Universal knots

Abstract

This paper contains detailed proofs of the results in the announcement "Universal knots'' [the authors, Bull. Amer. Math. Soc. (N.S.) 8 (1983), 449–450;]. The authors exhibit a knot K that is universal, i.e. every closed, orientable 3-manifold M can be represented as a covering of S3 branched over K, thereby giving an affirmative answer to a question of Thurston. The idea is to start with a 3-fold covering M→S3 branched over a knot and to change it to a covering M→S3 branched over a certain link L4 of four (unknotted) components. This shows that L4 is universal. Then a covering S3→S3 that is branched over a certain link L2 of two components with L4 in the preimage of L2, and a covering S3→S3 that is branched over K with L2 in the preimage of K, are constructed. This shows that L2 and K are universal. The knot K is rather complicated. In a later paper [Topology 24 (1985), no. 4, 499–504;] the authors show that the "figure eight'' knot is universal.