Skein Algebras of Three-Manifolds at 4th Roots of Unity

This paper introduces an algebra structure on the part of the skein module of an arbitrary $3$-manifold $M$ spanned by links that represent $0$ in $H_1(M;{\mathbb{Z}}_2)$ when the value of the parameter used in the Kauffman bracket skein relation is equal to $\pm \mathbf{i}$. It is proved that if $M$ has no $2$-torsion in $H_1(M;\mathbb{Z})$ then those algebras, $K_{\pm \mathbf{i}}^0(M)$, are naturally isomorphic to the corresponding algebras when the value of the parameter is $\pm 1$. This implies that the algebra $K_{\pm \mathbf{i}}^0(M)$ is the unreduced coordinate ring of the variety of $PS{L}_2(\mathbb{C})$-characters of ${\pi }_1(M)$ that lift to $S{L}_2(\mathbb{C})$-representations.