Analogue of Temperley-Lieb algebra of higher rank
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.