Surgeries on torus knots, rational balls, and cabling

TL;DR

This paper classifies surgeries on positive torus knots and their cables that bound rational homology balls, revealing boundedness properties and accumulation points related to these surgeries.

Contribution

It provides a classification of surgeries on torus knots and cables that bound rational homology balls, introducing bounds on the set of such surgeries for each knot.

Findings

S(K) is bounded for each knot K

n-surgery bounds a rational homology ball if n is an accumulation point of S(K)

The classification applies to positive integral surgeries on positive torus knots

Abstract

We classify which positive integral surgeries on positive torus knots bound rational homology balls. Additionally, for a given knot K we consider which cables K(p,q) admit integral surgeries that bound rational homology balls. For such cables, let S(K) be the set of corresponding rational numbers q/p. We show that S(K) is bounded for each K. Moreover, if n-surgery on K bounds a rational homology ball then n is an accumulation point for S(K).