Torus Knot Splice Base: Difference between revisions

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{{Knot Navigation Links|prev=<*PreviousKnot*>.jpg|next=<*NextKnot*>.jpg}}
{{Knot Navigation Links|prev=<*PreviousKnot*>|next=<*NextKnot*>|imageext=jpg}}


{| align=left
Visit [<*KnotilusURL[K]<>" "<>ThisKnot*>'s page] at [http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/html/start.html Knotilus]!
|- valign=top
|[[Image:<*ThisKnot*>.jpg]]
|Visit [<*KnotilusURL[K]<>" "<>ThisKnot*>'s page] at [http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/html/start.html Knotilus]!


Visit [http://www.math.toronto.edu/~drorbn/KAtlas/TorusKnots/<*m*>.<*n*>.html <*ThisKnot*>'s page] at the original [http://www.math.toronto.edu/~drorbn/KAtlas/index.html Knot Atlas]!
Visit [http://www.math.toronto.edu/~drorbn/KAtlas/TorusKnots/<*m*>.<*n*>.html <*ThisKnot*>'s page] at the original [http://www.math.toronto.edu/~drorbn/KAtlas/index.html Knot Atlas]!

{{<*ThisKnot*> Quick Notes}}
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{{<*ThisKnot*> Further Notes and Views}}


===Knot presentations===
===Knot presentations===
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===[[Khovanov Homology]]===
[[Khovanov Homology]]. The coefficients of the monomials <math>t^rq^j</math> are shown, along with their alternating sums <math>\chi</math> (fixed <math>j</math>, alternation over <math>r</math>). The squares with <font class=HLYellow>yellow</font> highlighting are those on the "critical diagonals", where <math>j-2r=s+1</math> or <math>j-2r=s+1</math>, where <math>s=</math><*s=KnotSignature[K]*> is the signature of <*ThisKnot*>. Nonzero entries off the critical diagonals (if any exist) are highlighted in <font class=HLRed>red</font>.

The coefficients of the monomials <math>t^rq^j</math> are shown, along with their alternating sums <math>\chi</math> (fixed <math>j</math>, alternation over <math>r</math>). The squares with <font class=HLYellow>yellow</font> highlighting are those on the "critical diagonals", where <math>j-2r=s+1</math> or <math>j-2r=s+1</math>, where <math>s=</math><*s=KnotSignature[K]*> is the signature of <*ThisKnot*>. Nonzero entries off the critical diagonals (if any exist) are highlighted in <font class=HLRed>red</font>.


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

Revision as of 21:30, 26 August 2005


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

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

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

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

[[Image:<*ThisKnot*>.jpg]] Visit [<*KnotilusURL[K]<>" "<>ThisKnot*>'s page] at Knotilus!

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

{{<*ThisKnot*> Quick Notes}}


{{<*ThisKnot*> Further Notes and Views}}

Knot presentations

Planar diagram presentation <*PD[K]*>
Gauss code <*List @@ GaussCode[K]*>
Dowker-Thistlethwaite code <*StringReplace[StringTake[ToString[DTCode[K]], {8, -2}], ","->""]*>

Polynomial invariants

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

Vassiliev invariants

V2 and V3 <*{Vassiliev[2][K], Vassiliev[3][K]}*>)

Khovanov Homology

The coefficients of the monomials are shown, along with their alternating sums (fixed , alternation over ). The squares with yellow highlighting are those on the "critical diagonals", where or , where <*s=KnotSignature[K]*> is the signature of <*ThisKnot*>. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Computer Talk

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

<*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]*>

In[1]:=    
<< KnotTheory`
<*InOut[1]; KnotTheoryWelcomeMessage[]*>