Power sum elements in the $G_2$ skein algebra
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866911110538461184 |
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| author | Beaumont-Gould, Bodie Brodsky, Erik Higgins, Vijay Hogan, Alaina Melby, Joseph M. Piazza, Joshua |
| author_facet | Beaumont-Gould, Bodie Brodsky, Erik Higgins, Vijay Hogan, Alaina Melby, Joseph M. Piazza, Joshua |
| contents | 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, $\mathcal{S}_q^{G_2}(Σ),$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_01773 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Power sum elements in the $G_2$ skein algebra Beaumont-Gould, Bodie Brodsky, Erik Higgins, Vijay Hogan, Alaina Melby, Joseph M. Piazza, Joshua Geometric Topology Quantum Algebra 57K31 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, $\mathcal{S}_q^{G_2}(Σ),$ 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. |
| title | Power sum elements in the $G_2$ skein algebra |
| topic | Geometric Topology Quantum Algebra 57K31 |
| url | https://arxiv.org/abs/2310.01773 |