Torus Knot Splice Base: Difference between revisions

Latest revision as of 16:09, 18 September 2005

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.

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, $q\leftrightarrow q^{-1}$ ): {<* 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.

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)

V_2,1 through V_6,9:

V_2,1

V_3,1

V_4,1

V_4,2

V_4,3

V_5,1

V_5,2

V_5,3

V_5,4

V_6,1

V_6,2

V_6,3

V_6,4

V_6,5

V_6,6

V_6,7

V_6,8

V_6,9

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

V_2,1 through V_6,9 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^{r}q^{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]]

@@ Line 1: / Line 1: @@
-<!-- Script generated - do not edit! -->
+<!-- <* (* -->{{Splice Base Notice}}<!-- *) *> -->
+<!-- <* (* --><!-- You can ignore this warning; it's for people trying to edit individual knot pages--><!-- *) *> -->
+<!--                       WARNING! WARNING! WARNING!
-<!-- <*K=Knot[ThisKnot]; {m,n}=List@@K;*> -->
+<!-- This page was generated from the splice template [[Torus Knot Splice Template]]. Please do not edit!
+<!-- Almost certainly, you want to edit [[Template:Torus Knot Page]], which actually produces this page.
-<span id="top"></span>
+<!-- The text below simply calls [[Template:Torus Knot Page]] setting the values of all the parameters appropriately.
+<!-- This page itself was created by running [[Media:KnotPageSpliceRobot.nb]] on [[Torus Knot Splice Template]]. -->
-{{Knot Navigation Links|name={{PAGENAME}}|ext=jpg}}
+<!-- <*{m,n}=List@@K;*> -->
+{{Torus Knot Page|
-{| align=left
+m = <*m*> |
-|- valign=top
+n = <*n*> |
-|[[Image:<*ThisKnot*>.jpg]]
+KnotilusURL = <*KnotilusURL[K]*> |
-|Visit [<*KnotilusURL[K]<>" "<>ThisKnot*>'s page] at [http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/html/start.html Knotilus]!
+braid_table     = <* BraidPlot[CollapseBraid[BR[K]], Mode -> "Wiki", Images -> {"BraidPart0.gif", "BraidPart1.gif",
+                     "BraidPart2.gif", "BraidPart3.gif", "BraidPart4.gif"}] *> |
-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]!
+same_alexander  = <*  alex = Alexander[K][t];
+                      others = DeleteCases[Select[AllKnots[], (alex === Alexander[#][t])&], K];
-{{:<*ThisKnot*> Quick Notes}}
+                      If[others === {}, "", StringJoin[("[["<>NameString[#]<>"]], ")& /@ others]]
-|}
+                  *> |
+same_jones      = <* J = Jones[K][q];
-<br style="clear:both" />
+                    others = DeleteCases[Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&], K];
+                     If[others === {}, "", StringJoin[("[["<>NameString[#]<>"]], ")& /@ others]]
-{{:<*ThisKnot*> Further Notes and Views}}
+                  *> |
+khovanov_table  = <*TabularKh[Kh[K][q, t], KnotSignature[K]+{1,-1}]*> |
-===Knot presentations===
+coloured_jones_2 = <*ColouredJones[K, 2][q]*> |
+coloured_jones_3 = <*ColouredJones[K, 3][q]*> |
-{|
+coloured_jones_4 = <*ColouredJones[K, 4][q]*> |
-|'''[[Planar Diagrams|Planar diagram presentation]]'''
+coloured_jones_5 = <*ColouredJones[K, 5][q]*> |
-|style="padding-left: 1em;" | <*PD[K]*>
+coloured_jones_6 = <*ColouredJones[K, 6][q]*> |
-|-
+coloured_jones_7 = <*ColouredJones[K, 7][q]*>
-|'''[[Gauss Codes|Gauss code]]'''
+}}
-|style="padding-left: 1em;" | <*List @@ GaussCode[K]*>
-|-
-|'''[[DT (Dowker-Thistlethwaite) Codes|Dowker-Thistlethwaite code]]'''
-|style="padding-left: 1em;" | <*StringReplace[StringTake[ToString[DTCode[K]], {8, -2}], ","->""]*>
-|}
-{{Polynomial Invariants|name=<*ThisKnot*>}}
-===[[Finite Type (Vassiliev) Invariants|Vassiliev invariants]]===
-{| style="margin-left: 1em;"
-|-
-|'''V<sub>2</sub> and V<sub>3</sub>'''
-|style="padding-left: 1em;" | <*{Vassiliev[2][K], Vassiliev[3][K]}*>
-|}
-===[[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>.
-<center><*TabularKh[Kh[K][q, t], s+{1,-1}]*></center>
-{{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]*>
-<*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]*>
-</table>

Torus Knot Splice Base: Difference between revisions

Latest revision as of 16:09, 18 September 2005

Contents

Knot presentations

Polynomial invariants

"Similar" Knots (within the Atlas)

Vassiliev invariants

Khovanov Homology

Computer Talk

Modifying This Page

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