Seifert surfaces of knots in S⁴

Daniel Ruberman

Vol. 145 (1990), No. 1, 97–116

DOI: 10.2140/pjm.1990.145.97

Abstract

This paper uses some ideas from 3-dimensional topology to study knots in S⁴. We show that the Poincaré conjecture implies the existence of a non-fibered knot whose complement fibers homotopically. In a different direction, we show that Gromov’s norm is an obstruction to a knot having a Seifert surface made out of Seifert fibered spaces, and hence to being ribbon. We also prove that any 3-manifold is invertibly homology cobordant to a hyperbolic 3-manifold, so that every knot in S⁴ has a hyperbolic Seifert surface.

Mathematical Subject Classification 2000

Primary: 57Q45

Secondary: 57N10, 57N13

Milestones

Received: 7 November 1988

Revised: 15 May 1989

Published: 1 September 1990