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7.7 The A2 Invariant

We compute the A2 (or quantum $ sl(3)$) invariant using the normalization and formulas of [Kh3], which in itself follows [Ku]:

In[1]

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.

Figure 8: The knots $ 10_{22}$ and $ 10_{35}$.
\begin{figure}\centering { \includegraphics[height=2cm]{figs/10.22.eps} \qquad\includegraphics[height=2cm]{figs/10.35.eps} } \end{figure}

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
q    + q    - q    + q   - -- + -- + q  - q  + q  - 2 q   + q   - q   + q   + 
                            4    2
                           q    q
 
     20
>   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}


Dror Bar-Natan 2005-09-14