Miller Institute Knot

Author

Eric W. Weisstein
May 25, 2007

This notebook downloaded from http://mathworld.wolfram.com/notebooks/KnotTheory/MillerInstituteKnot.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/MillerInstituteKnot.html.

©2007 Wolfram Research, Inc. except for portions noted otherwise

Initialization

knot={6,2};

Symbol

Alexander-Briggs Notation

KnotData[knot,"AlexanderBriggsNotation"]//TraditionalForm

Braid Word

KnotData[knot,"BraidWordNotation"]//TraditionalForm

StandardName

KnotData[knot,"StandardName"]

Figure

g1=Graphics[{Thickness[.015],KnotData[knot,"KnotDiagramData"]},ImageSize->200]

Show[KnotData[knot],ImageSize->200]

Graphics3D[{EdgeForm[],ClosedBraid[BraidWord[Link[knot]],Style->TubeStyle][[1]]},ImageSize->200,ViewPoint->{0,0,2},Boxed->False]

Show[ClosedBraid[BraidWord[Link[knot]],Style->TubeStyle],ImageSize->200]

Properties

Classes

KnotData[knot,"Classes"]

KnotClassString[knot]

Properties

props={"AlexanderBriggsList","AlexanderBriggsNotation","AlternateNames","ArfInvariant","BridgeIndex","ColoringNumberSet","ConcordanceOrder","ConwayNotation","ConwayString","ColoringNumberSet","CrossingNumber","ConwayNotation","ConwayString","CrossingNumber","DegreeThreeVassiliev","DegreeTwoVassiliev","Determinant","DowkerList","DowkerNotation","Genus","HyperbolicVolume","NakanishiIndex","Name","OzsvathSzaboTau","SeifertMatrix","Signature","SmoothFourGenus","StandardName","StickNumber","SuperbridgeIndex","ThurstonBennequin","TopologicalFourGenus","UnknottingNumber"};

(#->KnotData[knot,#])&/@props//Column

Polynomials

polys={{"AlexanderPolynomial",x},{"BLMHoPolynomial",x},{"BracketPolynomial",x},{"ConwayPolynomial",x},{"HOMFLYPolynomial",{ℓ,m}},{"JonesPolynomial",t},{"KauffmanPolynomial",{a,z}}};

(#[[1]]->KnotData[knot,#[[1]]][Sequence@@#[[2]]])&/@polys//Column//TraditionalForm

Other Knots with Same Polynomials

Function[poly,{poly,Select[DeleteCases[KnotData[All],knot],KnotData[#,poly]===KnotData[knot,poly]&]}]/@First/@polys//Column