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 H₁(M;Z₂) when the value of the parameter used in the Kauffman bracket skein relation is equal to
± i. It is proved that if M has no 2-torsion in H₁(M;Z) then those algebras,
K_{± i}⁰(M), are naturally isomorphic to the corresponding algebras when the value of the parameter is ± 1. This implies that the algebra K_{± i}⁰(M) is the unreduced coordinate ring of the variety of PSL₂(C)-characters of π₁(M) that lift to SL₂(C)-representations.

This material is based upon work supported by and while serving at the National Science Foundation. Any opinion, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.