On knots that are universal

Abstract

The authors construct a cover S3→S3 branched over the "figure eight" knot with preimage the "roman link" and a cover S3→S3 branched over the roman link with preimage containing the Borromean rings L. Since L is universal (i.e. every closed, orientable 3-manifold can be represented as a covering of S3 branched over L) it follows that the "figure eight'' knot is universal, thereby answering a question of Thurston in the affirmative. More generally, it is shown that every rational knot or link which is not toroidal is universal