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

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?

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.
• Thanks for the response. I did a bit of searching around and came to the same conclusion. In this thesis: liverpool.ac.uk/~su14/papers/aistonthesis.pdf the author proves how the quantum $\mathfrak{sl}_N$ invariants relate to the homfly polynomial. Since the Kauffman bracket ($\mathfrak{sl}_2$) can be used to compute the Jones polynomial, I need to find links that are not distinguished by the Jones polynomial, but are distinguished by the homfly polynomial. (1/2) – user530316 Mar 17 at 22:20