This is documentation for Mathematica 6, which was
based on an earlier version of the Wolfram Language.
View current documentation (Version 11.2)


KnotData[knot, "property"]
gives the specified property for a knot.
gives an image of the knot.
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.
  • KnotData[All] 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
"Genus"genus of knot complement
"HyperbolicVolume"hyperbolic volume
"NakanishiIndex"Nakanishi index
"OzsvathSzaboTau"Ozsvath-Szabo tau invariant
"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
  • 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.
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