Representations of the oriented skein category

TL;DR

This paper develops the representation theory of the oriented skein category, a key structure in knot theory, by analyzing its Grothendieck ring and constructing a graded lift through 2-representations, covering both semisimple and non-semisimple cases.

Contribution

It introduces a highest weight theory approach to the representation theory of $OS(z,t)$ and constructs a graded lift via 2-representations, extending understanding beyond semisimple cases.

Findings

Determined the Grothendieck ring for all parameter choices.

Constructed a graded lift as a 2-representation of a Kac-Moody 2-category.

Discussed the degenerate oriented Brauer category $OB( ext{delta})$.

Abstract

The oriented skein category $OS(z,t)$ is a ribbon category which underpins the definition of the HOMFLY-PT invariant of an oriented link, in the same way that the Temperley-Lieb category underpins the Jones polynomial. In this article, we develop its representation theory using a highest weight theory approach. This allows us to determine the Grothendieck ring of its additive Karoubi envelope for all possible choices of parameters, including the (already well-known) semisimple case, and all non-semisimple situations. Then we construct a graded lift of $OS(z,t)$ by realizing it as a 2-representation of a Kac-Moody 2-category. We also discuss the degenerate analog of $OS(z,t)$, which is the oriented Brauer category $OB(\delta)$.