The A2 Invariant: Difference between revisions
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{{Help1|n=2|s=A2Invariant}} |
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A2Invariant[L][q] computes the A2 (sl(3)) invariant of a knot or link L as a function of the variable q. |
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<!--$$Jones[Knot[10, 22]][q] == Jones[Knot[10, 35]][q]$$--> |
<!--$$Jones[Knot[10, 22]][q] == Jones[Knot[10, 35]][q]$$--> |
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<pre style="color: red; border: 0px; padding: 0em"><nowiki>Jones[Knot[10, 22]][q] == Jones[Knot[10, 35]][q]</nowiki></pre> |
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{{InOut2|n=3}}<pre style="border: 0px; padding: 0em"><nowiki>True</nowiki></pre> |
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<!--$$A2Invariant[Knot[10, 22]][q]$$--> |
<!--$$A2Invariant[Knot[10, 22]][q]$$--> |
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<pre style="color: red; border: 0px; padding: 0em"><nowiki>A2Invariant[Knot[10, 22]][q]</nowiki></pre> |
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{{InOut2|n=4}}<pre style="border: 0px; padding: 0em"><nowiki> -12 -8 -6 -4 2 4 6 8 10 12 14 18 |
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-1 + q + q + q - q + -- - q - 2 q + q - q + q + q + q |
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2 |
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q</nowiki></pre> |
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<!--$$A2Invariant[Knot[10, 35]][q]$$--> |
<!--$$A2Invariant[Knot[10, 35]][q]$$--> |
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<pre style="color: red; border: 0px; padding: 0em"><nowiki>A2Invariant[Knot[10, 35]][q]</nowiki></pre> |
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{{InOut2|n=5}}<pre style="border: 0px; padding: 0em"><nowiki> -14 -12 -10 -8 2 2 2 6 8 10 14 16 18 20 |
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q + q - q + q - -- + -- + q - q + q - 2 q + q - q + q + q |
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4 2 |
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q q</nowiki></pre> |
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The <math>A2</math> invariant attains <!--$all=Join[AllKnots[], AllLinks[]]; Length[Union[A2Invariant[#][q]& /@ all]]$--><!--END--> values on the <!--$Length[all]$--><!--END--> knots and links known to <code>KnotTheory</code>: |
The <math>A2</math> invariant attains <!--$all=Join[AllKnots[], AllLinks[]]; Length[Union[A2Invariant[#][q]& /@ all]]$--><!--The content to END was generated by WikiSplice: do not edit; see manual.-->2163<!--END--> values on the <!--$Length[all]$--><!--The content to END was generated by WikiSplice: do not edit; see manual.-->2226<!--END--> knots and links known to <code>KnotTheory</code>: |
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<!--$$all = Join[AllKnots[], AllLinks[]];$$--> |
<!--$$all = Join[AllKnots[], AllLinks[]];$$--> |
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<pre style="color: red; border: 0px; padding: 0em"><nowiki>all = Join[AllKnots[], AllLinks[]];</nowiki></pre> |
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<!--$$Length /@ {Union[A2Invariant[#][q]& /@ all], all}$$--> |
<!--$$Length /@ {Union[A2Invariant[#][q]& /@ all], all}$$--> |
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<pre style="color: red; border: 0px; padding: 0em"><nowiki>Length /@ {Union[A2Invariant[#][q]& /@ all], all}</nowiki></pre> |
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{{InOut2|n=7}}<pre style="border: 0px; padding: 0em"><nowiki>{2163, 2226}</nowiki></pre> |
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Revision as of 22:05, 26 August 2005
We compute the (or quantum ) invariant using the normalization and formulas of [Khovanov], which in itself follows [Kuperberg]:
(For In[1] see Setup)
In[2]:= ?A2Invariant
A2Invariant[L][q] computes the A2 (sl(3)) invariant of a knot or link L as a function of the variable q. |
As an example, let us check that the knots 10_22 and 10_35 have the same Jones polynomial but different invariants:
| In[3]:= |
Jones[Knot[10, 22]][q] == Jones[Knot[10, 35]][q] |
| Out[3]= | True |
| In[4]:= |
A2Invariant[Knot[10, 22]][q] |
| Out[4]= | -12 -8 -6 -4 2 4 6 8 10 12 14 18
-1 + q + q + q - q + -- - q - 2 q + q - q + q + q + q
2
q
|
| In[5]:= |
A2Invariant[Knot[10, 35]][q] |
| Out[5]= | -14 -12 -10 -8 2 2 2 6 8 10 14 16 18 20
q + q - q + q - -- + -- + q - q + q - 2 q + q - q + q + q
4 2
q q
|
The invariant attains 2163 values on the 2226 knots and links known to KnotTheory:
| In[6]:= |
all = Join[AllKnots[], AllLinks[]]; |
| In[7]:= |
Length /@ {Union[A2Invariant[#][q]& /@ all], all}
|
| Out[7]= | {2163, 2226}
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[Khovanov] ^ M. Khovanov, link homology I, arXiv:math.QA/0304375.
[Kuperberg] ^ G. Kuperberg, Spiders for rank 2 Lie algebras, Comm. Math. Phys. 180 (1996) 109-151, arXiv:q-alg/9712003.
