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Complement of a knot in three-sphere
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 the 3-sphere
Knot_complement
Fundamental group of a knot complement
the knot group is the fundamental group of its complement in S 3 {\displaystyle S^{3}} . Two equivalent knots have isomorphic knot groups, so the knot group
Knot_group
Simplest non-trivial closed knot with three crossings
In knot theory, a branch of mathematics, the trefoil knot is the simplest example of a nontrivial knot. The trefoil can be obtained by joining the two
Trefoil_knot
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
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)
Operation combining two oriented knots
framing integer. Given a knot in the 3-sphere, the knot complement is all the points of the 3-sphere not contained in the knot. A major theorem of Gordon
Knot_(mathematics)
Function of a knot that takes the same value for equivalent knots
invariants associated with the knot complement include the knot group which is just the fundamental group of the complement. The knot quandle is also a complete
Knot_invariant
Knot invariant
a knot in the 3-sphere. Let X be the infinite cyclic cover of the knot complement of K. This covering can be obtained by cutting the knot complement along
Alexander_polynomial
was asking what non-diagrammatic properties of the knot complement would characterize alternating knots. In November 2015, Joshua Evan Greene published a
Alternating_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
American mathematician (1946–2012)
the hyperbolic structure of the figure-eight knot complement. He showed that the figure-eight knot complement could be decomposed as the union of two regular
William_Thurston
Knot which lies on the surface of a torus in 3-dimensional space
boundary circle. The knot complement of the (p, q) -torus knot deformation retracts to the space X. Therefore, the knot group of a torus knot has the presentation
Torus_knot
Concept in mathematical knot theory
linear sum of colored Jones polynomial of surgery presentations of the knot complement. Finite type invariant Kontsevich invariant Kashaev's invariant
Quantum_invariant
1991 mathematical film
Not Knot is a 16-minute film on the mathematics of knot theory and low-dimensional topology, centered on and titled after the concept of a knot complement
Not_Knot
Topics referred to by the same term
given angle Knot complement Complement of a point, the dilation of a point in the centroid of a given triangle, with ratio −1/2 Complement (set theory)
Complement
Mathematical knot with crossing number 5
polynomial is not monic, the three-twist knot is not fibered. The three-twist knot is a hyperbolic knot, with its complement having a volume of approximately
Three-twist_knot
Two tame knots with homeomorphic complements are the same or mirror images
Gordon–Luecke theorem on knot complements states that if the complements of two tame knots are homeomorphic, then the knots are equivalent. In particular
Gordon–Luecke_theorem
Branch of topology
3-manifolds. The knot complement of a tame knot K is the three-dimensional space surrounding the knot. To make this precise, suppose that K is a knot in a three-manifold
Low-dimensional_topology
Mathematical knot with crossing number 6
The stevedore knot is a ribbon knot, and is therefore also a slice knot. The stevedore knot is a hyperbolic knot, with its complement having a volume
Stevedore_knot_(mathematics)
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
contact structure. Lissajous knot Ribbon knot Satellite knot Slice knot Torus knot Transverse knot Twist knot Virtual knot Wild knot Borromean rings, the simplest
List_of_knot_theory_topics
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
metres Beam: 10.2 metres Draught: 3.3 metres Speed: 32 knots Range: 2,400 miles at 14 knots Complement: 32 including 6 officers Armament: 1 × AK-176 76 mm
List of corvette classes in service
List_of_corvette_classes_in_service
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
Simplest nontrivial knot link
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. This space
Hopf_link
Group presentations useful in knot theory
The open subspace which is the complement of the knot, S 3 ∖ K {\displaystyle S^{3}\setminus K} is the knot complement. Its fundamental group π 1 ( S
Wirtinger_presentation
Three linked but pairwise separated rings
proved hyperbolic, in the 1970s, and this link complement was a central example in the video Not Knot, produced in 1991 by the Geometry Center. Hyperbolic
Borromean_rings
hyperbolic manifold,[L00] a bound on the hyperbolic volume of a knot complement of an alternating knot,[L04] and a proof that every diagram of the unknot can be
Marc_Lackenby
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
invariant for knots, as proven in [Waldhausen 1968]. The square knot and the granny knot are distinct knots, and have non-homeomorphic complements. However
Peripheral_subgroup
Smallest closed orientable hyperbolic 3-manifold
sibling manifold, or sister, of the figure-eight knot complement. The figure eight knot's complement and its sibling have the smallest volume of any orientable
Weeks_manifold
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
Mathematical knot with crossing number 6
The 62 knot is a hyperbolic knot, with its complement having a volume of approximately 4.40083. Surface of knot 6.2 Ways to assemble of knot 6.2 Example
62_knot
Determining whether a knot is the unknot
determine whether a knot is the unknot by testing all sequences of Pachner moves of this length, starting from the complement of the given knot, and determining
Unknotting_problem
this early period, knot theory primarily consisted of study into the knot group and homological invariants of the knot complement. In 1961 Wolfgang Haken
History_of_knot_theory
Topological invariant in knot theory
the knot K. Slice knots are known to have zero signature. Knot signatures can also be defined in terms of the Alexander module of the knot complement. Let
Signature_of_a_knot
Gieseking manifold has a double cover homeomorphic to the figure-eight knot complement. The underlying compact manifold has a Klein bottle boundary, and the
Gieseking_manifold
Mathematician and topologist
David Gabai as advisor (thesis: Extending fibrations of knot complements to ribbon disk complements). After completing her doctoral degree, Miller worked
Maggie_Miller_(mathematician)
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
Topological invariant of manifolds that can distinguish homotopy-equivalent manifolds
show that the (twisted) Alexander polynomial of knots is the Reidemeister torsion of its knot complement in S 3 {\displaystyle S^{3}} . (Milnor 1962) For
Analytic_torsion
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
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
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 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
Analog of the knot group
group of the link complement's fundamental group, since one can start with elements of the fundamental group, and then by knotting or unknotting components
Link_group
Operation used to modify three-dimensional topological spaces
_{i}} : every longitude is chosen so that it is null-homologous in the knot complement—equivalently, if it is the boundary of a Seifert surface. When the
Dehn_surgery
Concept in knot theory
the knot complement. Sometimes, this condition is included in the definition of Conway spheres. Gordon, Cameron McA.; Luecke, John (2006). "Knots with
Conway_sphere
Conjecture in knot theory relating quantum invariants and hyperbolic geometry
called knot theory, the volume conjecture is an open problem that relates quantum invariants of knots to the hyperbolic geometry of their complements. Let
Volume_conjecture
Kazakh navy warship class
vessels in the class have a displacement of 240 tons, a top speed of 30 knots, and are armed with "modernized anti-aircraft missile and artillery units
Kazakhstan-class_missile_boat
American mathematician (born 1956)
cusped orientable hyperbolic 3-manifolds are precisely the figure-eight knot complement and its sibling manifold. Adams has investigated and defined a variety
Colin_Adams_(mathematician)
Link formed from a finite number of twisted sections
result 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 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
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
polynomial of a knot is a generator of the Fitting ideal of the first homology of the infinite abelian cover of the knot complement. The zeroth Fitting
Fitting_ideal
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
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
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
American mathematician (1888–1971)
homology of a "cyclic covering" of the knot complement. From this invariant, he defined the first of the polynomial knot invariants. With Garland Briggs, he
James_Waddell_Alexander_II
Shape in hyperbolic geometry
the knot complements of hyperbolic links, which have a cusp for each component of the link. For example, the complement of the figure-eight knot is associated
Ideal_polyhedron
Mathemical concept
obtained by ( 5 , 1 ) {\displaystyle (5,1)} surgery on the figure-8 knot complement. It was introduced by Robert Meyerhoff (1987) as a possible candidate
Meyerhoff_manifold
US Navy Knox class frigate
designed speed of 27 knots (50 km/h; 31 mph). The Knox class had a range of 4,500 nautical miles (8,300 km; 5,200 mi) at a speed of 20 knots (37 km/h; 23 mph)
USS_Miller_(FF-1091)
Way to divide polygon into smaller parts
alternating knot or link complement has a subdivision rule, with some tiles that do not subdivide, corresponding to the boundary of the link complement. The
Finite_subdivision_rule
Two interlinked loops with five structural crossings
the Whitehead link can produce the sibling manifold of the complement of the figure-eight knot, and Dehn filling on both components can produce the Weeks
Whitehead_link
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)
topology topics. Knot (mathematics) Link (knot theory) Wild knots Examples of knots (and links) Unknot Trefoil knot Figure-eight knot (mathematics) Borromean
List of geometric topology topics
List_of_geometric_topology_topics
was done by Gromov. The figure-eight knot and the (-2, 3, 7) pretzel knot are the only two knots whose complements are known to have more than 6 exceptional
Hyperbolic_Dehn_surgery
Volume conjecture relating quantum invariants of knots to the hyperbolic geometry of their knot complements. Whitehead conjecture: every connected subcomplex
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
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
Austrian mathematician (1865–1945)
(2-forms) According to Hornich (1948). I.e. the fundamental group of a knot complement. According to Zaremba himself: see the "mixed boundary condition" entry
Wilhelm_Wirtinger
Inland buoy tender of the United States Coast Guard
Her maximum speed is 10.5 knots. She can travel 1,500 nautical miles at five knots without refueling. The ship's complement is eight enlisted personnel
USCGC_Elderberry
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
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)
American mathematician
behavior of knot invariants like bridge number when knots are combined by the connected sum operation, and the Kakimizu complexes of knot complements and other
Jennifer_Schultens
foliation of the knot complement, which can be certified with a taut sutured manifold hierarchy. Given an incompressible Seifert surface S for a knot K, then the
Incompressible_surface
Kriegsmarine battleship class
Scharnhorst's forward Seetakt radar. By 10:00, Scharnhorst, using her 4–6 knot speed advantage, broke off the engagement and resumed searching for the convoy
Scharnhorst-class_battleship
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
American mathematician
parabolics led him to discover the hyperbolic structure on the complement of the figure-eight knot and some others. This was one of the few examples of hyperbolic
Robert_Riley_(mathematician)
Ise-class battleship
horsepower (34,000 kW) and give the ships a speed of 23 knots (43 km/h; 26 mph). Ise reached 23.6 knots (43.7 km/h; 27.2 mph) from 56,498 shp (42,131 kW) during
Japanese_battleship_Ise
Symplectic topology tool
008. S2CID 17245314. Rasmussen, Jacob (2003). "Floer homology and knot complements". arXiv:math/0306378. Salamon, Dietmar; Wehrheim, Katrin (2008). "Instanton
Floer_homology
Class of Indian minehunters
configuration 2 × diesel engines and 2 × electric motors. Maximum speed: 20 knots Complement: <75 Armament: 1 × 76 mm naval gun, 2 × 30 mm CIWS or DEW, 2 × 12.7 mm
Future Mine Counter Measure Vessels (India)
Future_Mine_Counter_Measure_Vessels_(India)
Mathematical concept
S 3 ∖ K ) {\displaystyle \pi _{1}(S^{3}\setminus K)} of nontrivial knot complements fall into this category and therefore are not hyperbolic. This is also
Hyperbolic_group
PC-461-class submarine chaser of the U.S. Navy
that of World War I "four-piper" destroyers, but were half the size and complement of their big sisters. On 19 May 1954 she was decommissioned and transferred
USS_PC-1168
Dreadnought battleship class of the United States Navy
producing a series of studies with speeds of 20 knots (37 km/h; 23 mph), 20.5 knots (38 km/h; 24 mph), and 21 knots and main batteries that ranged from eight
Nevada-class_battleship
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
Knot that can't be tied in a string of constant diameter
In the 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
Wild_knot
American mathematician
Hatcher and William Thurston, Incompressible surfaces in 2-bridge knot complements, Inventiones Mathematicae 79 (1985), no. 2, 225–246. Allen Hatcher
Allen_Hatcher
Four patrol vessels
sufficient for an operational range of 1,000 nm at speeds of 24 knots and 2,000 nm at speeds of 15 knots, both with 10% remaining fuel. Electrical power is supplied
San_Juan-class_patrol_vessel
Class of dreadnought battleship
and the boiler rooms enlarged to increase speed by 0.5 knots (0.93 km/h; 0.58 mph) to 23 knots (43 km/h; 26 mph). To save weight the forecastle deck was
Ise-class_battleship
Short story by Pu Songling
his cousin and He, apparently having become heterosexual, tie the knot. Complementing the tale is a "Jesting Judgement" by Pu Songling; the poem echoes
Huang_Jiulang
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
Family of mathematical knots
In knot theory, a branch of mathematics, a twist knot is a knot obtained by repeatedly twisting a closed loop and then linking the ends together. (That
Twist_knot
1970s modification of Project 61 anti-submarine destroyers
8 m (16 ft) Propulsions - 2 shaft; COGAG; 4 gas-turbines, 35 knots maximum. Complement - 320 Armament - Surface-to-surface missiles: 4 SS-N-2C launchers;
Mod_Kashin-class_destroyer
French Navy hydrography survey vessel
(1840 kW). Radar : Navigational radar DECCA 1226. Electric plant : 620 kW. Complement : 4 officers, 18 petty officers, 15 crew members, 11 hydrographers. Armament:
French_ship_Lapérouse_(A791)
Replenishment oiler of the Spanish Navy
vessel a maximum speed of 20 knots (37 km/h; 23 mph) and a range of 13,450 nautical miles (24,909 km; 15,478 mi) at 20 knots. The replenishment oiler is
Spanish_oiler_Patiño
Leander-class cruiser
aboard Annapolis, required an immediate operation for appendicitis and the 7 knot speed of Annapolis would not enable it to reach Bermuda in time. The two
HMS_Orion_(85)
Math research center at the University of Minnesota
construction of mathematical movies including: Not Knot, exploring hyperbolic space rendering of knot complements. Outside In, a movie about sphere eversion based
Geometry_Center
30-pounder Parrott rifle, and five 24-pounder howitzers. She was crewed by a complement of 41 officers and enlisted men. The first of the Pawtuxet-class cutters
USRC_Wayanda
Ise-class battleship
horsepower (34,000 kW) and give the ships a speed of 23 knots (43 km/h; 26 mph). Hyūga reached 24 knots (44 km/h; 28 mph) from 63,211 shp (47,136 kW) during
Japanese_battleship_Hyūga
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
Decomposition of a compact oriented 3-manifold by dividing it into two handlebodies
Heegaard splittings of hyperbolic three-manifolds which are two-bridge knot complements. Computational methods can be used to determine or approximate the
Heegaard_splitting
KNOT COMPLEMENT
KNOT COMPLEMENT
Girl/Female
British, English
Fearless; Brave
Boy/Male
Arabic, Finnish
Knot
Girl/Female
Hindu, Indian, Kannada, Marathi, Sanskrit, Telugu
Knot
Female
Egyptian
, the wife of Necho I. (?).
Male
Danish
, knot.
Male
Scandinavian
Variant spelling of Scandinavian Knut, CNUT means "knot."Â
Boy/Male
Danish, Dutch, Finnish, German, Norse, Polish, Scandinavian, Swedish
Race; Kind; Knot
Male
Danish
, knot.
Surname or Lastname
English, German, and Dutch
English, German, and Dutch : variant spelling of Knopp.Polish : occupational name for a weaver, Polish knap (see Knapik).Jewish (Ashkenazic) : metonymic occupational name from Yiddish knop ‘button’ (see Knopf).
Male
Scandinavian
Scandinavian form of Old Norse Knútr, KNUT means "knot."Â
Boy/Male
Norse Scandinavian Teutonic
Knot.
Surname or Lastname
English
English : from the Middle English personal name Knut, of Scandinavian origin.German : variant of Knoth.
Boy/Male
American, British, Christian, English
From the Hills; Hill
Boy/Male
Norse
Knot.
Boy/Male
Finnish, German
Knot; White-haired
Boy/Male
Norse
Knot.
Boy/Male
Norse Scandinavian Swedish
Knot.
Girl/Female
Tamil
Knot
Boy/Male
English
From the hills.
Male
Norwegian
Norwegian variant form of Scandinavian Knut, KNUTE means "knot."Â
KNOT COMPLEMENT
KNOT COMPLEMENT
Girl/Female
German, Swedish
Mighty with a Spear; Strength of the Spear
Girl/Female
Scottish
From the east.
Boy/Male
Gaelic
Son of the handsome man.
Boy/Male
Hindu
Who is with light/glance
Girl/Female
Arabic, Muslim
Name of a Sahabiyah RA
Boy/Male
Danish, French, German, Swedish
Army of Power; People of Power; Commander of the Army
Male
Celtic
, high king paramount and commander.
Boy/Male
English
From the heath.
Boy/Male
Muslim
Boy/Male
Hindu, Indian, Telugu
Famous; Scholar; Lord Ganesh; One with Fame
KNOT COMPLEMENT
KNOT COMPLEMENT
KNOT COMPLEMENT
KNOT COMPLEMENT
KNOT COMPLEMENT
n.
A knob, lump, swelling, or protuberance.
v. t.
To form into a knot, or into knots; to tie together, as cord; to fasten by tying.
n.
A knoblike ornament or handle; as, the knob of a lock, door, or drawer.
n.
A rounded hill or mountain; as, the Pilot Knob.
n.
Something not easily solved; an intricacy; a difficulty; a perplexity; a problem.
v. i.
To form knots or joints, as in a cord, a plant, etc.; to become entangled.
v. t.
To unite closely; to connect; to engage; as, hearts knit together in love.
v. i.
To knit knots for fringe or trimming.
n.
A nautical mile, or 6080.27 feet; as, when a ship goes eight miles an hour, her speed is said to be eight knots.
n.
A kind of epaulet. See Shoulder knot.
n.
A cluster of persons or things; a collection; a group; a hand; a clique; as, a knot of politicians.
v. t.
To unite closely; to knit together.
n.
A portion of a branch of a tree that forms a mass of woody fiber running at an angle with the grain of the main stock and making a hard place in the timber. A loose knot is generally the remains of a dead branch of a tree covered by later woody growth.
v. t.
To punish with the knout.
n.
A division of the log line, serving to measure the rate of the vessel's motion. Each knot on the line bears the same proportion to a mile that thirty seconds do to an hour. The number of knots which run off from the reel in half a minute, therefore, shows the number of miles the vessel sails in an hour.
imp. & p. p.
of Knit
v. t.
To tie in or with, or form into, a knot or knots; to form a knot on, as a rope; to entangle.
n.
See Knop.
n.
A knob; a bud; a bunch; a button.