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The discovery of the Jones polynomial triggered mathematical developments in areas including knot theory and quantum algebra. One way to define the Jones polynomial is by using the braiding in the Temperley-Lieb category, which is a skein category corresponding to the representation category of quantum $sl_2$. In order to describe irreducible representations of quantum $sl_2$ with planar graphs in the Temperley-Lieb category, H. Wenzl gave explicit inductive formulas defining the Jones-Wenzl projectors.

G. Kuperberg defined Spiders for rank two simple Lie algebras as generalizations of the Temperley-Lieb category, and defined clasps as analogues of the Jones-Wenzl projectors. In this talk, I will review the background material and talk about how to give explicit inductive formulas for clasps in the $g_2$ case. I will also talk about clasp expansions in other cases and the applications of these clasp formulas. This is based on joint work with Elijah Bodish.(arXiv:2112.01007).