This is documentation for Mathematica 6, which was
based on an earlier version of the Wolfram Language.

KnotData

 KnotData[knot, "property"]gives the specified property for a knot. KnotData[knot]gives an image of the knot. KnotData["class"]gives a list of knots in the specified class.
• Prime knots with crossing numbers up to 10 can be specified in Alexander-Briggs {n, k} notation.
• Knots can also be specified in Dowker notation {i1, i2, i3, ...}, and in Conway notation "nnnn".
• Special knot specifications include:
 {"PretzelKnot",{n1,n2,...}} (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 "Trefoil" and "FigureEight".
• KnotData[] gives a list of classical named knots.
• gives a list of knots that have Alexander-Briggs notations.
• KnotData["Properties"] 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 {n, k} 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 {i1, i2, i3, ...} list "DowkerNotation" Dowker notation for display
• Naming-related properties include:
 "AlternateNames" alternate English names "Name" English or mathematical name "StandardName" standard Mathematica name
• KnotData[knot, "Classes"] gives a list of the classes in which knot occurs.
• KnotData["class"] gives a list of knots in the specified class.
• KnotData[knot, "class"] gives True or False depending on whether knot is 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
• Using KnotData may require internet connectivity.
The trefoil knot:
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The Alexander polynomial of the trefoil knot:
 Out[1]=
 Scope   (26)
 Applications   (5)
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