Comparing 2-handle additions to a genus 2 boundary component

Taylor, Scott

doi:10.1090/S0002-9947-2014-06253-3

Comparing 2-handle additions to a genus 2 boundary component
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Trans. Amer. Math. Soc. 366 (2014), 3747-3769 Request permission

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 a band sum. Additionally, we prove a collection of results comparing two 2-handle additions to a genus 2 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.

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