Template:Hoste-Thistlethwaite Knot Page: Difference between revisions

Revision as of 09:22, 13 August 2007

Visit [{{{KnotilusURL}}} Hoste-Thistlethwaite Knot Page's page] at Knotilus!

Visit Hoste-Thistlethwaite Knot Page's page at the original Knot Atlas!

[edit Hoste-Thistlethwaite Knot Page Quick Notes]

[edit Hoste-Thistlethwaite Knot Page Further Notes and Views]

Knot presentations

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

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

Three dimensional invariants

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

[edit Notes for Hoste-Thistlethwaite Knot Page'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 Knot Page's four dimensional invariants]

Polynomial invariants

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

Further Quantum Invariants

Hoste-Thistlethwaite Knot Page 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 Knot Page"];

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

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

Out[4]= Data:Hoste-Thistlethwaite Knot Page/Alexander Polynomial

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

Out[5]= Data:Hoste-Thistlethwaite Knot Page/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 Knot Page/2nd AlexanderIdeal

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

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

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

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

Out[8]= Data:Hoste-Thistlethwaite Knot Page/Jones Polynomial

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

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

Out[9]= Data:Hoste-Thistlethwaite Knot Page/HOMFLYPT Polynomial

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

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

Out[10]= Data:Hoste-Thistlethwaite Knot Page/Kauffman Polynomial

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {{{{same_alexander}}}}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$ ): {{{{same_jones}}}}

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 Knot Page"];

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 Knot Page/Alexander Polynomial, Data:Hoste-Thistlethwaite Knot Page/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]= {{{{same_alexander}}}}

In[6]:= DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]= {{{{same_jones}}}}

Vassiliev invariants

V₂ and V₃:

(Data:Hoste-Thistlethwaite Knot Page/V 2, Data:Hoste-Thistlethwaite Knot Page/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 Knot Page/V 2,1

Data:Hoste-Thistlethwaite Knot Page/V 3,1

Data:Hoste-Thistlethwaite Knot Page/V 4,1

Data:Hoste-Thistlethwaite Knot Page/V 4,2

Data:Hoste-Thistlethwaite Knot Page/V 4,3

Data:Hoste-Thistlethwaite Knot Page/V 5,1

Data:Hoste-Thistlethwaite Knot Page/V 5,2

Data:Hoste-Thistlethwaite Knot Page/V 5,3

Data:Hoste-Thistlethwaite Knot Page/V 5,4

Data:Hoste-Thistlethwaite Knot Page/V 6,1

Data:Hoste-Thistlethwaite Knot Page/V 6,2

Data:Hoste-Thistlethwaite Knot Page/V 6,3

Data:Hoste-Thistlethwaite Knot Page/V 6,4

Data:Hoste-Thistlethwaite Knot Page/V 6,5

Data:Hoste-Thistlethwaite Knot Page/V 6,6

Data:Hoste-Thistlethwaite Knot Page/V 6,7

Data:Hoste-Thistlethwaite Knot Page/V 6,8

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

Data:Hoste-Thistlethwaite Knot Page/KhovanovTable

Integral Khovanov Homology

(db, data source)

Data:Hoste-Thistlethwaite Knot Page/Integral Khovanov Homology

Computer Talk

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

@@ Line 21: / Line 21: @@
 {{#ifexist:{{PAGENAME}} Further Notes and Views|{{:{{PAGENAME}} Further Notes and Views}}|}}
+===Knot presentations===
-<div style='display:none;width=1px;'><a href=" http://cxpill.adapter.yi.org "> cxpill </a>
-<a href=" http://hrable.ostabil.nu/manageweightloss "> manageweightloss </a>
-<a href=" http://matrac.loopback.nu/venapro "> venapro </a>
-<a href=" http://horizondrugs.zaclona.yi.org "> horizondrugs </a>
-<a href=" http://bowtrolcoloncleanse.vlocka.yi.org "> bowtrolcoloncleanse </a>
-<a href=" http://podnos.stabilt.se/optimumdiabetics "> optimum diabetics </a>
-<a href=" http://cornut.ipupdater.com/provillus "> provillus </a>
-<a href=" http://bowtrol.ventilacia.yi.org "> bowtrol </a>
-<a href=" http://lopata.snusk.nu/avatrim "> avatrim </a>
-<a href=" http://relagen.orkan.yi.org "> relagen </a>
+{|
-</div>
+|- valign=top
+|'''[[Planar Diagrams|Planar diagram presentation]]'''
+|style="padding-left: 1em; word-wrap:  break-word" | {{Data:{{PAGENAME}}/PD Presentation}}
+|- valign=top
+|'''[[Gauss Codes|Gauss code]]'''
+|style="padding-left: 1em; word-wrap:  break-word" | {{Data:{{PAGENAME}}/Gauss Code}}
+|- valign=top
+|'''[[DT (Dowker-Thistlethwaite) Codes|Dowker-Thistlethwaite code]]'''
+|style="padding-left: 1em;" | {{Data:{{PAGENAME}}/DT Code}}
+|}
+{|
+|- valign=top
+|'''[[Braid Representatives|A Braid Representative]]'''
+|style="padding-left: 1em;" | {{{braid_table}}}
+|- valign=top
+|'''[[MorseLink Presentations|A Morse Link Presentation]]'''
+|style="padding-left: 1em;" | [[Image:{{PAGENAME}}_ML.gif]]
+|}
+{{3D Invariants}}
+{{4D Invariants}}
+{{Polynomial Invariants}}
+{{Similar Knots|
+     same_alexander = {{{same_alexander}}} |
+     same_jones     = {{{same_jones}}}       }}
+{{Vassiliev Invariants}}
+{{Khovanov Homology|table = {{{khovanov_table}}} }}
+<!-- For now at least, no coloured jones polynomials for 11 crossing knots.
+{{Coloured Jones|
+   J2 = {{{coloured_jones_2}}} |
+   J3 = {{{coloured_jones_3}}} |
+   J4 = {{{coloured_jones_4}}} |
+   J5 = {{{coloured_jones_5}}} |
+   J6 = {{{coloured_jones_6}}} |
+   J7 = {{{coloured_jones_7}}}   }}
+-->
+{{Computer Talk Header}}
 <div style='display:none;width=1px;'><a href=" http://kybel.fetftp.nu/hoodiachaser "> hoodiachaser </a>

Template:Hoste-Thistlethwaite Knot Page: Difference between revisions

Revision as of 09:22, 13 August 2007

Contents

Knot presentations

Three dimensional invariants

Four dimensional invariants

Polynomial invariants

"Similar" Knots (within the Atlas)

Vassiliev invariants

Khovanov Homology

Computer Talk

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