Rolfsen Splice Base

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[[Image:Data:Rolfsen Splice Base/Previous Knot.gif|80px|link=Data:Rolfsen Splice Base/Previous Knot]]

[[Data:Rolfsen Splice Base/Previous Knot]]

[[Image:Data:Rolfsen Splice Base/Next Knot.gif|80px|link=Data:Rolfsen Splice Base/Next Knot]]

[[Data:Rolfsen Splice Base/Next Knot]]

Contents

File:Rolfsen Splice Base.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit <*n*>&id=<*k*> Rolfsen Splice Base's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit Rolfsen Splice Base at Knotilus!


Knot presentations

Planar diagram presentation Data:Rolfsen Splice Base/PD Presentation
Gauss code Data:Rolfsen Splice Base/Gauss Code
Dowker-Thistlethwaite code Data:Rolfsen Splice Base/DT Code
Conway Notation Data:Rolfsen Splice Base/Conway Notation


Minimum Braid Representative A Morse Link Presentation An Arc Presentation
Data:Rolfsen Splice Base/BraidPlot
Length is Data:Rolfsen Splice Base/MinimalBraidLength, width is Data:Rolfsen Splice Base/MinimalBraidWidth,

Braid index is Data:Rolfsen Splice Base/BraidIndex

File:Rolfsen Splice Base ML.gif File:Rolfsen Splice Base AP.gif
Data:Rolfsen Splice Base/Arc Presentation

[edit Notes on presentations of Rolfsen Splice Base]


Three dimensional invariants

Symmetry type Data:Rolfsen Splice Base/Symmetry Type
Unknotting number Data:Rolfsen Splice Base/Unknotting Number
3-genus Data:Rolfsen Splice Base/3-Genus
Bridge index Missing
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:Rolfsen Splice Base/ThurstonBennequinNumber
Hyperbolic Volume Data:Rolfsen Splice Base/HyperbolicVolume
A-Polynomial See Data:Rolfsen Splice Base/A-polynomial

[edit Notes for Rolfsen Splice Base's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus Missing
Rasmussen s-Invariant Missing

[edit Notes for Rolfsen Splice Base's four dimensional invariants]

Polynomial invariants

Alexander polynomial Data:Rolfsen Splice Base/Alexander Polynomial
Conway polynomial Data:Rolfsen Splice Base/Conway Polynomial
2nd Alexander ideal (db, data sources) Data:Rolfsen Splice Base/2nd AlexanderIdeal
Determinant and Signature { Data:Rolfsen Splice Base/Determinant, Data:Rolfsen Splice Base/Signature }
Jones polynomial Data:Rolfsen Splice Base/Jones Polynomial
HOMFLY-PT polynomial (db, data sources) Data:Rolfsen Splice Base/HOMFLYPT Polynomial
Kauffman polynomial (db, data sources) Data:Rolfsen Splice Base/Kauffman Polynomial
The A2 invariant Data:Rolfsen Splice Base/QuantumInvariant/A2/1,0
The G2 invariant Data:Rolfsen Splice Base/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {<* alex = Alexander[K][t];

                     others = DeleteCases[Select[AllKnots[], (alex === Alexander[#][t])&], K];
                     If[others === {}, "", StringJoin[("[["<>NameString[#]<>"]], ")& /@ others]]
                 *>}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {<* J = Jones[K][q];

                   others = DeleteCases[Select[AllKnots[], (J === Jones[#][q]}

Vassiliev invariants

V2 and V3: (Data:Rolfsen Splice Base/V 2, Data:Rolfsen Splice Base/V 3)
V2,1 through V6,9:
V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9
Data:Rolfsen Splice Base/V 2,1 Data:Rolfsen Splice Base/V 3,1 Data:Rolfsen Splice Base/V 4,1 Data:Rolfsen Splice Base/V 4,2 Data:Rolfsen Splice Base/V 4,3 Data:Rolfsen Splice Base/V 5,1 Data:Rolfsen Splice Base/V 5,2 Data:Rolfsen Splice Base/V 5,3 Data:Rolfsen Splice Base/V 5,4 Data:Rolfsen Splice Base/V 6,1 Data:Rolfsen Splice Base/V 6,2 Data:Rolfsen Splice Base/V 6,3 Data:Rolfsen Splice Base/V 6,4 Data:Rolfsen Splice Base/V 6,5 Data:Rolfsen Splice Base/V 6,6 Data:Rolfsen Splice Base/V 6,7 Data:Rolfsen Splice Base/V 6,8 Data:Rolfsen Splice Base/V 6,9

V2,1 through V6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V2 and V3.

Khovanov Homology

The coefficients of the monomials t^rq^j are shown, along with their alternating sums \chi (fixed j, alternation over r). The squares with yellow highlighting are those on the "critical diagonals", where j-2r=s+1 or j-2r=s-1, where s=Data:Rolfsen Splice Base/Signature is the signature of Rolfsen Splice Base. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.    Data:Rolfsen Splice Base/KhovanovTable
Integral Khovanov Homology

(db, data source)

   Data:Rolfsen Splice Base/Integral Khovanov Homology

The Coloured Jones Polynomials