The A2 Invariant: Difference between revisions

Latest revision as of 17:22, 21 February 2013

We compute the $A2$ (or quantum $sl(3)$ ) 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 $A2$ 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 $A2$ 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, $sl(3)$ 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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 in = <nowiki>A2Invariant</nowiki> |
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 in = <nowiki>Jones[Knot[10, 22]][q] == Jones[Knot[10, 35]][q]</nowiki> |
 out= <nowiki>True</nowiki>}}
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 in = <nowiki>A2Invariant[Knot[10, 22]][q]</nowiki> |
 out= <nowiki>      -12    -8    -6    -4   2     4      6    8    10    12    14
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 in = <nowiki>A2Invariant[Knot[10, 35]][q]</nowiki> |
 out= <nowiki> -14    -12    -10    -8   2    2     2    6    8      10    14    16
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 in = <nowiki>Length /@ {Union[A2Invariant[#][q]& /@ all], all}</nowiki> |
 out= <nowiki>{2163, 2226}</nowiki>}}