Is the quantum $\mathfrak{sl}_3$ invariant stronger than the quantum $\mathfrak{sl}_2$ invariant?

Question

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!

Answer 1

The quantum $\mathfrak{sl}_3$ invariant is a special case of the HOMFLY-PT polynomial, which is essentially the $\mathfrak{sl}_N$ invariant. That polynomial has two variables $q$ and $t$. The $q$ variable corresponds to the deformation parameter, and is the same $q$ as in the $\mathfrak{sl}_2$ and $\mathfrak{sl}_3$ invariants, while $t = q^N$ encodes the rank of the Lie algebra (warning: there are many conventions for exactly which variables to use. Frequently $z = t^{1/2} - t^{-1/2}$ or similar is used instead.)

There are examples of knots not distinguished by Jones polynomial ($\mathfrak{sl}_2$ invariant) that are distinguished by their HOMFLY-PT polynomials. Your question about $\mathfrak{sl}_3$ invariants is related to this.

I don't have a reference for the previous claim at hand (I might find one in a bit) but it should at least give you some more search terms.