Boring split links

TL;DR

This paper studies the operation of boring on links in 3-manifolds, using sutured manifold theory to understand how it affects the topology of knots and links, especially in relation to rational tangle replacement.

Contribution

It introduces a new perspective on boring as a 2-handle attachment and establishes bounds on boundary components in rational tangle replacements, providing new proofs of existing theorems.

Findings

Boring can prevent obtaining split links or unknots if sufficiently complex.

Bounds on boundary components depend on the rational tangle replacement distance.

New proofs of Eudave-Mu oz and Scharlemann's band sum theorem are provided.

Abstract

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 Eudave-Mu\~noz and Scharlemann's band sum theorem.