Power sum elements in the G₂ skein algebra

2023/10/03 by Bodie Beaumont-Gould, Beaumont-Gould, Bodie, Erik Brodsky +9 · 1 citation
Mathematics · #57K31 #Advanced Algebra and Geometry #Algebraic structures and combinatorial models #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Quantum Algebra (math.QA)

paper · pdf · doi:10.48550/arxiv.2310.01773

openalex publication_date 2023/10/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the skein algebras of surfaces associated to the exceptional Lie group G₂, 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 G₂ skein algebra of a surface, S_q^G₂(Σ), when the quantum parameter q is a 2n-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.

Cited by

• Type C skein modules and transparent elements

Related

• Central elements in the SL_d-skein algebra of a surface
• Stated skein algebras of surfaces
• Monomial web basis for the SL(N) skein algebra of the twice punctured sphere
• A relationship between the Kauffman bracket skein algebras and Roger-Yang skein algebras of some small surfaces
• SU(3)-skein algebras and webs on surfaces