Vol. 241, No. 1, 2009

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ISSN: 0030-8730
Boring split links

Scott A. Taylor

Vol. 241 (2009), No. 1, 127–167
Abstract

Boring is an operation that converts a knot or two-component link in a 3-manifold into another knot or two-component link. It generalizes rational tangle replacement and can be described as a type of 2-handle attachment. Sutured manifold theory is used to study the existence of essential spheres and planar surfaces in the exteriors of knots and links obtained by boring a split link. It is shown, for example, that if the boring operation is complicated enough, a split link or unknot cannot be obtained by boring a split link. Particular attention is paid to rational tangle replacement. If a knot is obtained by rational tangle replacement on a split link, and a few minor conditions are satisfied, the number of boundary components of a meridional planar surface is bounded below by a number depending on the distance of the rational tangle replacement. This result is used to give new proofs of two results of Eudave-Muñoz and Scharlemann’s band sum theorem.

Keywords
3-manifold, sutured manifold, knot theory, tunnel number one, crossing change, rational tangle, 2-handle addition, handlebody
Mathematical Subject Classification 2000
Primary: 57M50, 57N10
Milestones
Received: 30 June 2008
Accepted: 13 January 2009
Published: 1 May 2009
Authors
Scott A. Taylor
Department of Mathematics
Colby College
5832 Mayflower Hill
Waterville, ME 04901
United States