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

TL;DR

This paper introduces the pop-switch planar algebra, a new algebraic structure that simplifies the representation of Jones-Wenzl idempotents by expressing them as sums of vertical strands, inspired by graph planar algebra concepts.

Contribution

The paper presents the pop-switch planar algebra, a novel framework that contains the Temperley-Lieb algebra and simplifies Jones-Wenzl idempotents representation.

Findings

Jones-Wenzl idempotents are isomorphic to sums of vertical strands in the new algebra.

The pop-switch planar algebra contains the Temperley-Lieb algebra as a subalgebra.

Simplifies the expression and understanding of 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.