# The A2 Invariant

We compute the ${\displaystyle A2}$ (or quantum ${\displaystyle sl(3)}$) invariant using the normalization and formulas of [Khovanov], which in itself follows :

(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 ${\displaystyle A2}$ 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 ${\displaystyle A2}$ 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, ${\displaystyle sl(3)}$ 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.