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A note on rational surgeries on a Hopf link

Velibor Bojković, Jovana Nikolić, Mladen Zekić

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.

A note on rational surgeries on a Hopf link

Abstract

It is clear that every rational surgery on a Hopf link in -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 -sphere.

Paper Structure

This paper contains 3 sections, 10 theorems, 17 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Rational calculus
  3. The main results

Key Result

Theorem 1.1

Rational surgery on a Hopf link with framing $\frac{p}{q}=[a_0;a_1,\hdots,a_n]$ and $\frac{r}{s}=[b_0;b_1,\hdots;b_m+1]$ is a lens space surgery and the resulting lens space is $L(a,b)$ where (if $\frac{a}{b}\in\mathbb{Q}$, we take $a$ and $b$ coprime, $b>0$).

Figures (4)

  • Figure 1: Hopf link with framing $m\in\mathbb{Z}$ and $\frac{p}{q}\in\mathbb{Q}$
  • Figure 2: Hopf link with rational surgery on both unknots
  • Figure 3: Equivalent surgery presentations of the homology sphere
  • Figure 4: Rolfsen move of the second kind performed on the Hopf link

Theorems & Definitions (23)

  • Theorem 1.1
  • Remark 1.2
  • Example 1.3
  • Example 1.4
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Corollary 2.3
  • proof
  • ...and 13 more