Narrow Results

The skein algebra of an oriented three-manifold is a classical limit of the Kauffman bracket skein module and gives the coordinate ring of the ${SL}_2(ℂ)$-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 $S^3$.

We prove that the Borromean Rings are the only Brunnian link of 3 or 4 components that can be built out of convex curves.

We prove that given any three polygonal unknots in ℝ³, then we may form the Borromean rings out of them through rigid motions of ℝ³ applied to the individual components together with possible scaling of the components. We also prove that if at least two of the unknots are planar, then we do not need scaling. This is true even for a set of three polygonal unknots that are arbitrarily close to three circles, which themselves cannot be used to form the Borromean rings.