knot T(3,5) has only constant n-colorings, but for the group G equal to the
Fox_n-coloring
Molecule whose structure resembles a knot
"Molecular 5-2 Knot". www.catenane.net. Retrieved 2023-12-03. Inomata, Yuuki; Sawada, Tomohisa; Fujita, Makoto (January 2020). "Metal-Peptide Torus Knots from Flexible
Molecular_knot
Group of vegetation programming and modeling software products
Emergent Game Technologies and the OGRE open-source rendering engine by Torus Knot.[citation needed] SpeedTree won a Scientific and Technical Academy Award
SpeedTree
Group whose operation is a composition of braids
§ Introduction). Example applications of braid groups include knot theory, where any knot may be represented as the closure of certain braids (a result
Braid_group
Prime knot with crossing number 10
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. In Dale
Perko_pair
Connected sum of two trefoil knots with opposite chirality
In knot theory, the square knot is a composite knot obtained by taking the connected sum of a trefoil knot with its reflection. It is closely related
Square_knot_(mathematics)
Mathematical invariant of a knot or link
of knot theory, the Jones polynomial is a knot polynomial discovered by Vaughan Jones in 1984. Specifically, it is an invariant of an oriented knot or
Jones_polynomial
Connected sum of two trefoil knots with same chirality
In knot theory, the granny knot is a composite knot obtained by taking the connected sum of two identical trefoil knots. It is closely related to the square
Granny_knot_(mathematics)
Computational organic chemist at Imperial College London
PMID 19050778. Rzepa, H. S. (2009). "Wormholes in chemical space connecting torus knot and torus link π-electron density topologies". Physical Chemistry Chemical
Henry_Rzepa
Normalized hyperbolic volume of the complement of a hyperbolic knot
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_volume
Branch of mathematics
a topologist cannot distinguish a coffee mug from a doughnut. A pliable torus (shaped like a doughnut) can be reshaped to a coffee mug by creating a dimple
Topology
Mathematical parameter of embeddings
his research showing that there is no upper bound on the distortion of torus knots, solving a problem originally posed by Mikhail Gromov. In the study of
Stretch_factor
Polynomials arising in knot theory
field of knot theory, the HOMFLY polynomial or HOMFLYPT polynomial, sometimes called the generalized Jones polynomial, is a 2-variable knot polynomial
HOMFLY_polynomial
Knot that bounds an embedded disk in 4-space
A slice knot is a mathematical knot in 3-dimensional space that bounds an embedded disk in 4-dimensional space. A knot K ⊂ S 3 {\displaystyle K\subset
Slice_knot
Three linked but pairwise separated rings
the "Ballantine rings". The first work of knot theory to include the Borromean rings was a catalog of knots and links compiled in 1876 by Peter Tait.
Borromean_rings
In knot theory, prime knots are those knots that are indecomposable under the operation of knot sum. The prime knots with ten or fewer crossings are listed
List_of_prime_knots
Peter Guthrie Tait in his study of knots. The Tait conjectures involve concepts in knot theory such as alternating knots, chirality, and writhe. All of the
Tait_conjectures
Mapping which preserves all topological properties of a given space
square and a circle are homeomorphic to each other, but a sphere and a torus are not. However, this description can be misleading. Some continuous deformations
Homeomorphism
longitude for each boundary torus, i.e. simple closed curves that are generators for the fundamental group of the torus. Let M ( u 1 , u 2 , … , u n
Hyperbolic_Dehn_surgery
American computer scientist
doubling systems can form an infinite number of topologically distinct torus knots and described the structure of their stable and unstable manifolds. From
Steve_Omohundro
Bridge number 2 In the mathematical field of knot theory, a 2-bridge knot is a knot which can be regular isotoped so that the natural height function given
2-bridge_knot
Continuous deformation between two continuous functions
embeddings, f and g, of the torus into R3. X is the torus, Y is R3, f is some continuous function from the torus to R3 that takes the torus to the embedded surface-of-a-doughnut
Homotopy
Knot invariant
a knot invariant which assigns a polynomial with integer coefficients to each knot type. James Waddell Alexander II discovered this, the first knot polynomial
Alexander_polynomial
Italian-British applied mathematician
explicit torus knot solutions to integrable equations of hydrodynamic type, and he contributed to determine new relations between energy of knotted fields
Renzo_L._Ricca
Collection of knots that do not intersect, but may be linked
mathematical knot theory, a link is a collection of knots that do not intersect, but which may be linked (or knotted) together. A knot can be described
Link_(knot_theory)
Function of a knot that takes the same value for equivalent knots
mathematical field of knot theory, a knot invariant is a quantity (in a broad sense) defined for each knot which is the same for equivalent knots. The equivalence
Knot_invariant
Particular knot energy
In mathematics, the Möbius energy of a knot is a particular knot energy, i.e., a functional on the space of knots. It was discovered by Jun O'Hara, who
Möbius_energy
One of three types of isotopy-preserving local changes to a knot diagram
In the mathematical area of knot theory, a Reidemeister move is any of three local moves on a link diagram. Kurt Reidemeister (1927) and, independently
Reidemeister_move
Operation on a knot producing a link with two components
Seifert surfaces. The Bing double of a knot K is defined by placing the Bing double of the unknot in the solid torus surrounding it, as shown in the figure
Bing_double
Mathematical knot
In knot theory, a branch of mathematics, a knot or link K {\displaystyle K} in the 3-dimensional sphere S 3 {\displaystyle S^{3}} is called fibered or
Fibered_knot
Attempt to classify and tabulate all possible knots
tabulate all possible knots. By 1998, all 1.7 million prime knots up to 16 crossings had been tabulated, and by 2020 all 350 million knots up to 19 crossings
Knot_tabulation
Two interlinked loops with five structural crossings
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
Whitehead_link
Process in mathematics of decomposing a topological space
the mapping torus of an Anosov map of a torus has a finite volume sol structure, but its JSJ decomposition cuts it open along one torus to produce a
JSJ_decomposition
Invariant of mathematical knots
abutting to their instanton knot Floer homology group and used this to show that Khovanov Homology (like the instanton knot Floer homology) detects the
Khovanov_homology
Invariant of a knot diagram
In knot theory, there are several competing notions of the quantity writhe, or Wr {\displaystyle \operatorname {Wr} } . In one sense, it is purely a property
Writhe
How many times curves wind around each other
the form of the linking integral. It is an important object of study in knot theory, algebraic topology, and differential geometry, and has numerous applications
Linking_number
Prominent bulge of the occipital bone at the back of the skull
Asturians and Basques. Although the terms occipital bun and occipital torus (torus occipitale), as well as all the other names used in this article, are
Occipital_bun
Planetary nebula in the constellation Aquarius
size of the inner disk is 8×19 arcmin in diameter (0.52 pc); the outer torus is 12×22 arcmin in diameter (0.77 pc); and the outer-most ring is about
Helix_Nebula
Type of group in mathematics
b^{-1}a^{m}b=a^{n}\rangle } where m , n ≠ 0 {\displaystyle m,n\neq 0} . Torus knot group G = ⟨ a , b ∣ a p = b q ⟩ {\displaystyle G=\langle a,b\mid a^{p}=b^{q}\rangle
One-relator_group
TORUS KNOT
TORUS KNOT
Male
Japanese
(å¾¹)Â Japanese name TORU means "penetrating; wayfarer." Compare with another form of Toru.
Boy/Male
Japanese
Sea.
Boy/Male
American, Australian, Gujarati, Indian, Kannada
Light
Boy/Male
Biblical English
Strength; rock; sharp.
Boy/Male
Egyptian
God of the sky.
Male
Egyptian
, Horus the Child.
Female
Egyptian
, house of Horus.
Male
Egyptian
, house of Horus.
Boy/Male
Hindu
Bull
Girl/Female
Greek
Descendant of Dorus.
Male
Egyptian
, Horus; the sun.
Male
Egyptian
, ("falcon"); son of Osiris and Isis.
Male
Egyptian
, Horus in Victory.
Biblical
strength; rock; sharp
Female
Egyptian
, house of Horus.
Male
Egyptian
, Horus the Supreme.
Boy/Male
American, British, English, Jamaican, Norse
Thunder Ruler; Form of Thor
Girl/Female
Hindu, Indian
Rhythm
Boy/Male
Egyptian
Disguise of Horus.
Girl/Female
Greek
Descendant of Dorus.
TORUS KNOT
TORUS KNOT
Boy/Male
Indian
Worshipers
Boy/Male
Arabic, Muslim
Beautiful; Radiant
Girl/Female
Biblical
Perfection, truth.
Boy/Male
Hindu
Brilliant, Another name for the city of benaras, Balaji
Boy/Male
Hindu, Indian, Tamil, Traditional
Aim; Peaceful; Satisfaction
Girl/Female
English American
Abbreviation of Thomasina and Tamara.
Boy/Male
French
Chubby cheeks.
Girl/Female
American, British, English
From the Meadow of the Royal Fortress; Cyneburg's Field
Boy/Male
Muslim
Nick name of abdur - Rehman
Girl/Female
Danish
Feminine of Neils.
TORUS KNOT
TORUS KNOT
TORUS KNOT
TORUS KNOT
TORUS KNOT
n.
A torus.
n.
Tufa. See under Tufa, and Toph.
n.
pl. of Sorus.
a.
Torose.
n.
Tonicity, or tone; as, muscular tonus.
n.
Same as Torus.
n.
A lage molding used in the bases of columns. Its profile is semicircular. See Illust. of Molding.
n.
Tophus.
n.
The berry or fruit of any tree of the genus Morus; also, the tree itself. See Morus.
v. t.
A turn; a revolution; as, the tours of the heavenly bodies.
n.
See 3d Tore, 2.
n.
One of the fruit dots, or small clusters of sporangia, on the back of the fronds of ferns.
pl.
of Sorus
n.
A genus of trees, some species of which produce edible fruit; the mulberry. See Mulberry.
n.
The receptacle of a flower; a torus.
pl.
of Torus
n.
One of the ventral parapodia of tubicolous annelids. It usually has the form of an oblong thickening or elevation of the integument with rows of uncini or hooks along the center. See Illust. under Tubicolae.
n.
The state of healthy tension or partial contraction of muscle fibers while at rest; tone; tonus.
n.
A heavy silk with a dull finish; as, gros de Naples; gros de Tours.
n.
The receptacle, or part of the flower on which the carpels stand.