BUILT-IN MATHEMATICA SYMBOL

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.

MORE INFORMATION

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.

KnotData 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

KnotData gives a hyperlink to more information about the knot with the specified name.

Using KnotData may require internet connectivity.

EXAMPLES

CLOSE ALL

Basic Examples (2)

The trefoil knot:

The Alexander polynomial of the trefoil knot:

The trefoil knot:

In[1]:=

Out[1]=

The Alexander polynomial of the trefoil knot:

In[1]:=

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 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:

Generalizations & Extensions (4)

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:

Properties & Relations (13)

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:

Possible Issues (2)

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:

Neat Examples (6)

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: