Parameterized diagrammatic categories and their decategorification

Determine how introducing a parameter analogous to the Temperley–Lieb parameter $\delta$ into the diagrammatic categories constructed from representation graphs changes their combinatorics, representation theory, and decategorifications.

Background

The Temperley–Lieb category admits a scalar parameter $\delta$, with $\delta=2$ yielding the category related to SU(2). The paper suggests investigating analogous parameterizations for the newly constructed diagrammatic categories.

The authors leave unresolved what structural effects such a parameter would have, explicitly mentioning changes to combinatorics, representation theory, and decategorification as possible objects of study.