Any knot complement covers at most one knot complement

Shicheng Wang and Ying Qing Wu

Vol. 158 (1993), No. 2, 387–395

DOI: 10.2140/pjm.1993.158.387

Abstract

It follows from Culler, Gordon, Luecke and Shalen’s Cyclic Surgery Theorem that any knot complement is covered by at most two knot complements. Gonzales-Acuna and Whitten proved a result on the other direction: A given knot complement can cover at most finitely many knot complements. This paper is to show that the best possible result in this direction holds: A given knot complement can nontrivially cover at most one knot complement. Moreover, if the knot is not a torus knot, then the covering map is unique up to equivalence.

Mathematical Subject Classification 2000

Primary: 57M25

Milestones

Received: 5 June 1991

Revised: 14 February 1992

Published: 1 April 1993