# A graphical categorification of the two-variable Chebyshev polynomials of the second kind

## TL;DR

This paper provides a graphical categorification of two-variable Chebyshev polynomials of the second kind using the $A_2$ spider, connecting diagrammatic calculus with representation theory of $U_q( ext{sl}_3)$ and introducing a natural $q$-deformation.

## Contribution

It introduces a novel diagrammatic categorification of two-variable Chebyshev polynomials via the $A_2$ spider, linking skein theory with quantum group representations.

## Key findings

- Recursive formula satisfied by $A_2$ clasps
- Definition of a $q$-deformation of the polynomials
- Diagrammatic approach based on skein theory

## Abstract

We show that the $A_2$ clasps in the Karoubi envelope of $A_2$ spider satisfy the recursive formula of the two-variable Chebyshev polynomials of the second kind associated with a root system of type $A_2$. The $A_2$ spider is a diagrammatic description of the representation category for $U_q(\mathfrak{sl}_3)$ and the $A_2$ clasps are projectors. Our categorification also gives a natural definition of a $q$-deformation of the two-variable Chebyshev polynomials. This paper is constructed based only on the linear skein theory and graphical calculus.