Template:Polynomial Invariants: Difference between revisions

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<div style="border: solid pink 1px">
{| style="width: 35em; float: right; font-size: 95%; margin: 0 0 1em 1em; border-left: 1px solid #aaa; border-collapse: collapse;"
===Polynomial invariants===
|-
<table width=98%>
| style="padding: 0.5em; background: #F7D79C; font-size: larger;" | '''{{{name}}}'''
<tr valign=top>
|-
<td>'''[[The Alexander-Conway Polynomial|Alexander polynomial]]'''</td>
| style="text-align: center; padding: 0.5em 0;" | {{{image}}}
<td>{{Data:{{PAGENAME}}/Alexander Polynomial}}</td>
|-
</tr>
|
<tr valign=top>
{| style="margin-left: 1em; border-top: 1px #AAA solid;"
<td>'''[[The Alexander-Conway Polynomial|Conway polynomial]]'''</td>
|-
<td>{{Data:{{PAGENAME}}/Conway Polynomial}}</td>
|'''Planar diagram presentation'''
</tr>
|style="padding-left: 1em;" | {{Data:{{{name}}}/PD Presentation}}
<tr valign=top>
|-
<td>'''[[The Alexander-Conway Polynomial|2nd Alexander ideal]] ([[Data:{{PAGENAME}}/2nd AlexanderIdeal|db]], [[Alexander ideals data sources|data sources]])'''</td>
|'''Gauss code'''
<td>{{Data:{{PAGENAME}}/2nd AlexanderIdeal}}</td>
|style="padding-left: 1em;" | {{Data:{{{name}}}/Gauss Code}}
</tr>
|-
<tr valign=top>
|'''Dowker-Thistlethwaite code'''
<td>'''[[The Determinant and the Signature|Determinant and Signature]]'''</td>
|style="padding-left: 1em;" | {{Data:{{{name}}}/DT Code}}
<td>{ {{Data:{{PAGENAME}}/Determinant}}, {{Data:{{PAGENAME}}/Signature}} }</td>
|-
</tr><tr valign=top>
|'''Jones polynomial'''
<td>'''[[The Jones Polynomial|Jones polynomial]]'''</td>
|style="padding-left: 1em;" | {{Data:{{{name}}}/Jones Polynomial}}
<td>{{Data:{{PAGENAME}}/Jones Polynomial}}</td>
|-
</tr><tr valign=top>
|'''Alexander polynomial'''
<td>'''[[The HOMFLY-PT Polynomial|HOMFLY-PT polynomial]]''' ([[Data:{{PAGENAME}}/HOMFLYPT Polynomial|db]], [[HOMFLY-PT polynomial data sources|data sources]])</td>
|style="padding-left: 1em;" | {{Data:{{{name}}}/Alexander Polynomial}}
<td>{{Data:{{PAGENAME}}/HOMFLYPT Polynomial}}</td>
|-
</tr><tr valign=top>
|'''Conway polynomial'''
<td>'''[[The Kauffman Polynomial|Kauffman polynomial]]''' ([[Data:{{PAGENAME}}/Kauffman Polynomial|db]], [[Kauffman polynomial data sources|data sources]])</td>
|style="padding-left: 1em;" | {{Data:{{{name}}}/Conway Polynomial}}
<td>{{Data:{{PAGENAME}}/Kauffman Polynomial}}</td>
|-
</tr>
|'''Determinant'''
<!--<tr valign=top>
|style="padding-left: 1em;" | {{Data:{{{name}}}/Determinant}}
<td>'''[[User:Ben/KRhomology|Khovanov-Rozansky polynomial]]'''</td>
|-
<td>{{Data:{{PAGENAME}}/Khovanov Rozansky Polynomial}}</td>
|'''Signature'''
</tr>-->
|style="padding-left: 1em;" | {{Data:{{{name}}}/Signature}}
<tr valign=top>
<td>'''[[Quantum knot invariants|The A2 invariant]]'''</td>
<td>{{Data:{{PAGENAME}}/QuantumInvariant/A2/1,0}}</td>
</tr>
<tr valign=top>
<td>'''[[Quantum knot invariants|The G2 invariant]]'''</td>
<td>{{Data:{{PAGENAME}}/QuantumInvariant/G2/1,0}}</td>
</tr>
</table>

<div class="NavFrame"><div class="NavHead">Further Quantum Invariants</div>
<div class="NavContent">{{:{{PAGENAME}} Quantum Invariants|quantum invariants}}.
</div></div>

<div class="NavFrame"><div class="NavHead">Computer Talk</div>
<div class="NavContent">The above data is available with the [http://www.wolfram.com/ Mathematica] package [[The Mathematica Package KnotTheory`|<code>KnotTheory`</code>]], as shown in the (simulated) Mathematica session below. Your input (in <font color=red>red</font>) 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 [http://katlas.math.toronto.edu/svn/KnotAtlas/PolynomialInvariantsSession.nb PolynomialInvariantsSession.nb].

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

{{In|n=1|in=<nowiki>AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`</nowiki>}}

{{Print|Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.
Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.}}

{|
|- align=left valign=top
|width=70px|<code style="color: blue; border: 0px; padding: 0em">In[3]:=</code>
|<code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);">K = Knot["{{PAGENAME}}"];</code>
|}
|}

{{In|n=4|in=<nowiki>Alexander[K][t]</nowiki>}}

{{Message|KnotTheory::loading: Loading precomputed data in PD4Knots`.}}

{{FreeStyleOut|4}}
| {{Data:{{PAGENAME}}/Alexander Polynomial}}
|}
|}

{{In|n=5|in=<nowiki>Conway[K][z]</nowiki>}}

{{FreeStyleOut|5}}
| {{Data:{{PAGENAME}}/Conway Polynomial}}
|}

{{In|n=6|in=<nowiki>Alexander[K, 2][t]</nowiki>}}

{{Message|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.}}

{{FreeStyleOut|6}}
| {{Data:{{PAGENAME}}/2nd AlexanderIdeal}}
|}

{{In|n=7|in=<nowiki>{KnotDet[K], KnotSignature[K]}</nowiki>}}

{{FreeStyleOut|7}}
| { {{Data:{{PAGENAME}}/Determinant}}, {{Data:{{PAGENAME}}/Signature}} }
|}

{{In|n=8|in=<nowiki>Jones[K][q]</nowiki>}}

{{Message|KnotTheory::loading: Loading precomputed data in Jones4Knots`.}}

{{FreeStyleOut|8}}
| {{Data:{{PAGENAME}}/Jones Polynomial}}
|}

{{In|n=9|in=<nowiki>HOMFLYPT[K][a, z]</nowiki>}}

{{Message|KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.}}

{{FreeStyleOut|9}}
| {{Data:{{PAGENAME}}/HOMFLYPT Polynomial}}
|}

{{In|n=10|in=<nowiki>Kauffman[K][a, z]</nowiki>}}

{{Message|KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.}}

{{FreeStyleOut|10}}
| {{Data:{{PAGENAME}}/Kauffman Polynomial}}
|}
</div></div>

Latest revision as of 12:53, 1 December 2007

Polynomial invariants

Alexander polynomial Data:Polynomial Invariants/Alexander Polynomial
Conway polynomial Data:Polynomial Invariants/Conway Polynomial
2nd Alexander ideal (db, data sources) Data:Polynomial Invariants/2nd AlexanderIdeal
Determinant and Signature { Data:Polynomial Invariants/Determinant, Data:Polynomial Invariants/Signature }
Jones polynomial Data:Polynomial Invariants/Jones Polynomial
HOMFLY-PT polynomial (db, data sources) Data:Polynomial Invariants/HOMFLYPT Polynomial
Kauffman polynomial (db, data sources) Data:Polynomial Invariants/Kauffman Polynomial
The A2 invariant Data:Polynomial Invariants/QuantumInvariant/A2/1,0
The G2 invariant Data:Polynomial Invariants/QuantumInvariant/G2/1,0
Further Quantum Invariants
Polynomial Invariants 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["Polynomial Invariants"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= Data:Polynomial Invariants/Alexander Polynomial
In[5]:= Conway[K][z]
Out[5]= Data:Polynomial Invariants/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:Polynomial Invariants/2nd AlexanderIdeal
In[7]:= {KnotDet[K], KnotSignature[K]}
Out[7]= { Data:Polynomial Invariants/Determinant, Data:Polynomial Invariants/Signature }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= Data:Polynomial Invariants/Jones Polynomial
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= Data:Polynomial Invariants/HOMFLYPT Polynomial
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= Data:Polynomial Invariants/Kauffman Polynomial
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