Power sum elements in the $G_2$ skein algebra

TL;DR

This paper explores the $G_2$ skein algebra of surfaces, identifying special polynomials and constructing central elements at roots of unity, advancing understanding of quantum invariants related to the exceptional Lie group.

Contribution

It introduces two polynomials, $P_n$ and $Q_n$, and constructs a family of central elements in the $G_2$ skein algebra at roots of unity, using skein-theoretic techniques.

Findings

Identification of polynomials $P_n$ and $Q_n$

Construction of central elements at roots of unity

Proof of uniqueness of transparent polynomials

Abstract

We study the skein algebras of surfaces associated to the exceptional Lie group $G_2,$ using Kuperberg webs. We identify two 2-variable polynomials, $P_n(x,y)$ and $Q_n(x,y),$ and use threading operations along knots to construct a family of central elements in the $G_2$ skein algebra of a surface, $\mathcal{S}_q^{G_2}(\Sigma),$ when the quantum parameter $q$ is a $2n\text{-th}$ root of unity. We verify these elements are central using elementary skein-theoretic arguments. We also obtain a result about the uniqueness of the so-called transparent polynomials $P_n$ and $Q_n.$ Our methods involve a detailed study of the skein modules of the annulus and the twice-marked annulus.