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Families of cosmetic surgeries

Qiuyu Ren

Abstract

We construct infinite families of chirally cosmetic surgeries on chiral hyperbolic knots and purely cosmetic surgeries on hyperbolic manifolds with multiple cusps, disproving conjectures that these phenomena do not appear, including Problem 1.12(d) in the K3 problem list. We also give some hints regarding why chirally cosmetic surgeries appear to be more common than purely cosmetic surgeries on $1$-cusped manifolds.

Families of cosmetic surgeries

Abstract

We construct infinite families of chirally cosmetic surgeries on chiral hyperbolic knots and purely cosmetic surgeries on hyperbolic manifolds with multiple cusps, disproving conjectures that these phenomena do not appear, including Problem 1.12(d) in the K3 problem list. We also give some hints regarding why chirally cosmetic surgeries appear to be more common than purely cosmetic surgeries on -cusped manifolds.

Paper Structure

This paper contains 4 sections, 4 theorems, 3 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The general construction
  3. Purely cosmetic surgeries on multi-cusped hyperbolic manifolds
  4. Chirally cosmetic surgeries on chiral knots

Key Result

Theorem 1.4

For any $n\ge3$, there exists an $n$-cusped hyperbolic manifold $M$ and slopes $s,s'$ on the first cusp, such that the Dehn fillings $M(s,\infty,\cdots,\infty)$ and $M(s',\infty,\cdots,\infty)$ are hyperbolic and purely cosmetic, and that no homeomorphism of $M$ sends one set of slopes to the other.

Figures (1)

  • Figure 1: A $3$-component amphichiral hyperbolic Brunnian link with symmetry group $\mathbb{Z}/6$. The image was created by the program KnotJob schutz_knotjob_2025.

Theorems & Definitions (9)

  • Conjecture 1.1: gordon1990dehn
  • Conjecture 1.2: futer2025excluding
  • Conjecture 1.3: ito2021noteichihara2023constraints
  • Theorem 1.4
  • Theorem 1.5
  • Proposition 2.2
  • proof
  • Proposition 4.1
  • proof : Proof of Proposition \ref{['prop:brun']}