The A2 Invariant

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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.

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 $A2$ 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 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 $A2$ 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.