$\rm{PGL}_2$ webs at roots of unity

Abstract: We study webs on surfaces for the $\rm{PGL}_2$ skein theory specialized at roots of unity. We construct the $\rm{PGL}_2$ analogue of Chebyshev polynomials and use them to construct central elements in the skein algebra. We then focus on the cases of the 3-punctured sphere and the closed torus. We compute the Frobenius homomorphism for these two surfaces and show that the $\rm{PGL}_2$ Frobenius homomorphism for the torus is not injective when the quantum parameter $q<sup>2$ is a root of unity of even order $n.$