Torus Knot Splice Base

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Planar Diagram: <* PD[K] *>

<a href="../Manual/TubePlot.html"><img src="<*m*>.<*n*>_240.jpg" border=0 alt="T(<*m*>,<*n*>)"> TubePlot</a>

The <*m(n-1)*>-Crossing Torus Knot T(<*m*>,<*n*>) <*Include["$knotaka.html"]*> Visit <a class=external href="<*KnotilusURL[K=TorusKnot[m, n]]*>">T(<*m*>,<*n*>)'s page</a> at <a class=external href="http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/html/start.html">Kno tilus</a>! <a href="../Manual/Acknowledgement.html">Acknowledgement</a>

<a href="../Manual/GaussCode.html">Gauss Code</a>:

<*List @@ GaussCode[K]*>

<a href="../Manual/BR.html">Braid Representative</a>:

<* BraidPlot[CollapseBraid[BR[K]], Mode -> "HTML"] *>

<a href="../Manual/AlexanderConway.html">Alexander Polynomial</a>:

<*PolyPrint[alex = Alexander[K][t], t]*>

<a href="../Manual/AlexanderConway.html">Conway Polynomial</a>:

<*PolyPrint[Conway[K][z], z]*>

Other knots with the same <a href="../Manual/AlexanderConway.html">Alexander/Conway Polynomial</a>:

{<* others = DeleteCases[Select[AllKnots[], (alex === Alexander[#][t])&], Knot[n,Type,k]]; If[others === {}, "", StringJoin[(ToString[#, FormatType -> HTMLForm]<>", ")& /@ others] ] *>...}

<a href="../Manual/DetAndSignature.html">Determinant and Signature</a>:

<*{KnotDet[K], s=KnotSignature[K]}*>

<a href="../Manual/Jones.html">Jones Polynomial</a>:

<*PolyPrint[J = Jones[K][q], q]*>

Other knots (up to mirrors) with the same <a href="../Manual/Jones.html">Jones Polynomial</a>:

{<* others = DeleteCases[Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])& ], Knot[n,Type,k]]; If[others === {}, "", StringJoin[(ToString[#, FormatType -> HTMLForm]<>", ")& /@ others] ] *>...}

<* If[Crossings[K]<=18, Include["ColouredJones.mhtml"] ,""] *>

<a href="../Manual/A2Invariant.html">A2 (sl(3)) Invariant</a>:

<*PolyPrint[A2Invariant[K][q], q]*>

<a href="../Manual/Kauffman.html">Kauffman Polynomial</a>:

<*PolyPrint[Kauffman[K][a, z], {a, z}]*>

<a href="../Manual/Vassiliev.html">V2 and V3, the type 2 and 3 Vassiliev invariants</a>:

<* {Vassiliev[2][K], Vassiliev[3][K]} *>

<a href="../Manual/KhovanovHomology.html">Khovanov Homology</a>. The coefficients of the monomials t^rq^j are shown, along with their alternating sums χ (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=<*s*> is the signature of T(<*m*>,<*n*>). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

 <*TabularKh[Kh[K][q, t], s+{1,-1}]*>

<* ComputerTalkHeader *>

<*GraphicsBox["`1`.`2`_240.jpg", "TubePlot[TorusKnot[`1`, `2`]]", m, n]*> <*InOut["Crossings[``]", K]*> <*InOut["PD[``]", K]*> <*InOut["GaussCode[``]", K]*> <*InOut["BR[``]", K]*> <*InOut["alex = Alexander[``][t]", K]*> <*InOut["Conway[``][z]", K]*> <*InOut["Select[AllKnots[], (alex === Alexander[#][t])&]"]*> <*InOut["{KnotDet[`1`], KnotSignature[`1`]}", K]*> <*InOut["J=Jones[``][q]", K]*> <*InOut[

 "Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]"

]*> <* If[Crossings[K]<=18, Include["ColouredJonesM.mhtml"] ,""] *> <*InOut["A2Invariant[``][q]", K]*> <*InOut["Kauffman[``][a, z]", K]*> <*InOut["{Vassiliev[2][`1`], Vassiliev[3][`1`]}", K ]*> <*InOut["Kh[``][q, t]", K]*>

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<a href="/~drorbn/">Dror Bar-Natan</a>: <a href="../index.html">The Knot Atlas</a>: <a href="index.html">Torus Knots</a>: <a href="#top">The Torus Knot T(<*m*>,<*n*>)</a>

<a href="<*prevm*>.<*prevn*>.html"><img border=0 width=120 height=120 src="<*prevm*>.<*prevn*>_120.jpg" alt="T(<*prevm*>,<*prevn*>)"> T(<*prevm*>,<*prevn*>)</a> <a href="<*nextm*>.<*nextn*>.html"><img border=0 width=120 height=120 src="<*nextm*>.<*nextn*>_120.jpg" alt="T(<*nextm*>,<*nextn*>)"> T(<*nextm*>,<*nextn*>)</a>

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