A note on rational surgeries on a Hopf link

TL;DR

This paper explicitly computes which lens spaces result from rational surgeries on a Hopf link in the 3-sphere, using continued fractions, and recovers known criteria for when such surgeries yield the 3-sphere.

Contribution

It provides an explicit calculation method for identifying the resulting lens space from rational surgeries on a Hopf link, enhancing understanding of such surgeries.

Findings

Explicit formulas for lens spaces from surgeries

Recovery of known criteria for 3-sphere surgeries

Application of continued fractions in surgery analysis

Abstract

It is clear that every rational surgery on a Hopf link in $3$-sphere is a lens space surgery. In this note we give an explicit computation which lens space is a resulting manifold. The main tool we use is the calculus of continued fractions. As a corollary, we recover the (well known) result on the criterion for when rational surgery on a Hopf link gives the $3$-sphere.