Power sum elements in the G2 skein algebra

Beaumont-Gould, Bodie¹ ; Brodsky, Erik² ; Higgins, Vijay² ; Hogan, Alaina³ ; Melby, Joseph M² ; Piazza, Joshua⁴

¹Department of Mathematics, Lewis and Clark College, Portland, OR, United States
²Department of Mathematics, Michigan State University, East Lansing, MI, United States
³Department of Mathematics, Grand Valley State University, Allendale, MI, United States
⁴Department of Mathematics and Computer Science, Wheaton College, Wheaton, IL, United States

Algebraic and Geometric Topology, Tome 25 (2025) no. 4, pp. 2477-2505

Résumé

We study the skein algebras of surfaces associated to the exceptional Lie group G2, using Kuperberg webs. We identify two 2-variable polynomials, Pn(x,y) and Qn(x,y), and use threading operations along knots to construct a family of central elements in the G2 skein algebra of a surface, 𝒮qG2(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 Pn and Qn. Our methods involve a detailed study of the skein modules of the annulus and the twice-marked annulus.

Détail

DOI : 10.2140/agt.2025.25.2477

Keywords: skein algebras of surfaces, Kuperberg webs, power sum polynomials

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