Comparing 2-handle additions to a genus 2 boundary component

TL;DR

This paper investigates the effects of 2-handle additions to genus 2 boundary components in 3-manifolds, proving several conjectures and providing new insights into knot theory and sutured manifold topology.

Contribution

It offers new proofs of existing conjectures and introduces a theorem on 2-handle attachments, advancing understanding of 3-manifold topology and knot theory.

Findings

Knots from band attachments satisfy the cabling conjecture

Unknotting number one knots are prime

Genus is superadditive under band sum

Abstract

We prove that knots obtained by attaching a band to a split link satisfy the cabling conjecture. We also give new proofs that unknotting number one knots are prime and that genus is superadditive under band sum. Additionally, we prove a collection of results comparing two 2-handle additions to a genus two boundary component of a compact, orientable 3-manifold. These results give a near complete solution to a conjecture of Scharlemann and provide evidence for a conjecture of Scharlemann and Wu. The proofs make use of a new theorem concerning the effects of attaching a 2-handle to a suture in the boundary of a sutured manifold.