${\mathrm{P}\mathrm{G}\mathrm{L}}_2$ skein algebras of small surfaces

We study the ${\mathrm{P}\mathrm{G}\mathrm{L}}_2$ 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 ${\mathrm{P}\mathrm{G}\mathrm{L}}_2$ 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 ${\mathrm{P}\mathrm{G}\mathrm{L}}_2$ 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 $\mathbb{N}[q,q^{-1}]\mathrm{.}$ We compute the kernels of natural maps between the ${\mathrm{P}\mathrm{G}\mathrm{L}}_2$ and Kauffman bracket skein algebras associated to the once-punctured torus and closed torus. We compute the center of the ${\mathrm{P}\mathrm{G}\mathrm{L}}_2$ skein algebra of the torus, which depends on the specialization of the quantum parameter $q\mathrm{.}$