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

Abstract

We study webs on surfaces for the ${\mathrm{PGL}}_2$ skein theory specialized at roots of unity. We construct the ${\mathrm{PGL}}_2$ 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 ${\mathrm{PGL}}_2$ Frobenius homomorphism for the torus is not injective when the quantum parameter $q^2$ is a root of unity of even order $n.$

arXiv comment26 pages