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KnotData

KnotData
gives the specified property for a knot.
KnotData[knot]
gives an image of the knot.
KnotData
gives 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:
In[1]:=
Click for copyable input
Out[1]=
 
The Alexander polynomial of the trefoil knot:
In[1]:=
Click for copyable input
Out[1]=
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 more information about 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:
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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