Notable mathematical knots.

Entity["Knot",name] or

represents an entity of type "Knot".

[prop]

gives the value of a specified property.

[{propi,}]

gives the value of a list of properties.

EntityClass["Knot",{propispeci,}]

represents a class of entities with values of propi defined by speci.

Sample Entities

Sample Entity Classes

Properties

  • AlexanderBriggsNotationAlexanderBriggs notation
    AlexanderPolynomialAlexander polynomial
    AlternateNamesalternate names
    Alternatingalternating
    Amphichiralamphichiral
    ArfInvariantArf invariant
    BLMHoPolynomialBLM/Ho polynomial
    BoundaryMeshRegionboundary mesh representation
    BracketPolynomialbracket polynomial
    BraidDiagrambraid diagram
    BraidImagebraid image
    BraidIndexbraid index
    BraidWordNotationbraid word
    BridgeIndexbridge index
    Chiralchiral
    Classesclasses
    ColoringNumberSetcoloring number set
    Compositecomposite
    ConcordanceOrderconcordance order
    ConwayNotationConway notation
    ConwayPolynomialConway polynomial
    CrossingNumbercrossing number
    DegreeThreeVassilievdegree3 Vassiliev invariant
    DegreeTwoVassilievdegree2 Vassiliev invariant
    Determinantdeterminant
    DowkerNotationDowker notation
    EntityClassesentity classes
    Genusgenus of complement
    HOMFLYPolynomialHOMFLY polynomial
    Hyperbolichyperbolic
    HyperbolicVolumehyperbolic volume of complement
    Imageimage
    Invertibleinvertible
    JonesPolynomialJones polynomial
    KauffmanPolynomialKauffman polynomial
    KnotDiagramdiagram
    MeshRegionmesh representation
    NakanishiIndexNakanishi index
    Namename
    Nonalternatingnonalternating
    Nonhyperbolicnonhyperbolic
    Noninvertiblenoninvertible
    Nonsatellitenonsatellite
    Nontorusnontorus
    OzsvathSzaboTauOzsváthSzabó τinvariant
    Primeprime
    Regionregion
    Satellitesatellite
    SeifertMatrixSeifert matrix
    Signaturesignature
    SmoothFourGenussmooth 4-genus
    StickNumberstick number
    SuperbridgeIndexsuperbridge index
    ThurstonBennequinThurstonBennequin number
    TopologicalFourGenustopological 4genus
    Torustorus
    UnknottingNumberunknotting 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]:
  • "Source"source information for the property
    "Date"the date associated with the entity-property value (if any)
    "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]:
  • "Qualifiers"the list of possible qualifiers for the property
    "QualifierValues"the list of possible values that can be given to each qualifier
    "DefaultQualifierValues"the list of default values for the property's qualifiers
    "Description"a brief textual description of the property
    "Definition"a detailed textual definition of the property
    "PhysicalQuantity"the physical quantity associated with the entity-property value
    "Unit"the unit associated with the entity-property value
  • A qualifier value of Automatic indicates that an applicable format of values can be used; e.g. for the "Date" qualifier, this includes a proper date or date span.

Examples

Basic Examples  (4)

Use for entity discovery:

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Find a property value for an entity:

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Retrieve a dataset of all available properties for an entity:

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Find knots with crossing number 7:

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