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https://katlas.org/index.php?title=The_A2_Invariant&diff=1693808
The A2 Invariant
2009-05-27T08:20:34Z
<p>AleltAcelb: </p>
<hr />
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http://www.textc4tvarouorri.com <br />
{{Manual TOC Sidebar}}<br />
<br />
We compute the <math>A2</math> (or quantum <math>sl(3)</math>) invariant using the normalization and formulas of {{ref|Khovanov}}, which in itself follows {{ref|Kuperberg}}:<br />
<br />
{{Startup Note}}<br />
<br />
<!--$$?A2Invariant$$--><br />
<!--Robot Land, no human edits to "END"--><br />
{{HelpLine|<br />
n = 1 |<br />
in = <nowiki>A2Invariant</nowiki> |<br />
out= <nowiki>A2Invariant[L][q] computes the A2 (sl(3)) invariant of a knot or link L as a function of the variable q.</nowiki>}}<br />
<!--END--><br />
<br />
{| align=center<br />
|[[Image:10_22.gif|thumb|180px|<center>[[10_22]]</center>]]<br />
|[[Image:10_35.gif|thumb|none|<center>[[10_35]]</center>|180px]]<br />
|}<br />
<br />
As an example, let us check that the knots [[10_22]] and [[10_35]] have the same Jones polynomial but different <math>A2</math> invariants:<br />
<br />
<!--$$Jones[Knot[10, 22]][q] == Jones[Knot[10, 35]][q]$$--><br />
<!--Robot Land, no human edits to "END"--><br />
{{InOut|<br />
n = 2 |<br />
in = <nowiki>Jones[Knot[10, 22]][q] == Jones[Knot[10, 35]][q]</nowiki> |<br />
out= <nowiki>True</nowiki>}}<br />
<!--END--><br />
<br />
<!--$$A2Invariant[Knot[10, 22]][q]$$--><br />
<!--Robot Land, no human edits to "END"--><br />
{{InOut|<br />
n = 3 |<br />
in = <nowiki>A2Invariant[Knot[10, 22]][q]</nowiki> |<br />
out= <nowiki> -12 -8 -6 -4 2 4 6 8 10 12 14<br />
-1 + q + q + q - q + -- - q - 2 q + q - q + q + q + <br />
2<br />
q<br />
<br />
18<br />
q</nowiki>}}<br />
<!--END--><br />
<br />
<!--$$A2Invariant[Knot[10, 35]][q]$$--><br />
<!--Robot Land, no human edits to "END"--><br />
{{InOut|<br />
n = 4 |<br />
in = <nowiki>A2Invariant[Knot[10, 35]][q]</nowiki> |<br />
out= <nowiki> -14 -12 -10 -8 2 2 2 6 8 10 14 16<br />
q + q - q + q - -- + -- + q - q + q - 2 q + q - q + <br />
4 2<br />
q q<br />
<br />
18 20<br />
q + q</nowiki>}}<br />
<!--END--><br />
<br />
The <math>A2</math> invariant attains <!--$all=Join[AllKnots[], AllLinks[]]; Length[Union[A2Invariant[#][q]& /@ all]]$--><!--Robot Land, no human edits to "END"-->2163<!--END--> values on the <!--$Length[all]$--><!--Robot Land, no human edits to "END"-->2226<!--END--> knots and links known to <code>KnotTheory</code>:<br />
<br />
<!--$$all = Join[AllKnots[], AllLinks[]];$$--><br />
<!--Robot Land, no human edits to "END"--><br />
{{In|<br />
n = 5 |<br />
in = <nowiki>all = Join[AllKnots[], AllLinks[]];</nowiki>}}<br />
<!--END--><br />
<br />
<!--$$Length /@ {Union[A2Invariant[#][q]& /@ all], all}$$--><br />
<!--Robot Land, no human edits to "END"--><br />
{{InOut|<br />
n = 6 |<br />
in = <nowiki>Length /@ {Union[A2Invariant[#][q]& /@ all], all}</nowiki> |<br />
out= <nowiki>{2163, 2226}</nowiki>}}<br />
<!--END--><br />
<br />
{{note|Khovanov}} M. Khovanov, ''<math>sl(3)</math> link homology I'', {{arXiv|math.QA/0304375}}.<br />
<br />
{{note|Kuperberg}} G. Kuperberg, ''Spiders for rank 2 Lie algebras'', Comm. Math. Phys. '''180''' (1996) 109-151, {{arXiv|q-alg/9712003}}.</div>
AleltAcelb
https://katlas.org/index.php?title=The_A2_Invariant&diff=1693657
The A2 Invariant
2009-05-21T13:34:13Z
<p>AleltAcelb: </p>
<hr />
<div>http://www.textc4tvarouorri.com <br />
{{Manual TOC Sidebar}}<br />
<br />
We compute the <math>A2</math> (or quantum <math>sl(3)</math>) invariant using the normalization and formulas of {{ref|Khovanov}}, which in itself follows {{ref|Kuperberg}}:<br />
<br />
{{Startup Note}}<br />
<br />
<!--$$?A2Invariant$$--><br />
<!--Robot Land, no human edits to "END"--><br />
{{HelpLine|<br />
n = 1 |<br />
in = <nowiki>A2Invariant</nowiki> |<br />
out= <nowiki>A2Invariant[L][q] computes the A2 (sl(3)) invariant of a knot or link L as a function of the variable q.</nowiki>}}<br />
<!--END--><br />
<br />
{| align=center<br />
|[[Image:10_22.gif|thumb|180px|<center>[[10_22]]</center>]]<br />
|[[Image:10_35.gif|thumb|none|<center>[[10_35]]</center>|180px]]<br />
|}<br />
<br />
As an example, let us check that the knots [[10_22]] and [[10_35]] have the same Jones polynomial but different <math>A2</math> invariants:<br />
<br />
<!--$$Jones[Knot[10, 22]][q] == Jones[Knot[10, 35]][q]$$--><br />
<!--Robot Land, no human edits to "END"--><br />
{{InOut|<br />
n = 2 |<br />
in = <nowiki>Jones[Knot[10, 22]][q] == Jones[Knot[10, 35]][q]</nowiki> |<br />
out= <nowiki>True</nowiki>}}<br />
<!--END--><br />
<br />
<!--$$A2Invariant[Knot[10, 22]][q]$$--><br />
<!--Robot Land, no human edits to "END"--><br />
{{InOut|<br />
n = 3 |<br />
in = <nowiki>A2Invariant[Knot[10, 22]][q]</nowiki> |<br />
out= <nowiki> -12 -8 -6 -4 2 4 6 8 10 12 14<br />
-1 + q + q + q - q + -- - q - 2 q + q - q + q + q + <br />
2<br />
q<br />
<br />
18<br />
q</nowiki>}}<br />
<!--END--><br />
<br />
<!--$$A2Invariant[Knot[10, 35]][q]$$--><br />
<!--Robot Land, no human edits to "END"--><br />
{{InOut|<br />
n = 4 |<br />
in = <nowiki>A2Invariant[Knot[10, 35]][q]</nowiki> |<br />
out= <nowiki> -14 -12 -10 -8 2 2 2 6 8 10 14 16<br />
q + q - q + q - -- + -- + q - q + q - 2 q + q - q + <br />
4 2<br />
q q<br />
<br />
18 20<br />
q + q</nowiki>}}<br />
<!--END--><br />
<br />
The <math>A2</math> invariant attains <!--$all=Join[AllKnots[], AllLinks[]]; Length[Union[A2Invariant[#][q]& /@ all]]$--><!--Robot Land, no human edits to "END"-->2163<!--END--> values on the <!--$Length[all]$--><!--Robot Land, no human edits to "END"-->2226<!--END--> knots and links known to <code>KnotTheory</code>:<br />
<br />
<!--$$all = Join[AllKnots[], AllLinks[]];$$--><br />
<!--Robot Land, no human edits to "END"--><br />
{{In|<br />
n = 5 |<br />
in = <nowiki>all = Join[AllKnots[], AllLinks[]];</nowiki>}}<br />
<!--END--><br />
<br />
<!--$$Length /@ {Union[A2Invariant[#][q]& /@ all], all}$$--><br />
<!--Robot Land, no human edits to "END"--><br />
{{InOut|<br />
n = 6 |<br />
in = <nowiki>Length /@ {Union[A2Invariant[#][q]& /@ all], all}</nowiki> |<br />
out= <nowiki>{2163, 2226}</nowiki>}}<br />
<!--END--><br />
<br />
{{note|Khovanov}} M. Khovanov, ''<math>sl(3)</math> link homology I'', {{arXiv|math.QA/0304375}}.<br />
<br />
{{note|Kuperberg}} G. Kuperberg, ''Spiders for rank 2 Lie algebras'', Comm. Math. Phys. '''180''' (1996) 109-151, {{arXiv|q-alg/9712003}}.</div>
AleltAcelb