Notice. The Knot Atlas is now recovering from a major crash. Hopefully all functionality will return slowly over the next few days. --Drorbn (talk) 21:23, 4 July 2013 (EDT)

Torus Knot Splice Base

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Stop hand.png This page is a 'splice base'.
It is used to generate knot pages for each knot in a certain knot table. Be careful editting! Changes will not be reflected on individual knot pages until the 'splicer' is run again.

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

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

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

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

Contents

File:Torus Knot Splice Base.jpg See other torus knots

Visit [<*KnotilusURL[K]*> Torus Knot Splice Base's page] at Knotilus!

Visit <*m*>.<*n*>.html Torus Knot Splice Base's page at the original Knot Atlas!

Edit Torus Knot Splice Base Quick Notes


Edit Torus Knot Splice Base Further Notes and Views


Knot presentations

Planar diagram presentation Data:Torus Knot Splice Base/PD Presentation
Gauss code Data:Torus Knot Splice Base/Gauss Code
Dowker-Thistlethwaite code Data:Torus Knot Splice Base/DT Code
Braid presentation <* BraidPlot[CollapseBraid[BR[K]], Mode -> "Wiki", Images -> {"BraidPart0.gif", "BraidPart1.gif",
                    "BraidPart2.gif", "BraidPart3.gif", "BraidPart4.gif"}] *>

Polynomial invariants

Alexander polynomial Data:Torus Knot Splice Base/Alexander Polynomial
Conway polynomial Data:Torus Knot Splice Base/Conway Polynomial
2nd Alexander ideal (db, data sources) Data:Torus Knot Splice Base/2nd AlexanderIdeal
Determinant and Signature { Data:Torus Knot Splice Base/Determinant, Data:Torus Knot Splice Base/Signature }
Jones polynomial Data:Torus Knot Splice Base/Jones Polynomial
HOMFLY-PT polynomial (db, data sources) Data:Torus Knot Splice Base/HOMFLYPT Polynomial
Kauffman polynomial (db, data sources) Data:Torus Knot Splice Base/Kauffman Polynomial
The A2 invariant Data:Torus Knot Splice Base/QuantumInvariant/A2/1,0
The G2 invariant Data:Torus Knot 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:Torus Knot Splice Base/V 2, Data:Torus Knot 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:Torus Knot Splice Base/V 2,1 Data:Torus Knot Splice Base/V 3,1 Data:Torus Knot Splice Base/V 4,1 Data:Torus Knot Splice Base/V 4,2 Data:Torus Knot Splice Base/V 4,3 Data:Torus Knot Splice Base/V 5,1 Data:Torus Knot Splice Base/V 5,2 Data:Torus Knot Splice Base/V 5,3 Data:Torus Knot Splice Base/V 5,4 Data:Torus Knot Splice Base/V 6,1 Data:Torus Knot Splice Base/V 6,2 Data:Torus Knot Splice Base/V 6,3 Data:Torus Knot Splice Base/V 6,4 Data:Torus Knot Splice Base/V 6,5 Data:Torus Knot Splice Base/V 6,6 Data:Torus Knot Splice Base/V 6,7 Data:Torus Knot Splice Base/V 6,8 Data:Torus Knot 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:Torus Knot Splice Base/Signature is the signature of Torus Knot Splice Base. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.    Data:Torus Knot Splice Base/KhovanovTable
Integral Khovanov Homology

(db, data source)

   Data:Torus Knot Splice Base/Integral Khovanov Homology

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

Read me first: Modifying Knot Pages

See/edit the Torus Knot Page master template (intermediate).

See/edit the Torus Knot_Splice_Base (expert).

Back to the top.

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

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

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

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