Hoste-Thistlethwaite Splice Base: Difference between revisions

Revision as of 09:44, 30 August 2005

File:Hoste-Thistlethwaite Splice Base.gif (Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit Hoste-Thistlethwaite Splice Base at Knotilus!
	[edit Hoste-Thistlethwaite Splice Base Quick Notes]

[edit Hoste-Thistlethwaite Splice Base Further Notes and Views]

Knot presentations

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

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation	File:Hoste-Thistlethwaite Splice Base ML.gif

Three dimensional invariants

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

[edit Notes for Hoste-Thistlethwaite 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 Hoste-Thistlethwaite Splice Base's four dimensional invariants]

Polynomial invariants

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

Further Quantum Invariants

Hoste-Thistlethwaite 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["Hoste-Thistlethwaite Splice Base"];

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

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

Out[4]= Data:Hoste-Thistlethwaite Splice Base/Alexander Polynomial

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

Out[5]= Data:Hoste-Thistlethwaite 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:Hoste-Thistlethwaite Splice Base/2nd AlexanderIdeal

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

Out[7]= { Data:Hoste-Thistlethwaite Splice Base/Determinant, Data:Hoste-Thistlethwaite Splice Base/Signature }

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

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

Out[8]= Data:Hoste-Thistlethwaite Splice Base/Jones Polynomial

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

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

Out[9]= Data:Hoste-Thistlethwaite Splice Base/HOMFLYPT Polynomial

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

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

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

Data:Hoste-Thistlethwaite Splice Base/V 3,1

Data:Hoste-Thistlethwaite Splice Base/V 4,1

Data:Hoste-Thistlethwaite Splice Base/V 4,2

Data:Hoste-Thistlethwaite Splice Base/V 4,3

Data:Hoste-Thistlethwaite Splice Base/V 5,1

Data:Hoste-Thistlethwaite Splice Base/V 5,2

Data:Hoste-Thistlethwaite Splice Base/V 5,3

Data:Hoste-Thistlethwaite Splice Base/V 5,4

Data:Hoste-Thistlethwaite Splice Base/V 6,1

Data:Hoste-Thistlethwaite Splice Base/V 6,2

Data:Hoste-Thistlethwaite Splice Base/V 6,3

Data:Hoste-Thistlethwaite Splice Base/V 6,4

Data:Hoste-Thistlethwaite Splice Base/V 6,5

Data:Hoste-Thistlethwaite Splice Base/V 6,6

Data:Hoste-Thistlethwaite Splice Base/V 6,7

Data:Hoste-Thistlethwaite Splice Base/V 6,8

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

Data:Hoste-Thistlethwaite Splice Base/KhovanovTable

Integral Khovanov Homology

(db, data source)

Data:Hoste-Thistlethwaite 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 Hoste-Thistlethwaite Knot Page master template (intermediate).

See/edit the Hoste-Thistlethwaite_Splice_Base (expert).

Back to the top.

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

[[Data:Hoste-Thistlethwaite Splice Base/Previous Knot]]

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

[[Data:Hoste-Thistlethwaite Splice Base/Next Knot]]

@@ Line 1: / Line 1: @@
-<!-- <* (* -->{{Splice Template Notice}}<!-- *) *> -->
+<!-- <* (* -->{{Splice Template Notice}}<!-- *) *> --> <!--
+<*K=Knot[ThisKnot];{n,k}=List@@K;*> -->
-<!-- -->
+{{Hoste-Thistlethwaite Knot Page|
-<!-- <*K=Knot[ThisKnot];{n,k}=Drop[List@@K,{2}];*> -->
+n = <*n*> |
-<!-- -->
+t = <*If[AlternatingQ[K],"a","n"]*> |
-<!-- provide an anchor so we can return to the top of the page -->
+k = <*k*> |
-<span id="top"></span>
+KnotilusURL = <*KnotilusURL[K]*> |
-<!-- -->
+same_alexander  = <*  alex = Alexander[K][t];
-<!-- this relies on transclusion for next and previous links -->
+                      others = DeleteCases[Select[AllKnots[], (alex === Alexander[#][t])&], K];
-{{Knot Navigation Links|ext=gif}}
+                      If[others === {}, "", StringJoin[("[["<>NameString[#]<>"]], ")& /@ others]]
-{{Hoste-Thistlethwaite Knot Page Header|n=<*n*>|t=<*If[AlternatingQ[K],"a","n"]*>|k=<*k*>|KnotilusURL=<*KnotilusURL[K]*>}}
+                  *> |
+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]]
-{{:{{PAGENAME}} Further Notes and Views}}
+                  *> |
+khovanov_table  = <*TabularKh[Kh[K][q, t], KnotSignature[K]+{1,-1}]*> |
-{{Knot Presentations}}
+coloured_jones_2 = <*ColouredJones[K, 2][q]*> |
-{{3D Invariants}}
+coloured_jones_3 = <*ColouredJones[K, 3][q]*> |
-{{4D Invariants}}
+coloured_jones_4 = <*ColouredJones[K, 4][q]*> |
-{{Polynomial Invariants}}
+coloured_jones_5 = <*ColouredJones[K, 5][q]*> |
-{{Vassiliev Invariants}}
+coloured_jones_6 = <*ColouredJones[K, 6][q]*> |
+coloured_jones_7 = <*ColouredJones[K, 7][q]*> |
-{{Khovanov Homology|table=<*TabularKh[Kh[K][q, t], KnotSignature[K]+{1,-1}]*>}}
+computer_talk =
+         <table>
-{{Computer Talk Header}}
+         <tr valign=top>
+         <td><pre style="color: blue; border: 0px; padding: 0em">In[1]:=&nbsp;&nbsp;&nbsp;&nbsp;</pre></td>
-<table>
+         <td align=left><pre style="color: red; border: 0px; padding: 0em">&lt;&lt; KnotTheory`</pre></td>
-<tr valign=top>
+         </tr>
-<td><pre style="color: blue; border: 0px; padding: 0em">In[1]:=&nbsp;&nbsp;&nbsp;&nbsp;</pre></td>
+         <tr valign=top><td colspan=2><*InOut[1]; KnotTheoryWelcomeMessage[]*></td></tr>
-<td align=left><pre style="color: red; border: 0px; padding: 0em">&lt;&lt; KnotTheory`</pre></td>
+         <*InOut["Crossings[``]", K]*>
-</tr>
+         <*InOut["PD[``]", K]*>
-<tr valign=top><td colspan=2><pre style="border: 0px; padding: 0em"><*InOut[1]; KnotTheoryWelcomeMessage[]*></pre></td></tr>
-<*InOut["Crossings[``]", K]*>
+         <*InOut["GaussCode[``]", K]*>
-<*InOut["PD[``]", K]*>
+         <*InOut["BR[``]", K]*>
-<*InOut["GaussCode[``]", K]*>
+         <*InOut["alex = Alexander[``][t]", K]*>
-<*InOut["BR[``]", K]*>
+         <*InOut["Conway[``][z]", K]*>
-<*InOut["alex = Alexander[``][t]", K]*>
+         <*InOut["Select[AllKnots[], (alex === Alexander[#][t])&]"]*>
-<*InOut["Conway[``][z]", K]*>
+         <*InOut["{KnotDet[`1`], KnotSignature[`1`]}", K]*>
-<*InOut["Select[AllKnots[], (alex === Alexander[#][t])&]"]*>
+         <*InOut["J=Jones[``][q]", K]*>
+         <*InOut[
-<*InOut["{KnotDet[`1`], KnotSignature[`1`]}", K]*>
+           "Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]"
-<*InOut["J=Jones[``][q]", K]*>
+         ]*>
-<*InOut[
+         <* If[Crossings[K]<=18, Include["ColouredJonesM.mhtml"] ,""] *>
-  "Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]"
+         <*InOut["A2Invariant[``][q]", K]*>
-]*>
+         <*InOut["Kauffman[``][a, z]", K]*>
-<* If[Crossings[K]<=18, Include["ColouredJonesM.mhtml"] ,""] *>
-<*InOut["A2Invariant[``][q]", K]*>
+         <*InOut["{Vassiliev[2][`1`], Vassiliev[3][`1`]}", K ]*>
-<*InOut["Kauffman[``][a, z]", K]*>
+         <*InOut["Kh[``][q, t]", K]*>
+         </table> }}
-<*InOut["{Vassiliev[2][`1`], Vassiliev[3][`1`]}", K ]*>
-<*InOut["Kh[``][q, t]", K]*>
-</table>
-<* (* <!-- *) *> [[Category:Knot Page]] <* (* --> *) *>

Hoste-Thistlethwaite Splice Base: Difference between revisions

Revision as of 09:44, 30 August 2005

Contents

Knot presentations

Three dimensional invariants

Four dimensional invariants

Polynomial invariants

"Similar" Knots (within the Atlas)

Vassiliev invariants

Khovanov Homology

Computer Talk

Modifying This Page

Navigation menu

Page actions

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