PGL2 skein algebras of small surfaces
Abstract
We study the PGL2 skein algebras of small surfaces, focusing on the closed torus and the once-punctured torus. We provide explicit finite presentations by generators and relations for the PGL2 skein algebra of the torus and once-punctured torus, analogous to those of Bullock and Przytycki for the Kauffman bracket skein algebras. We give explicit bases of the PGL2 skein algebras of the torus and once-punctured torus and describe the multiplicative structure of the basis for the torus, analogous to the product-to-sum formula of Frohman and Gelca. We twist the basis for the torus to obtain one with structure constants in ℕ[q,q−1]. We compute the kernels of natural maps between the PGL2 and Kauffman bracket skein algebras associated to the once-punctured torus and closed torus. We compute the center of the PGL2 skein algebra of the torus, which depends on the specialization of the quantum parameter q.
Related
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BibTeX
@misc{ellison2026pgl2,
title = {{$\rmPGL_2$ skein algebras of small surfaces}},
author = {Cole Ellison and Vijay Higgins and Mark Sandey and Eden Sylvester},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01075},
eprint = {2610.01075},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01075},
}
RIS
TY - UNPB
TI - PGL_2 skein algebras of small surfaces
AU - Cole Ellison
AU - Vijay Higgins
AU - Mark Sandey
AU - Eden Sylvester
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.01075
UR - https://arxiv.org/abs/2610.01075
AB - We study the PGL_2 skein algebras of small surfaces, focusing on the closed torus and the once-punctured torus. We provide explicit finite presentations by generators and relations for the PGL_2 skein algebra of the torus and once-punctured torus, analogous to those of Bullock and Przytycki for the Kauffman bracket skein algebras. We give explicit bases of the PGL_2 skein algebras of the torus and once-punctured torus and describe the multiplicative structure of the basis for the torus, analogous to the product-to-sum formula of Frohman and Gelca. We twist the basis for the torus to obtain one with structure constants in ℕ[q,q^−1]. We compute the kernels of natural maps between the PGL_2 and Kauffman bracket skein algebras associated to the once-punctured torus and closed torus. We compute the center of the PGL_2 skein algebra of the torus, which depends on the specialization of the quantum parameter q.
ER -