Rolfsen Splice Base: Difference between revisions

Latest revision as of 14:55, 23 June 2006

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:Rolfsen Splice Base/Previous Knot.gif|80px|link=Data:Rolfsen Splice Base/Previous Knot]]

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

[[Image:Data:Rolfsen Splice Base/Next Knot.gif|80px|link=Data:Rolfsen Splice Base/Next Knot]]

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

File:Rolfsen Splice Base.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit <*n*>&id=<*k*> Rolfsen Splice Base's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit Rolfsen Splice Base at Knotilus!

[edit Rolfsen Splice Base Quick Notes]

[edit Rolfsen Splice Base Further Notes and Views]

Knot presentations

Planar diagram presentation	Data:Rolfsen Splice Base/PD Presentation
Gauss code	Data:Rolfsen Splice Base/Gauss Code
Dowker-Thistlethwaite code	Data:Rolfsen Splice Base/DT Code
Conway Notation	Data:Rolfsen Splice Base/Conway Notation

Minimum Braid Representative	A Morse Link Presentation	An Arc Presentation
Data:Rolfsen Splice Base/BraidPlot Length is Data:Rolfsen Splice Base/MinimalBraidLength, width is Data:Rolfsen Splice Base/MinimalBraidWidth, Braid index is Data:Rolfsen Splice Base/BraidIndex	File:Rolfsen Splice Base ML.gif	File:Rolfsen Splice Base AP.gif Data:Rolfsen Splice Base/Arc Presentation

[edit Notes on presentations of Rolfsen Splice Base]

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.org/wiki/KnotTheory.

In[3]:= K = Knot["Rolfsen Splice Base"];

In[4]:= PD[K]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]= Data:Rolfsen Splice Base/PD Presentation

In[5]:= GaussCode[K]

Out[5]= Data:Rolfsen Splice Base/Gauss Code

In[6]:= DTCode[K]

Out[6]= Data:Rolfsen Splice Base/DT Code

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];

In[8]:= ConwayNotation[K]

Out[8]= Data:Rolfsen Splice Base/Conway Notation

In[9]:= br = BR[K]

KnotTheory::credits: The minimum braids representing the knots with up to 10 crossings were provided by Thomas Gittings. See arXiv:math.GT/0401051.

Out[9]= Data:Rolfsen Splice Base/BraidWord

In[10]:= {First[br], Crossings[br], BraidIndex[K]}

KnotTheory::credits: The braid index data known to KnotTheory` is taken from Charles Livingston's http://www.indiana.edu/~knotinfo/.

KnotTheory::loading: Loading precomputed data in IndianaData`.

Out[10]= { Data:Rolfsen Splice Base/MinimalBraidWidth, Data:Rolfsen Splice Base/MinimalBraidLength, Data:Rolfsen Splice Base/BraidIndex }

In[11]:= Show[BraidPlot[br]]

Data:Rolfsen Splice Base/BraidPlot

Out[11]= -Graphics-

In[12]:= Show[DrawMorseLink[K]]

KnotTheory::credits: "MorseLink was added to KnotTheory` by Siddarth Sankaran at the University of Toronto in the summer of 2005."

KnotTheory::credits: "DrawMorseLink was written by Siddarth Sankaran at the University of Toronto in the summer of 2005."

File:Rolfsen Splice Base ML.gif

Out[12]= -Graphics-

In[13]:= ap = ArcPresentation[K]

Out[13]= ArcPresentationData:Rolfsen Splice Base/Arc Presentation

In[14]:= Draw[ap]

File:Rolfsen Splice Base AP.gif

Out[14]= -Graphics-

Three dimensional invariants

Symmetry type	Data:Rolfsen Splice Base/Symmetry Type
Unknotting number	Data:Rolfsen Splice Base/Unknotting Number
3-genus	Data:Rolfsen Splice Base/3-Genus
Bridge index	Missing
Super bridge index	Missing
Nakanishi index	Missing
Maximal Thurston-Bennequin number	Data:Rolfsen Splice Base/ThurstonBennequinNumber
Hyperbolic Volume	Data:Rolfsen Splice Base/HyperbolicVolume
A-Polynomial	See Data:Rolfsen Splice Base/A-polynomial

[edit Notes for Rolfsen Splice Base's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus	Missing
Topological 4 genus	Missing
Concordance genus	Missing
Rasmussen s-Invariant	Missing

[edit Notes for Rolfsen Splice Base's four dimensional invariants]

Polynomial invariants

Alexander polynomial	Data:Rolfsen Splice Base/Alexander Polynomial
Conway polynomial	Data:Rolfsen Splice Base/Conway Polynomial
2nd Alexander ideal (db, data sources)	Data:Rolfsen Splice Base/2nd AlexanderIdeal
Determinant and Signature	{ Data:Rolfsen Splice Base/Determinant, Data:Rolfsen Splice Base/Signature }
Jones polynomial	Data:Rolfsen Splice Base/Jones Polynomial
HOMFLY-PT polynomial (db, data sources)	Data:Rolfsen Splice Base/HOMFLYPT Polynomial
Kauffman polynomial (db, data sources)	Data:Rolfsen Splice Base/Kauffman Polynomial
The A2 invariant	Data:Rolfsen Splice Base/QuantumInvariant/A2/1,0
The G2 invariant	Data:Rolfsen Splice Base/QuantumInvariant/G2/1,0

Further Quantum Invariants

Rolfsen 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["Rolfsen Splice Base"];

In[4]:= Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]= Data:Rolfsen Splice Base/Alexander Polynomial

In[5]:= Conway[K][z]

Out[5]= Data:Rolfsen 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:Rolfsen Splice Base/2nd AlexanderIdeal

In[7]:= {KnotDet[K], KnotSignature[K]}

Out[7]= { Data:Rolfsen Splice Base/Determinant, Data:Rolfsen Splice Base/Signature }

In[8]:= Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]= Data:Rolfsen Splice Base/Jones Polynomial

In[9]:= HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]= Data:Rolfsen Splice Base/HOMFLYPT Polynomial

In[10]:= Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]= Data:Rolfsen 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["Rolfsen 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:Rolfsen Splice Base/Alexander Polynomial, Data:Rolfsen 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:Rolfsen Splice Base/V 2, Data:Rolfsen 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:Rolfsen Splice Base/V 2,1

Data:Rolfsen Splice Base/V 3,1

Data:Rolfsen Splice Base/V 4,1

Data:Rolfsen Splice Base/V 4,2

Data:Rolfsen Splice Base/V 4,3

Data:Rolfsen Splice Base/V 5,1

Data:Rolfsen Splice Base/V 5,2

Data:Rolfsen Splice Base/V 5,3

Data:Rolfsen Splice Base/V 5,4

Data:Rolfsen Splice Base/V 6,1

Data:Rolfsen Splice Base/V 6,2

Data:Rolfsen Splice Base/V 6,3

Data:Rolfsen Splice Base/V 6,4

Data:Rolfsen Splice Base/V 6,5

Data:Rolfsen Splice Base/V 6,6

Data:Rolfsen Splice Base/V 6,7

Data:Rolfsen Splice Base/V 6,8

Data:Rolfsen 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:Rolfsen Splice Base/Signature is the signature of Rolfsen Splice Base. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

Data:Rolfsen Splice Base/KhovanovTable

Integral Khovanov Homology

(db, data source)

Data:Rolfsen Splice Base/Integral Khovanov Homology

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the

n+1

-dimensional representation of

sl(2)

)

$n$	$J_{n}$
2	<ColouredJones[K, 2][q]>
3	<ColouredJones[K, 3][q]>
4	<ColouredJones[K, 4][q]>
5	<ColouredJones[K, 5][q]>
6	<ColouredJones[K, 6][q]>
7	<ColouredJones[K, 7][q]>

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session, or any of the Computer Talk sections above.

Modifying This Page

Read me first: Modifying Knot Pages

See/edit the Rolfsen Knot Page master template (intermediate).

See/edit the Rolfsen_Splice_Base (expert).

Back to the top.

[[Image:Data:Rolfsen Splice Base/Previous Knot.gif|80px|link=Data:Rolfsen Splice Base/Previous Knot]]

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

[[Image:Data:Rolfsen Splice Base/Next Knot.gif|80px|link=Data:Rolfsen Splice Base/Next Knot]]

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

@@ Line 1: / Line 1: @@
-<!-- <* (* -->{{Splice Template Notice}}<!-- *) *> -->
+<!-- <* (* -->{{Splice Base Notice}}<!-- *) *> -->
+<!-- <*{n,k}=(K /. Knot[n_,k_]:>{n,k});*> -->
+{{Rolfsen Knot Page|
-<!-- <*K=Knot[ThisKnot];{n,k}=List@@K;*> -->
+n = <*n*> |
+k = <*k*> |
-<!-- provide an anchor so we can return to the top of the page -->
+same_alexander  = <*  alex = Alexander[K][t];
-<span id="top"></span>
+                      others = DeleteCases[Select[AllKnots[], (alex === Alexander[#][t])&], K];
+                      If[others === {}, "", StringJoin[("[["<>NameString[#]<>"]], ")& /@ others]]
-<!-- this relies on transclusion for next and previous links -->
+                  *> |
-{{Knot Navigation Links|ext=gif}}
+same_jones      = <* J = Jones[K][q];
+                    others = DeleteCases[Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&], K];
-{{Rolfsen Knot Page Header|n=<*n*>|k=<*k*>|KnotilusURL=<*KnotilusURL[K]*>}}
+                     If[others === {}, "", StringJoin[("[["<>NameString[#]<>"]], ")& /@ others]]
+                  *> |
-<br style="clear:both" />
+coloured_jones_2 = <*ColouredJones[K, 2][q]*> |
+coloured_jones_3 = <*ColouredJones[K, 3][q]*> |
-{{:{{PAGENAME}} Further Notes and Views}}
+coloured_jones_4 = <*ColouredJones[K, 4][q]*> |
+coloured_jones_5 = <*ColouredJones[K, 5][q]*> |
-{{Knot Presentations}}
+coloured_jones_6 = <*ColouredJones[K, 6][q]*> |
-{{3D Invariants}}
+coloured_jones_7 = <*ColouredJones[K, 7][q]*>
-{{4D Invariants}}
+}}
-{{Polynomial Invariants}}
-{{Vassiliev Invariants}}
-===[[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>{{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>.
-<center><*TabularKh[Kh[K][q, t], KnotSignature[K]+{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>
-<* (* <!-- *) *> {{Category:Knot Page}} <* (* --> *) *>

$n$	$J_{n}$
2	<ColouredJones[K, 2][q]>
3	<ColouredJones[K, 3][q]>
4	<ColouredJones[K, 4][q]>
5	<ColouredJones[K, 5][q]>
6	<ColouredJones[K, 6][q]>
7	<ColouredJones[K, 7][q]>

Rolfsen Splice Base: Difference between revisions

Latest revision as of 14:55, 23 June 2006

Contents

Knot presentations

Three dimensional invariants

Four dimensional invariants

Polynomial invariants

"Similar" Knots (within the Atlas)

Vassiliev invariants

Khovanov Homology

The Coloured Jones Polynomials

Computer Talk

Modifying This Page

Navigation menu

Page actions

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