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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
Prime knot named for John Horton Conway
the knot is not a smoothly slice knot, though it is topologically slice (the Kinoshita–Terasaka knot is both). Weisstein, Eric W. "Conway's Knot". mathworld
Conway_knot
Simplest non-trivial closed knot with three crossings
(2,3)-torus knot. It is also the knot obtained by closing the braid σ13. The trefoil is an alternating knot. However, it is not a slice knot, meaning it
Trefoil_knot
Type of mathematical knot
ribbon knot if f | M : M → R {\displaystyle f_{|M}\colon M\to \mathbb {R} } has no interior local maxima. Every ribbon knot is known to be a slice knot. A
Ribbon_knot
Study of mathematical knots
smoothly slice knots are referred to as slice. There are other types of knots, such as rationally slice, which are not necessarily smoothly slice.) A ribbon
Knot_theory
Mathematical knot with crossing number 6
not fibered. 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
Stevedore_knot_(mathematics)
Property of knots in mathematics
In mathematics, the slice genus of a smooth knot K in S3 (sometimes called its Murasugi genus or 4-ball genus) is the least integer g such that K is the
Slice_genus
Topics referred to by the same term
Communications Engine Slice category, in category theory, a special case of a comma category Slice genus, in knot theory Slice knot, in knot theory Slice sampling
Slice
Specific knot in knot theory with 11 crossings
In knot theory, the Kinoshita–Terasaka knot is a particular prime knot with 11 crossings. It is named after Japanese mathematicians Shinichi Kinoshita
Kinoshita–Terasaka_knot
American mathematician
known for solving a long-standing problem in knot theory by proving that the Conway knot is not smoothly slice. Piccirillo was raised in Greenwood, Maine
Lisa_Piccirillo
Connected sum of two trefoil knots with opposite chirality
groups. Unlike the granny knot, the square knot is a ribbon knot, and it is therefore also a slice knot. Weisstein, Eric W. "Square Knot". MathWorld.
Square_knot_(mathematics)
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
Knot invariant named after Cahit Arf
the Arf invariant of a slice knot vanishes. Kauffman (1987) p.74 Kauffman (1987) pp.75–78 Jones, Vaughan F. R. (1990), "Knot theory and statistical mechanics"
Arf_invariant_of_a_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)
Family of mathematical knots
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} half-twists
Twist_knot
Operation on a knot producing a link with two components
namesake, the American mathematician R. H. Bing. The Bing double of a slice knot is a slice link, though it is unknown whether the converse is true. The components
Bing_double
Connected sum of two trefoil knots with same chirality
Unlike the square knot, the granny knot is not a ribbon knot or a slice knot. The Alexander polynomial of the granny knot is Δ ( t ) = ( t − 1 + t − 1 ) 2
Granny_knot_(mathematics)
Link equivalence relation weaker than isotopy but stronger than homotopy
submanifolds to be not just abstractly cobordant, but "cobordant in N". Slice knot Habegger, Nathan; Masbaum, Gregor (2000), "The Kontsevich integral and
Link_concordance
Greek myth; metaphor for tangled problem
difference how the knot was loosed. Sources from antiquity disagree on his solution. In one version of the story, he drew his sword and sliced it in half with
Gordian_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
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
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
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
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
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)
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
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
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
Topological invariant in knot theory
of 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
Signature_of_a_knot
Embedding of the circle in three dimensional Euclidean space
mathematics, a knot is an embedding of the circle (S1) into three-dimensional Euclidean space, R3 (also known as E3). Often two knots are considered equivalent
Knot_(mathematics)
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 given
2-bridge_knot
Mathematical knot
Martin; Thompson, Abigail (2010). "Fibered knots and potential counterexamples to the property 2R and slice-ribbon conjectures". Geometry & Topology. 14
Fibered_knot
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
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
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
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
Postulate in particle physics
tangled knot, traced out by the one electron. Any given moment in time is represented by a slice across spacetime, and would meet the knotted line a great
One-electron_universe
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
Mathematics determining that the Conway knot is not a smoothly slice knot, answering an unsolved problem in knot theory first proposed over fifty years
Timeline of women in mathematics in the United States
Timeline_of_women_in_mathematics_in_the_United_States
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
Prime knot with crossing number 10
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
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
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
American mathematician (1913–1973)
responsible for introducing several basic phrases to knot theory: the phrases slice knot, ribbon knot, and Seifert circle all appear in print for the first
Ralph_Fox
Mathematics determining that the Conway knot is not a smoothly slice knot, answering an unsolved problem in knot theory first proposed over fifty years
Timeline of women in mathematics
Timeline_of_women_in_mathematics
Invariant of mathematical knots
using Khovanov homology. This integer valued invariant of a knot gives a bound on the slice genus, and is sufficient to prove the Milnor conjecture. In
Khovanov_homology
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
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
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)
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
Smallest number of edges of an equivalent polygonal path for a knot
of knots, the stick number is a knot invariant that intuitively gives the smallest number of straight "sticks" stuck end to end needed to form a knot. Specifically
Stick_number
Town in Maine, United States
Piccirillo, Mathematician known for determining that the Conway knot is not a slice knot Addison Emery Verrill, Yale University professor of zoology, born
Greenwood,_Maine
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
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
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
Simplest nontrivial knot link
In mathematical knot theory, the Hopf link is the simplest nontrivial link with more than one component. It consists of two circles linked together exactly
Hopf_link
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
Invariant of a quadratic form over a field of characteristic 2
direct summand), and so is a knot invariant. It is additive under connected sum, and vanishes on slice knots, so is a knot concordance invariant. The intersection
Arf_invariant
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
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
Experimental SWATH vessel
HSV Sea Slice was an experimental vessel, built by Lockheed Martin, for the United States Navy, later used in commercial service. Based on a variant of
Sea_Slice
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
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
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
American mathematician
Christopher William (2015). "Counterexamples to Kauffman's conjectures on slice knots". Advances in Mathematics. 274: 263–284. arXiv:1303.4418. doi:10.1016/j
Tim_Cochran
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
Binding around the end of a rope to prevent it from fraying
A whipping knot or whipping is a binding of marline twine or whipcord around the end of a rope to prevent its natural tendency to fray. Some whippings
Whipping_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
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
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
School in Bethel, Oxford, Maine, United States
Piccirillo - Mathematician known for determining that the Conway knot is not a slice knot Anna Willard - Professional Runner District property valuations
Telstar_High_School
Type of knot
middle rope is sliced. This allows climbers rappelling down cliff faces to keep most of the rope used for the rappel, by tying the knot at the top, and
Sheepshank
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
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)
Boolean functions (Hao Huang, 2019). Deciding whether the Conway knot is a slice knot (Lisa Piccirillo, 2020). List of conjectures List of unsolved problems
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
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)
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
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
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
Determining whether a knot is the unknot
algorithmically recognizing the unknot, given some representation of a knot, e.g., a knot diagram. There are several types of unknotting algorithms. A major
Unknotting_problem
mathematical knots and links. See also list of knots, list of geometric topology topics. 01 knot/Unknot - a simple un-knotted closed loop 31 knot/Trefoil knot -
List of mathematical knots and links
List_of_mathematical_knots_and_links
Mathematical theory of knots
"Thurston–Bennequin number", The Knot Atlas. Lee Rudolph (1997). "The slice genus and the Thurston–Bennequin invariant of a knot". Proceedings of the American
Thurston–Bennequin_number
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
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
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
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
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
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
Minimum number of times a specific knot must be passed through itself to become untied
In the mathematical area of knot theory, the unknotting number of a knot is the minimum number of times the knot must be passed through itself (crossing
Unknotting_number
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
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
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
Pizza chain in New York City, US
pizza slices, with the classic cheese slice being the staple. Over time, the menu has expanded to include pepperoni slices, whole pies, garlic knots, and
2_Bros._Pizza
Necklace used to hold ID cards or other items
typically had a lanyard consisting of a string loop tied together with a diamond knot. It helped secure the item and gave an extended grip over a small handle
Lanyard
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
In the 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
Bridge_number
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
In the mathematical field of knot theory, the crosscap number of a knot K is the minimum of C ( K ) ≡ 1 − χ ( S ) , {\displaystyle C(K)\equiv 1-\chi (S)
Crosscap_number
Notation used to describe knots based on operations on tangles
In knot theory, Conway notation, invented by John Horton Conway, is a way of describing knots that makes many of their properties clear. It composes a
Conway_notation_(knot_theory)
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SLICE KNOT
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