Planar surfaces in knot manifolds

Howard Wilson Lambert

Vol. 39 (1971), No. 3, 727–733

DOI: 10.2140/pjm.1971.39.727

Abstract

Let K be a knot manifold, that is the 3-sphere S^s minus an open regular neighborhood of a polygonal simple closed curve in ∕S^s. Whether K can be embedded in S⁸ differently or in a homotopy 3-sphere different from S⁸ (if such really exist) leads in a natural way to the question of which planar surfaces can be embedded in K. Geometric conditions are imposed on the embedded planar surfaces which are sufficient to imply that K is not knotted, that is K is homeomorphic to a disk cross S¹.

Mathematical Subject Classification

Primary: 55A25

Milestones

Received: 30 September 1970

Revised: 14 February 1971

Published: 1 December 1971