The A2 Invariant: Difference between revisions

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<!--$$?A2Invariant$$-->
<!--$$?A2Invariant$$-->
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{{HelpLine|
{{Help1|n=2|s=A2Invariant}}
n = 2 |
⚫
A2Invariant[L][q] computes the A2 (sl(3)) invariant of a knot or link L as a function of the variable q.
in = <nowiki>A2Invariant</nowiki> |
{{Help2}}
⚫
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>}}
<!--END-->
<!--END-->


{| align=center
{| align=center
|[[Image:5_1.gif|thumb|180px|<center>[[10_22]]</center>]]
|[[Image:10_22.gif|thumb|180px|<center>[[10_22]]</center>]]
|[[Image:10_132.gif|thumb|none|<center>[[10_35]]</center>|180px]]
|[[Image:10_35.gif|thumb|none|<center>[[10_35]]</center>|180px]]
|}
|}


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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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{{InOut1|n=3}}
{{InOut|
n = 3 |
<pre style="color: red; border: 0px; padding: 0em"><nowiki>Jones[Knot[10, 22]][q] == Jones[Knot[10, 35]][q]</nowiki></pre>
in = <nowiki>Jones[Knot[10, 22]][q] == Jones[Knot[10, 35]][q]</nowiki> |
{{InOut2|n=3}}<pre style="border: 0px; padding: 0em"><nowiki>True</nowiki></pre>
out= <nowiki>True</nowiki>}}
{{InOut3}}
<!--END-->
<!--END-->


<!--$$A2Invariant[Knot[10, 22]][q]$$-->
<!--$$A2Invariant[Knot[10, 22]][q]$$-->
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{{InOut1|n=4}}
{{InOut|
n = 4 |
<pre style="color: red; border: 0px; padding: 0em"><nowiki>A2Invariant[Knot[10, 22]][q]</nowiki></pre>
in = <nowiki>A2Invariant[Knot[10, 22]][q]</nowiki> |
{{InOut2|n=4}}<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
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
2
q</nowiki></pre>
q
{{InOut3}}
18
q</nowiki>}}
<!--END-->
<!--END-->


<!--$$A2Invariant[Knot[10, 35]][q]$$-->
<!--$$A2Invariant[Knot[10, 35]][q]$$-->
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{{InOut1|n=5}}
{{InOut|
n = 5 |
<pre style="color: red; border: 0px; padding: 0em"><nowiki>A2Invariant[Knot[10, 35]][q]</nowiki></pre>
in = <nowiki>A2Invariant[Knot[10, 35]][q]</nowiki> |
{{InOut2|n=5}}<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
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
4 2
q q</nowiki></pre>
q q
{{InOut3}}
18 20
q + q</nowiki>}}
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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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{{In1|n=6}}
{{In|
n = 6 |
<pre style="color: red; border: 0px; padding: 0em"><nowiki>all = Join[AllKnots[], AllLinks[]];</nowiki></pre>
in = <nowiki>all = Join[AllKnots[], AllLinks[]];</nowiki>}}
{{In2}}
<!--END-->
<!--END-->


<!--$$Length /@ {Union[A2Invariant[#][q]& /@ all], all}$$-->
<!--$$Length /@ {Union[A2Invariant[#][q]& /@ all], all}$$-->
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{{InOut1|n=7}}
{{InOut|
n = 7 |
<pre style="color: red; border: 0px; padding: 0em"><nowiki>Length /@ {Union[A2Invariant[#][q]& /@ all], all}</nowiki></pre>
in = <nowiki>Length /@ {Union[A2Invariant[#][q]& /@ all], all}</nowiki> |
{{InOut2|n=7}}<pre style="border: 0px; padding: 0em"><nowiki>{2163, 2226}</nowiki></pre>
out= <nowiki>{2163, 2226}</nowiki>}}
{{InOut3}}
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<!--END-->


Latest revision as of 17:22, 21 February 2013


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[2]:= ?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[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 -1 + q + q + q - q + -- - q - 2 q + q - q + q + q + 2 q 18 q
In[5]:= A2Invariant[Knot[10, 35]][q]
Out[5]= -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

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}

[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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