Knot

Notable mathematical knots.

Entity["Knot",name] or

represents an entity of type "Knot".

[prop]

gives the value of a specified property.

[{prop_i,…}]

gives the value of a list of properties.

EntityClass["Knot",{prop_ispec_i,…}]

represents a class of entities with values of prop_i defined by spec_i.

Sample Entities

Sample Entity Classes

Properties

	AlexanderBriggsNotation	Alexander‐Briggs notation
	AlexanderPolynomial	Alexander polynomial
	AlternateNames	alternate names
	Alternating	alternating
	Amphichiral	amphichiral
	ArfInvariant	Arf invariant
	BLMHoPolynomial	BLM/Ho polynomial
	BoundaryMeshRegion	boundary mesh representation
	BracketPolynomial	bracket polynomial
	BraidDiagram	braid diagram
	BraidImage	braid image
	BraidIndex	braid index
	BraidWordNotation	braid word
	BridgeIndex	bridge index
	Chiral	chiral
	Classes	classes
	ColoringNumberSet	coloring number set
	Composite	composite
	ConcordanceOrder	concordance order
	ConwayNotation	Conway notation
	ConwayPolynomial	Conway polynomial
	CrossingNumber	crossing number
	DegreeThreeVassiliev	degree‐3 Vassiliev invariant
	DegreeTwoVassiliev	degree‐2 Vassiliev invariant
	Determinant	determinant
	DowkerNotation	Dowker notation
	EntityClasses	entity classes
	Genus	genus of complement
	HOMFLYPolynomial	HOMFLY polynomial
	Hyperbolic	hyperbolic
	HyperbolicVolume	hyperbolic volume of complement
	Image	image
	Invertible	invertible
	JonesPolynomial	Jones polynomial
	KauffmanPolynomial	Kauffman polynomial
	KnotDiagram	diagram
	MeshRegion	mesh representation
	NakanishiIndex	Nakanishi index
	Name	name
	Nonalternating	nonalternating
	Nonhyperbolic	nonhyperbolic
	Noninvertible	noninvertible
	Nonsatellite	nonsatellite
	Nontorus	nontorus
	OzsvathSzaboTau	Ozsváth‐Szabó τ‐invariant
	Prime	prime
	Region	region
	Satellite	satellite
	SeifertMatrix	Seifert matrix
	Signature	signature
	SmoothFourGenus	smooth 4-genus
	StickNumber	stick number
	SuperbridgeIndex	superbridge index
	ThurstonBennequin	Thurston‐Bennequin number
	TopologicalFourGenus	topological 4‐genus
	Torus	torus
	UnknottingNumber	unknotting number

Details

"Knot" entities include prime knots up to 10 crossings, as well as some special infinite families.
"Knot" entity classes include common mathematical types of knots, such as "Alternating", "Chiral" and "Torus", together with the negations of these.
Mathematical properties are available for most "Knot" entities that are concisely representable and either well known or straightforward to compute. Properties for a small number of special parametrized family members are computed on the fly.
Some properties are available for the "Knot" entity type as a whole and can be given using the form EntityValue["Knot",property]. Such properties include:

	"Properties"	the list of available properties
	"PropertyCanonicalNames"	the standard names of available properties
	"SampleEntities"	a sample list of available entities (typically of length 10)
	"SampleEntityClasses"	a sample list of available entity classes (typically of length 10)
	"EntityCount"	number of entities available
	"Entities"	the list of available entities
	"EntityCanonicalNames"	the standard names of available entities
	"EntityClasses"	the list of available entity classes
	"EntityClassCanonicalNames"	the standard names of available entity classes
	"PropertyClasses"	the list of available property classes
	"PropertyClassCanonicalNames"	the standard names of available property classes
	"PropertyCount"	number of properties available

The following annotations can be used in the third argument of EntityValue["Knot",property,annotation]:

	"EntityAssociation"	an association of entities and entity-property values
	"PropertyAssociation"	an association of properties and entity-property values
	"EntityPropertyAssociation"	an association in which the specified entities are keys, and values are a nested association of properties and entity-property values
	"PropertyEntityAssociation"	an association in which the specified properties are keys, and values are a nested association of entities and entity-property values
	"Dataset"	a dataset in which the specified entities are keys, and values are an association of property names and entity-property values

The following annotations can be used in the second argument of EntityValue[property,annotation]:
"Description" a brief textual description of the property

Examples

Basic Examples (4)

Use for entity discovery:

Find a property value for an entity:

Retrieve a dataset of all available properties for an entity:

Find knots with crossing number 7:

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