Torus Knot Splice Base: Difference between revisions

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<!-- <* (* -->{{Splice Template Notice}}<!-- *) *> -->
<!-- <* (* -->{{Splice Base Notice}}<!-- *) *> -->
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<!-- <*K=Knot[ThisKnot]; {m,n}=List@@K;*> -->
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{{Knot Navigation Links|ext=jpg}}
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<!-- <*{m,n}=List@@K;*> -->

{{Torus Knot Page|
{| align=left
m = <*m*> |
|- valign=top
n = <*n*> |
|[[Image:{{PAGENAME}}.jpg]]
|{{Torus Knot Site Links|m=<*m*>|n=<*n*>|KnotilusURL=<*KnotilusURL[K]*>}}
KnotilusURL = <*KnotilusURL[K]*> |
braid_table = <* BraidPlot[CollapseBraid[BR[K]], Mode -> "Wiki", Images -> {"BraidPart0.gif", "BraidPart1.gif",

"BraidPart2.gif", "BraidPart3.gif", "BraidPart4.gif"}] *> |
{{:{{PAGENAME}} Quick Notes}}
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same_alexander = <* alex = Alexander[K][t];
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|}
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others = DeleteCases[Select[AllKnots[], (alex === Alexander[#][t])&], K];

If[others === {}, "", StringJoin[("[["<>NameString[#]<>"]], ")& /@ others]]
<br style="clear:both" />
*> |

same_jones = <* J = Jones[K][q];
{{:{{PAGENAME}} Further Notes and Views}}
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others = DeleteCases[Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&], K];

If[others === {}, "", StringJoin[("[["<>NameString[#]<>"]], ")& /@ others]]
{{Knot Presentations}}
*> |
{{Polynomial Invariants}}
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khovanov_table = <*TabularKh[Kh[K][q, t], KnotSignature[K]+{1,-1}]*> |
{{Vassiliev Invariants}}
coloured_jones_2 = <*ColouredJones[K, 2][q]*> |

coloured_jones_3 = <*ColouredJones[K, 3][q]*> |
===[[Khovanov Homology]]===
coloured_jones_4 = <*ColouredJones[K, 4][q]*> |

coloured_jones_5 = <*ColouredJones[K, 5][q]*> |
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>{{Data:{{PAGENAME}}/Signature}} is the signature of {{PAGENAME}}. Nonzero entries off the critical diagonals (if any exist) are highlighted in <font class=HLRed>red</font>.
coloured_jones_6 = <*ColouredJones[K, 6][q]*> |

coloured_jones_7 = <*ColouredJones[K, 7][q]*>
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<center><*TabularKh[Kh[K][q, t], KnotSignature[K]+{1,-1}]*></center>
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}}

{{Computer Talk Header}}

<table>
<tr valign=top>
<td><pre style="color: blue; border: 0px; padding: 0em">In[1]:=&nbsp;&nbsp;&nbsp;&nbsp;</pre></td>
<td align=left><pre style="color: red; border: 0px; padding: 0em">&lt;&lt; KnotTheory`</pre></td>
</tr>
<tr valign=top><td colspan=2><pre style="border: 0px; padding: 0em"><*InOut[1]; KnotTheoryWelcomeMessage[]*></pre></td></tr>
<*InOut["Crossings[``]", K]*>
<*InOut["PD[``]", K]*>
<*InOut["GaussCode[``]", K]*>
<*InOut["BR[``]", K]*>
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<*InOut["alex = Alexander[``][t]", K]*>
<*InOut["Conway[``][z]", K]*>
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<*InOut["Select[AllKnots[], (alex === Alexander[#][t])&]"]*>
<*InOut["{KnotDet[`1`], KnotSignature[`1`]}", K]*>
<*InOut["J=Jones[``][q]", K]*>
<*InOut[
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"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]*>
</table>

<* (* <!-- *) *> {{Category:Knot Page}} <* (* --> *) *>
<* (* {{Category:Splice Template}} *) *>

Latest revision as of 16:09, 18 September 2005

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]]

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

Visit Torus Knot Splice Base at Knotilus!

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
Further Quantum Invariants
Torus Knot Splice Base Quantum Invariants.
Computer Talk
The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:= AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`
Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.

Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:= K = Knot["Torus Knot Splice Base"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= Data:Torus Knot Splice Base/Alexander Polynomial
In[5]:= Conway[K][z]
Out[5]= Data:Torus Knot Splice Base/Conway Polynomial
In[6]:= Alexander[K, 2][t]
KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.
Out[6]= Data:Torus Knot Splice Base/2nd AlexanderIdeal
In[7]:= {KnotDet[K], KnotSignature[K]}
Out[7]= { Data:Torus Knot Splice Base/Determinant, Data:Torus Knot Splice Base/Signature }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= Data:Torus Knot Splice Base/Jones Polynomial
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= Data:Torus Knot Splice Base/HOMFLYPT Polynomial
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= Data:Torus Knot Splice Base/Kauffman Polynomial

"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, ): {<* J = Jones[K][q]; 

                   others = DeleteCases[Select[AllKnots[], (J === Jones[#][q]}
Computer Talk
The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:= AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`
Loading KnotTheory` version of May 31, 2006, 14:15:20.091.

Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:= K = Knot["Torus Knot Splice Base"];
In[4]:= {A = Alexander[K][t], J = Jones[K][q]}
KnotTheory::loading: Loading precomputed data in PD4Knots`.
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[4]= { Data:Torus Knot Splice Base/Alexander Polynomial, Data:Torus Knot Splice Base/Jones Polynomial }
In[5]:= DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]
KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.
KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.
Out[5]= {<* alex = Alexander[K][t]; 
                     others = DeleteCases[Select[AllKnots[], (alex === Alexander[#][t])&], K];
                     If[others === {}, "", StringJoin[("[["<>NameString[#]<>"]], ")& /@ others]]
                 *>}
In[6]:= DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]
KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
Out[6]= {<* 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 are shown, along with their alternating sums (fixed , alternation over ). The squares with yellow highlighting are those on the "critical diagonals", where or , where 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]]

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