Both the $\mathfrak{sl}_2$ and $\mathfrak{sl}_3$ quantum framed link invariants can be computed using linear skeins. The first being computed using the Kauffman bracket and the second using a similar bracket which involves trivalent graphs. The recursive rules prescribed to compute these two invariants via skeins are very similar with the $\mathfrak{sl}_3$ involving a longer list of computational rules.

I want to know if the quantum $\mathfrak{sl}_3$ invariant does a better job compare to the $\mathfrak{sl}_2$ invariant when it comes to distinguishing links? Are there well known examples of links with the same $\mathfrak{sl}_2$ invariant, but differing $\mathfrak{sl}_3$ invariant?

Thanks in advance!