Obstructing Reducible Surgeries: Slice Genus and Thickness Bounds

TL;DR

This paper investigates reducible surgeries on knots, establishing bounds related to slice genus and thickness, and confirming the Cabling Conjecture for certain knots using Heegaard Floer homology techniques.

Contribution

It introduces new bounds on thickness and slice genus for knots with reducible surgeries, advancing the understanding of knot surgery properties and verifying the Cabling Conjecture for thin knots.

Findings

Thickness bounds for L-space knots with reducible surgeries

Lower bounds on slice genus for knots with reducible surgeries

Verification of the Cabling Conjecture for thin knots

Abstract

In this paper, we study reducible surgeries on knots in $S^3$. We develop thickness bounds for L-space knots that admit reducible surgeries, and lower bounds on the slice genus for general knots that admit reducible surgeries. The L-space knot thickness bounds allow us to finish off the verification of the Cabling Conjecture for thin knots, which was mostly worked out in \cite{DeY21b}. We also provide a new upper bound on reducing slopes for fibered, hyperbolic slice knots and on multiple reducing slopes for slice knots. Our techniques involve the $d$-invariants and mapping cone formula from Heegaard Floer homology.