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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!

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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.

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    $\begingroup$ 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) $\endgroup$ – user530316 Mar 17 at 22:20
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    $\begingroup$ This paper sciencedirect.com/science/article/pii/S0040938302000125 provides a potential set of examples. I'm currently trying to decide if they have distinct homfly polynomials. (2/2) $\endgroup$ – user530316 Mar 17 at 22:21

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