Sliced skein algebras and geometric Kauffman bracket

Abstract

The sliced skein algebra of a closed surface of genus g with m punctures, $S={\Sigma }_{g,m}$, is the quotient of the Kauffman bracket skein algebra $S_{\xi }\left(S\right)$ corresponding to fixing the scalar values of its peripheral curves. We show that the sliced skein algebra of a finite type surface is a domain if the ground ring is a domain. When the quantum parameter ξ is a root of unity we calculate the center of the sliced skein algebra and its PI-degree. Among applications we show that any smooth point of a sliced character variety is an Azumaya point of the skein algebra $S_{\xi }\left(S\right)$.

For any $S{L}_2\left(C\right)$-representation ρ of the fundamental group of an oriented connected 3-manifold M and a root of unity ξ with the order of ${\xi }^2$ odd, we introduce the ρ-reduced skein module $S_{\xi ,\rho }\left(M\right)$. We show that $S_{\xi ,\rho }\left(M\right)$ has dimension 1 when M is closed and ρ is irreducible. We also show that if ρ is irreducible the ρ-reduced skein module of a handlebody, as a module over the skein algebra of its boundary, is simple and has the dimension equal to the PI-degree of the skein algebra of its boundary.