The skein algebra of the Borromean rings complement

The skein algebra of an oriented \(3\)-manifold is a classical limit of the Kauffman bracket skein module and gives the coordinate ring of the \(SL_2(\mathbb{C})\)-character variety. In this paper we determine the quotient of a polynomial ring which is isomorphic to the skein algebra of a group with three generators and two relators. As an application, we give an explicit formula for the skein algebra of the Borromean rings complement in \(S^3\).