The pop-switch planar algebra and the Jones-Wenzl idempotents

Abstract

The Jones-Wenzl idempotents are elements of the Temperley-Lieb planar algebra that are important, but complicated to write down. We will present a new planar algebra, the pop-switch planar algebra, which contains the Temperley-Lieb planar algebra. It is motivated by Jones' idea of the graph planar algebra of type $A_n$. In the tensor category of idempotents of the pop-switch planar algebra, the $n$th Jones-Wenzl idempotent is isomorphic to a direct sum of $n+1$ diagrams consisting of only vertical strands.