T(5,4)

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Further T(5,4) views

Planar Diagram: X[17, 25, 18, 24] X[10, 26, 11, 25] X[3, 27, 4, 26] X[11, 19, 12, 18] X[4,\

 20, 5, 19] X[27, 21, 28, 20] X[5, 13, 6, 12] X[28, 14, 29, 13] X[21, 15,\

 22, 14] X[29, 7, 30, 6] X[22, 8, 23, 7] X[15, 9, 16, 8] X[23, 1, 24, 30]\

 X[16, 2, 17, 1] X[9, 3, 10, 2] 

<a href="../Manual/TubePlot.html"><img src="m.n_240.jpg" border=0 alt="T(m,n)"> TubePlot</a>

The m (-1 + n)-Crossing Torus Knot T(m,n) Include[$knotaka.html] Visit <a class=external href="KnotilusURL[GaussCode[PD[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>:

{PD[TorusKnot[m, n]]}

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

BraidPlot[CollapseBraid[BR[TorusKnot[m, n]]], Mode -> HTML]

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

PolyPrint[1, t]

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

PolyPrint[1, z]

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

{ToString[Knot[0, 1], FormatType -> HTMLForm]<>, <> ToString[Knot[11, NonAlternating, 34], FormatType -> HTMLForm]<>, <> ToString[Knot[11, NonAlternating, 42], FormatType -> HTMLForm]<>, ...}

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

{1, 0}

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

Sqrt[q] TorusKnot[m, n] PolyPrint[-(-----------------------), q] 1 + q

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

{...}

Include[ColouredJones.mhtml]

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

PolyPrint[TorusKnot[m, n], q]

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

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

{0, 0}

<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=0 is the signature of T(m,n). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

 TabularKh[$Failed[q, t], {1, -1}]

ComputerTalkHeader

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

  Jones[#][q])&]]

Include[ColouredJonesM.mhtml] InOut[A2Invariant[``][q], TorusKnot[m, n]] InOut[Kauffman[``][a, z], TorusKnot[m, n]] InOut[{Vassiliev[2][`1`], Vassiliev[3][`1`]}, TorusKnot[m, n]] InOut[Kh[``][q, t], TorusKnot[m, n]]

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