The A2 Invariant

2009-01-05T05:08:24Z

OrbasLitrt: boceltcsit

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{{Manual TOC Sidebar}}

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}}:

{{Startup Note}}



{{HelpLine|
n = 1 |
in = <nowiki>A2Invariant</nowiki> |
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>}}


{| align=center
|[[Image:10_22.gif|thumb|180px|<center>[[10_22]]</center>]]
|[[Image:10_35.gif|thumb|none|<center>[[10_35]]</center>|180px]]
|}

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:



{{InOut|
n = 2 |
in = <nowiki>Jones[Knot[10, 22]][q] == Jones[Knot[10, 35]][q]</nowiki> |
out= <nowiki>True</nowiki>}}




{{InOut|
n = 3 |
in = <nowiki>A2Invariant[Knot[10, 22]][q]</nowiki> |
out= <nowiki> -12 -8 -6 -4 2 4 6 8 10 12 14
-1 + q + q + q - q + -- - q - 2 q + q - q + q + q +
2
q

18
q</nowiki>}}




{{InOut|
n = 4 |
in = <nowiki>A2Invariant[Knot[10, 35]][q]</nowiki> |
out= <nowiki> -14 -12 -10 -8 2 2 2 6 8 10 14 16
q + q - q + q - -- + -- + q - q + q - 2 q + q - q +
4 2
q q

18 20
q + q</nowiki>}}


The <math>A2</math> invariant attains 2163 values on the 2226 knots and links known to <code>KnotTheory</code>:



{{In|
n = 5 |
in = <nowiki>all = Join[AllKnots[], AllLinks[]];</nowiki>}}




{{InOut|
n = 6 |
in = <nowiki>Length /@ {Union[A2Invariant[#][q]& /@ all], all}</nowiki> |
out= <nowiki>{2163, 2226}</nowiki>}}


{{note|Khovanov}} M. Khovanov, ''<math>sl(3)</math> link homology I'', {{arXiv|math.QA/0304375}}.

{{note|Kuperberg}} G. Kuperberg, ''Spiders for rank 2 Lie algebras'', Comm. Math. Phys. '''180''' (1996) 109-151, {{arXiv|q-alg/9712003}}.

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The A2 Invariant