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 AlexanderBriggs notation {n,k} .
  • 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 AlexanderBriggs 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
  • Region-related representations include:
  • "BoundaryMeshRegion"boundary mesh representation
    "MeshRegion"mesh representation
    "Region"geometric region
  • 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"OzsvathSzabo tau invariant
    "SmoothFourGenus"smooth 4-genus
    "StickNumber"stick number
    "SuperbridgeIndex"superbridge index
    "ThurstonBennequin"ThurstonBennequin 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"AlexanderBriggs {n,k} list
    "AlexanderBriggsNotation"AlexanderBriggs 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 Wolfram Language 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
  • KnotData[name,"Information"] gives a hyperlink to more information about the knot with the specified name.
  • Using KnotData may require internet connectivity.


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Basic Examples  (2)

The trefoil knot:

Click for copyable input

The Alexander polynomial of the trefoil knot:

Click for copyable input

Scope  (26)

Generalizations & Extensions  (4)

Applications  (5)

Properties & Relations  (13)

Possible Issues  (2)

Neat Examples  (6)

Introduced in 2007
Updated in 2016