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Type of knot in knot theory
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
2-bridge_knot
Family of mathematical knots
Every twist knot is also a 2-bridge knot. Of the twist knots, only the unknot and the stevedore knot are slice knots. A twist knot with n {\displaystyle n}
Twist_knot
Bridge in Meixi Lake District, Changsha
Lucky knot bridge (or knot bridge or knot footbridge) spans the Dragon King Harbor River in Meixi Lake District, Changsha, China. The 185 m long and 24
Lucky_Knot_Bridge
Simplest non-trivial closed knot with three crossings
3)-torus knot lying on torus ( r − 2 ) 2 + z 2 = 1 {\displaystyle (r-2)^{2}+z^{2}=1} : x = ( 2 + cos 3 t ) cos 2 t y = ( 2 + cos 3 t ) sin 2 t z =
Trefoil_knot
mathematical field of knot theory, the bridge number, also called the bridge index, is an invariant of a knot defined as the minimal number of bridges required in
Bridge_number
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
Study of mathematical knots
In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope,
Knot_theory
Knot defined by parametric equations defining Lissajous curves
and figure-eight knot are not Lissajous. No torus knot can be Lissajous. No fibered 2-bridge knot can be Lissajous. Bogle, M. G. V.; Hearst, J. E.; Jones
Lissajous_knot
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
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
Embedding of the circle in three dimensional Euclidean space
term knot is also applied to embeddings of S j in Sn, especially in the case j = n − 2. The branch of mathematics that studies knots is known as knot theory
Knot_(mathematics)
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
Type of mathematical knot
mathematical theory of knots, a satellite knot is a knot that contains an incompressible, non boundary-parallel torus in its complement. Every knot is either hyperbolic
Satellite_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)
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
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
Non-trivial knot which cannot be written as the knot sum of two non-trivial knots
In knot theory, a prime knot or prime link is a knot that is, in a certain sense, indecomposable. Specifically, it is a non-trivial knot which cannot
Prime_knot
Integer-valued knot invariant; least number of crossings in a knot diagram
mathematical area of knot theory, the crossing number of a knot is the smallest number of crossings of any diagram of the knot. It is a knot invariant. By way
Crossing_number_(knot_theory)
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
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)
Mathematical knot with crossing number 7
case they are not knots at all.) The 74 knot is a Lissajous knot, representable for example by the parametric equation x = cos ( 2 t + 0.22 ) y = cos
74_knot
Binding hitch knot
The constrictor knot is one of the most effective binding knots. Simple and secure, it is a harsh knot that can be difficult or impossible to untie once
Constrictor_knot
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
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
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
Mathematical knot with crossing number 5
In knot theory, the cinquefoil knot, also known as Solomon's seal knot or the pentafoil knot, is one of two knots with crossing number five, the other
Cinquefoil_knot
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
Type of mathematical knot
(−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
(−2,3,7)_pretzel_knot
American mathematician
William Thurston, Incompressible surfaces in 2-bridge knot complements, Inventiones Mathematicae 79 (1985), no. 2, 225–246. Allen Hatcher, A proof of the Smale
Allen_Hatcher
Complement of a knot in three-sphere
In mathematics, the knot complement of a tame knot K is the space where the knot is not. If a knot is embedded in the 3-sphere, then the complement is
Knot_complement
Prime knot with crossing number 10
McCoy: On 2-bridge knots with differing smooth and topological slice genera, Proc. Amer. Math. Soc. 144, p. 5435–5442, 2016. "10_161", The Knot Atlas. Pictures
Perko_pair
Mathematical invariant of a knot or link
link which assigns to each oriented knot or link a Laurent polynomial in the variable t 1 / 2 {\displaystyle t^{1/2}} with integer coefficients. Suppose
Jones_polynomial
Three linked but pairwise separated rings
Intelligencer, 37 (2): 20–25, doi:10.1007/s00283-014-9499-4, MR 3356112, S2CID 558993 "Borromean rings", The Knot Atlas. Rolfsen, Dale (1990), Knots and Links
Borromean_rings
Type of mathematical knot
In the mathematical area of knot theory, a ribbon knot is a knot that bounds a self-intersecting disk with only ribbon singularities. Intuitively, this
Ribbon_knot
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
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
219–238. Teragaito, Masakazu and Hirasawa, Mikami. "Crosscap numbers of 2-bridge knots," Arxiv:math.GT/0504446. J.Uhing. "Zur Kreuzhaubenzahl von Knoten",
Crosscap_number
Knot that can't be tied in a string of constant diameter
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
Knot that is not equivalent to its mirror image
field of knot theory, a chiral knot is a knot that is not equivalent to its mirror image (when identical while reversed). An oriented knot that is equivalent
Chiral_knot
Type of knot
A miller's knot (also sack knot or bag knot) is a binding knot used to secure the opening of a sack or bag. Historically, large sacks often contained
Miller's_knot
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)
2025 ship collision in New York City, U.S.
from 3.3 knots to 5.1 knots, and the harbor pilot called for nearby tugboat assistance." Videos show the ship's three masts impacting the bridge one by
Cuauhtémoc–Brooklyn Bridge collision
Cuauhtémoc–Brooklyn_Bridge_collision
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
Simplest nontrivial knot link
Hopf link is a (2,2)-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
Hopf_link
Type of knot
The half hitch is a simple hitch knot, where the working end of a line is brought over and under the standing part. Insecure on its own, it is a valuable
Half_hitch
Fundamental group of a knot complement
a knot is an embedding of a circle into 3-dimensional Euclidean space. The knot group of a knot K is defined as the fundamental group of the knot complement
Knot_group
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
2024 bridge collapse near Baltimore, Maryland, US
bridge, as part of its emergency procedures. At 1:28:45 a.m., the ship struck the southwest pier of the central truss arch span, at roughly 8 knots (9
Francis Scott Key Bridge collapse
Francis_Scott_Key_Bridge_collapse
Type of mathematical link
knot (the figure-eight knot) 52 knot (the three-twist knot) 61 knot (the stevedore knot) 62 knot 63 knot 74 knot 10 161 knot (the "Perko pair" knot)
Hyperbolic_link
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)
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
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
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
Minimum number of times a specific knot must be passed through itself to become untied
first few knots: Trefoil knot unknotting number 1 Figure-eight knot unknotting number 1 Cinquefoil knot unknotting number 2 Three-twist knot unknotting
Unknotting_number
Link formed from a finite number of twisted sections
In the mathematical theory of knots, a pretzel link is a special kind of link. It consists of a finite number of tangles made of two intertwined circular
Pretzel_link
Mathematical tool for studying knots
tool used to study knots. A central question in the mathematical theory of knots is whether two knot diagrams represent the same knot. One way to answer
Skein_relation
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
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)
Property in knot theory
In the mathematical field of knot theory, the tricolorability of a knot is the ability of a knot to be colored with three colors subject to certain rules
Tricolorability
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
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
Knot invariant named after Cahit Arf
In the mathematical field of knot theory, the Arf invariant of a knot, named after Cahit Arf, is a knot invariant obtained from a quadratic form associated
Arf_invariant_of_a_knot
Encyclopedic website dedicated to knot theory
The Knot Atlas is a website, an encyclopedia rather than atlas, dedicated to knot theory. It and its predecessor were created by mathematician Dror Bar-Natan
The_Knot_Atlas
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
Genus of parasitic worms
Root-knot nematodes are plant-parasitic nematodes from the genus Meloidogyne. They exist in soil in areas with hot climates or short winters. About 2000
Root-knot_nematode
Orientable surface whose boundary is a knot or link
boundary is a given knot or link. Such surfaces can be used to study the properties of the associated knot or link. For example, many knot invariants are most
Seifert_surface
Type of invariant in Knot theory
mathematical theory of knots, a finite type invariant, or Vassiliev invariant (so named after Victor Anatolyevich Vassiliev), is a knot invariant that can
Finite_type_invariant
Link that consists of finitely many unlinked unknots
unlink in Wiktionary, the free dictionary. In the mathematical field of knot theory, an unlink is a link that is equivalent (under ambient isotopy) to
Unlink
Notation used to describe knots based on operations on tangles
clear. It composes a knot using certain operations on tangles to construct it. In Conway notation, the tangles are generally algebraic 2-tangles. This means
Conway_notation_(knot_theory)
Polynomial invariant of framed links
In the mathematical field of knot theory, the bracket polynomial (also known as the Kauffman bracket) is a polynomial invariant of framed links. Although
Bracket_polynomial
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
knot/Cinquefoil knot, (5,2)-torus knot, Solomon's seal knot, pentafoil knot - a prime knot with crossing number five which can be arranged as a {5/2}
List of mathematical knots and links
List_of_mathematical_knots_and_links
Invariant of mathematical knots
relation to knot polynomials from the view point of Khovanov homology. The skein relation for three links L 1 , L 2 {\displaystyle L_{1},L_{2}} and L 3
Khovanov_homology
Suspension bridge in Kobe, Japan
The Akashi Kaikyo Bridge (Japanese: 明石海峡大橋, Hepburn: Akashi Kaikyō Ōhashi) is a suspension bridge that links the city of Kobe on the Japanese island of
Akashi_Kaikyo_Bridge
Analog of the knot group
In knot theory, an area of mathematics, the link group of a link is an analog of the knot group of a knot. They were described by John Milnor in his Ph
Link_group
Every knot or link can be represented as a closed braid
In mathematics Alexander's theorem states that every knot or link can be represented as a closed braid; that is, a braid in which the corresponding ends
Alexander's_theorem
Class of mathematical knot with special properties
theory of knots, a Berge knot (named after mathematician John Berge) or doubly primitive knot is any member of a particular family of knots in the 3-sphere
Berge_knot
Operation on a knot
of knots, a flype is a kind of manipulation of knot and link diagrams used in the Tait flyping conjecture. It consists of twisting a part of a knot, a
Flype
as knot theory, an invertible knot is a knot that can be continuously deformed to itself, but with its orientation reversed. A non-invertible knot is
Invertible_knot
of knot theory, a knot polynomial is a knot invariant in the form of a polynomial whose coefficients encode some of the properties of a given knot. The
Knot_polynomial
Kind of operation in knot theory
field of knot theory, a mutation is an operation on a knot that can produce different knots. Suppose K is a knot given in the form of a knot diagram.
Mutation_(knot_theory)
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
Interlinked multi-loop construction where cutting one loop frees all the others
In knot theory, a branch of topology, a Brunnian link is a nontrivial link that becomes a set of trivial unlinked circles if any one component is removed
Brunnian_link
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
American film by Mike Flanagan
of the True Knot, a cult that feeds on people with psychic powers. Ferguson said she had difficulty filming the scene where the True Knot attacks a young
Doctor_Sleep_(2019_film)
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
1890 short story by Ambrose Bierce
"An Occurrence at Owl Creek Bridge" is an 1890 short story by American writer and American Civil War veteran Ambrose Bierce, described as "one of the most
An Occurrence at Owl Creek Bridge
An_Occurrence_at_Owl_Creek_Bridge
Village in Mallorca, Spain
in 1933, which features many hairpin bends and a 270° spiral bridge called the tie knot. The climb is officially called the Coll del Reis or the Coll
Sa_Calobra
Two-variable polynomial knot invariant
In knot theory, the Kauffman polynomial is a 2-variable knot polynomial due to Louis Kauffman. It is initially defined on a link diagram as F ( K ) (
Kauffman_polynomial
Generalization of knots in 3-dimensional Euclidean space
problems in mathematics In knot theory, a virtual knot is a generalization of knots in 3-dimensional Euclidean space, R3, to knots in thickened surfaces Σ
Virtual_knot
Bridge in New York City
The Brooklyn Bridge is a cable-stayed suspension bridge in New York City, spanning the East River between the boroughs of Manhattan and Brooklyn. Opened
Brooklyn_Bridge
Method for tying a scarf for wear
The Hoxton knot, Chelsea knot, French loop, Parisian scarf knot or Snug Tug is a method of arranging a scarf about the neck. The scarf is doubled back
Hoxton_knot
Former railway station in England
The station opened on 5 December 1870 as the terminus of the Garstang and Knot-End Railway when it opened the 7 miles 1 chain (11.3 km) long line from Garstang
Pilling_railway_station
Japanese manga series by Marcey Naito
Tying the Knot with an Amagami Sister (Japanese: 甘神さんちの縁結び, Hepburn: Amagami-san Chi no Enmusubi; lit. 'Matchmaking of the Amagami Household') is a Japanese
Tying the Knot with an Amagami Sister
Tying_the_Knot_with_an_Amagami_Sister
Link of three loops with ten crossings
In the mathematical theory of knots, L10a140 is the name in the Thistlethwaite link table of a link of three loops, which has ten crossings between the
L10a140_link
Canadian bridge from Prince Edward Island to the mainland
The Confederation Bridge (French: Pont de la Confédération) is a box girder bridge carrying the Trans-Canada Highway across the Abegweit Passage of the
Confederation_Bridge
In knot theory, a knot move or operation is a change or changes which preserve crossing number. Operations are used to investigate whether knots are equivalent
Knot_operation
Proteins with backbone entangled in a knot
the knots discovered in proteins are deep trefoil (31) knots. Figure eight knots (41), three-twist knots (52), Stevedore knots (61) and Septoil knot (71)
Knotted_protein
Mathematical invariants used to classify plane curves
In mathematics, particularly in topology and knot theory, Arnold invariants are invariants introduced by Vladimir Arnold in 1994 for studying the topology
Arnold_invariants
Concept in knot theory
In mathematical knot theory, a Conway sphere, named after John Horton Conway, is a 2-sphere intersecting a given knot or link in a 3-manifold transversely
Conway_sphere
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