Boring Split Links

Boring is an operation which 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 EudaveMuñoz and Scharlemann’s band sum theorem.