PGL2 skein algebras of small surfaces
Cole Ellison, Vijay Higgins, Mark Sandey, Eden Sylvester
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.
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