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Abstract
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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.
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Keywords
3-manifold, sutured manifold, knot
theory, tunnel number one, crossing change, rational tangle,
2-handle addition, handlebody
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Mathematical Subject Classification
Primary: 57M50, 57N10
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Milestones
Received: 30 June 2008
Accepted: 13 January 2009
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