This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

KnotData

 KnotDatagives the specified property for a knot. KnotData[knot]gives an image of the knot. KnotDatagives a list of knots in the specified class.
• Prime knots with crossing numbers up to 10 can be specified in Alexander-Briggs notation .
• Knots can also be specified in Dowker notation , and in Conway notation .
• Special knot specifications include:
 {"PretzelKnot",{n1,n2,...}} (, , ...)-pretzel knot {"TorusKnot",{m,n}} (m, n)-torus knot (m, n coprime)
• Knots with standard names can be specified by their names, such as and .
• KnotData gives a list of classical named knots.
• KnotData[All] gives a list of knots that have Alexander-Briggs notations.
• KnotData gives a list of possible properties for knots.
• Graphical representations for knots include:
 "Image" 3D image of the knot "ImageData" graphics data for the 3D knot image "KnotDiagram" 2D diagram of the knot "KnotDiagramData" graphics data for the 2D knot diagram
• Invariants for knots include:
 "ArfInvariant" Arf invariant "BraidIndex" braid index "BridgeIndex" bridge index "ColoringNumberSet" list of colorable numbers "ConcordanceOrder" concordance order "CrossingNumber" crossing number "DegreeThreeVassiliev" degree-3 Vassiliev invariant "DegreeTwoVassiliev" degree-2 Vassiliev invariant "Determinant" determinant "Genus" genus of knot complement "HyperbolicVolume" hyperbolic volume "NakanishiIndex" Nakanishi index "OzsvathSzaboTau" Ozsvath-Szabo tau invariant "Signature" signature "SmoothFourGenus" smooth 4-genus "StickNumber" stick number "SuperbridgeIndex" superbridge index "ThurstonBennequin" Thurston-Bennequin number "TopologicalFourGenus" topological 4-genus "UnknottingNumber" unknotting number
• Polynomial invariants given as pure functions include:
 "AlexanderPolynomial" Alexander polynomial "BLMHoPolynomial" BLMHo polynomial "BracketPolynomial" normalized bracket polynomial "ConwayPolynomial" Conway polynomial "HOMFLYPolynomial" HOMFLY polynomial "JonesPolynomial" Jones polynomial "KauffmanPolynomial" Kauffman polynomial
• Other properties include:
 "SeifertMatrix" Seifert matrix "SpaceCurve" space curve function for a knot embedding
• Graphical representations for knots as braids include:
 "BraidDiagram" 2D diagram of the knot as a braid "BraidDiagramData" graphics data for the 2D braid diagram "BraidImage" 3D image of the knot as a braid "BraidImageData" graphics data for the 3D braid image
• Notations for knots include:
 "AlexanderBriggsList" Alexander-Briggs list "AlexanderBriggsNotation" Alexander-Briggs notation for display "BraidWord" braid word as a list "BraidWordNotation" braid word in algebraic notation "ConwayNotation" Conway notation for display "ConwayString" Conway notation as a string "DowkerList" Dowker list "DowkerNotation" Dowker notation for display
• Naming-related properties include:
 "AlternateNames" alternate English names "Name" English or mathematical name "StandardName" standard Mathematica name
• KnotData gives a list of the classes in which knot occurs.
• KnotData gives a list of knots in the specified class.
• Basic classes of knots include:
 "AlmostAlternating" almost alternating "Alternating" alternating "Amphichiral" amphichiral "Chiral" chiral "Hyperbolic" hyperbolic "Invertible" invertible "Nonalternating" non-alternating "Prime" prime "Ribbon" ribbon "Satellite" satellite "Slice" slice "Torus" torus "Twist" twist
• Negative classes of knots include:
 "Composite" not prime "NonalmostAlternating" not almost alternating "Nonhyperbolic" not hyperbolic "Noninvertible" not invertible "Nonribbon" not ribbon "Nonsatellite" not satellite "Nonslice" not slice "Nontorus" not torus "Nontwist" not twist
• KnotData gives a hyperlink to more information about the knot with the specified name.
• Using KnotData may require internet connectivity.
The trefoil knot:
The Alexander polynomial of the trefoil knot:
The trefoil knot:
 Out[1]=

The Alexander polynomial of the trefoil knot:
 Out[1]=
 Scope   (26)
Obtain a list of classical named knots:
Obtain a list of knots that have Alexander-Briggs notations:
A knot can be specified by its standard Mathematica name:
Knots can also be specified in Alexander-Briggs notation:
Conway notation:
Dowker notation:
A torus knot is specified by a pair of coprime integers:
A pretzel knot is specified by the number of crossings of its tangles:
Find the English name of a knot:
A list of alternate names can also be found:
Find the list of knot classes:
Find the list of knots belonging to a class:
Test whether an element belongs to a class:
Get a list of classes in which a knot belongs:
A list of knots which are noninvertible and alternating:
Get a list of possible properties:
Get a list of available properties for a particular knot:
Image of a knot:
Diagram of a knot:
Get different notational forms:
Get different notational forms, useful for input:
A property value can be any valid Mathematica expression:
Polynomial invariants are given as pure functions:
A space curve for a knot is given as Function or InterpolatingFunction:
3D images of knots are Graphics3D objects:
Get the 3D primitives for the :
2D diagrams of knots are Graphics objects:
Get the 2D primitives for the :
A property that is not applicable to a knot has the value Missing:
A property that is not available for a knot has the value Missing:
A property that is unknown for a knot has the value Missing:
Specify a list of properties for a knot:
Braid index of a knot:
Braid word as a list:
Braid word notation:
Braid image:
 Applications   (5)
The 20 amphichiral knots having 10 or fewer crossings:
Number of prime knots per crossing numbers:
The trefoil is a tricolorable knot:
Tricolorable trefoil:
The two-bridge knots are exactly the rational knots:
Number of rational knots per crossing numbers:
A stick knot:
Graphics data of knots can be used in Graphics and Graphics3D:
A 3D image:
A braid image:
Alexander polynomials are symmetric:
Alexander polynomials of oriented knots take values or at :
Alexander polynomials can be expressed in terms of the Seifert matrix:
Conway polynomials are modified versions of Alexander polynomials:
Identities for Jones polynomials:
A tori knot has a mirror :
The tori and are equivalent:
Kauffman polynomials are generalizations of Jones polynomials:
Kauffman polynomials extend BLMHo polynomials to two variables:
Kauffman and normalized bracket polynomials:
Relation between normalized bracket polynomials and Jones polynomials:
The Arf invariant of a knot is related to the Alexander polynomial:
The Perko pair is represented by the unique knot :
Exactly 165 distinct prime knots with 10 crossings are listed:
Braids of an even pretzel knot are shifted to end with a braid having an even crossing number:
A tabbed list of knot diagrams:
Random-colored torus knots:
A torus knot sits on a torus:
The knot :
A pretzel knot rendered with spheres:
Color a knot:
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