The A2 Invariant: Difference between revisions

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<!--$$?A2Invariant$$-->
<!--$$?A2Invariant$$-->
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{{Help1|n=2|s=A2Invariant}}
{{Help1|n=1|s=A2Invariant}}
A2Invariant[L][q] computes the A2 (sl(3)) invariant of a knot or link L as a function of the variable q.
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>
<pre style="color: red; border: 0px; padding: 0em"><nowiki>Jones[Knot[10, 22]][q] == Jones[Knot[10, 35]][q]</nowiki></pre>
{{InOut2|n=3}}<pre style="border: 0px; padding: 0em"><nowiki>True</nowiki></pre>
{{InOut2|n=2}}<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>
<pre style="color: red; border: 0px; padding: 0em"><nowiki>A2Invariant[Knot[10, 22]][q]</nowiki></pre>
{{InOut2|n=4}}<pre style="border: 0px; padding: 0em"><nowiki> -12 -8 -6 -4 2 4 6 8 10 12 14 18
{{InOut2|n=3}}<pre style="border: 0px; padding: 0em"><nowiki> -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
-1 + q + q + q - q + -- - q - 2 q + q - q + q + q + q
2
2
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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>
<pre style="color: red; border: 0px; padding: 0em"><nowiki>A2Invariant[Knot[10, 35]][q]</nowiki></pre>
{{InOut2|n=5}}<pre style="border: 0px; padding: 0em"><nowiki> -14 -12 -10 -8 2 2 2 6 8 10 14 16 18 20
{{InOut2|n=4}}<pre style="border: 0px; padding: 0em"><nowiki> -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
q + q - q + q - -- + -- + q - q + q - 2 q + q - q + q + q
4 2
4 2
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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>:
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>:


<!--$$all = Join[AllKnots[], AllLinks[]];$$-->
<!--$$all = Join[AllKnots[], AllLinks[]];$$-->
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<pre style="color: red; border: 0px; padding: 0em"><nowiki>all = Join[AllKnots[], AllLinks[]];</nowiki></pre>
<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>
<pre style="color: red; border: 0px; padding: 0em"><nowiki>Length /@ {Union[A2Invariant[#][q]& /@ all], all}</nowiki></pre>
{{InOut2|n=7}}<pre style="border: 0px; padding: 0em"><nowiki>{2163, 2226}</nowiki></pre>
{{InOut2|n=6}}<pre style="border: 0px; padding: 0em"><nowiki>{2163, 2226}</nowiki></pre>
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Revision as of 19:44, 27 August 2005


KnotTheory`/Navigation 

The Mathematica Package KnotTheory`

We compute the (or quantum ) invariant using the normalization and formulas of [Khovanov], which in itself follows [Kuperberg]:

(For In[1] see Setup)

In[1]:= ?A2Invariant

A2Invariant[L][q] computes the A2 (sl(3)) invariant of a knot or link L as a function of the variable q.

10_22
10_35

As an example, let us check that the knots 10_22 and 10_35 have the same Jones polynomial but different invariants:

In[2]:= 
Jones[Knot[10, 22]][q] == Jones[Knot[10, 35]][q]
Out[2]= 
True
In[3]:= 
A2Invariant[Knot[10, 22]][q]
Out[3]= 
      -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[4]:= 
A2Invariant[Knot[10, 35]][q]
Out[4]= 
 -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[5]:= 
all = Join[AllKnots[], AllLinks[]];
In[6]:= 
Length /@ {Union[A2Invariant[#][q]& /@ all], all}
Out[6]= 
{2163, 2226}

[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.

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