In Piunikhin’s Turaev-Viro and Kauffman invariants for 3-manifolds coincide, the exact relation between Kauffman algebra and the representation of the quantum group $U_q(sl_2)$ is shown.
This seems to be a coincidence due to low rank $n=2$, because all irreducible representations can be obtained as symmetric products of the lowest nontrivial one, which is not the case for higher $n$. Also, the smallest nontrivial one is small enough to impose Kauffman bracket relation!
What then is an analogue of TL-algebra that corresponds to the representations of $U_q(sl_n)$?
There are a lot of references. Here is a paper by Sussan on link invariants associated to $sl_k$. Another pair of papers (a) & (b) by Cautis-Kamnitzer also work on this problem.
More recent approaches using the idea of "categorification" are described by Lauda, see results by Khovanov-Lauda (and references therein) and this paper by Webster.