Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Purely cosmetic surgeries and Casson--Walker--Lescop invariants

Kazuhiro Ichihara, In Dae Jong, Yasuyoshi Tsutsumi

Abstract

Using the rational surgery formula for the Casson--Walker--Lescop invariant of links in the $3$-sphere, we show that any null-homologous knot in a rational homology sphere admits at most two pairs of integral purely cosmetic surgeries. We also present constraints for null-homologous knots in certain $3$-manifolds with the first Betti number one or two to admit purely cosmetic surgeries. As another application, we show that, for a null-homologous knot in some $3$-manifolds, including $S^2 \times S^1$, there are at most two knots which are inequivalent to the given one, but whose exteriors are orientation-preservingly homeomorphic to that of the given one.

Purely cosmetic surgeries and Casson--Walker--Lescop invariants

Abstract

Using the rational surgery formula for the Casson--Walker--Lescop invariant of links in the -sphere, we show that any null-homologous knot in a rational homology sphere admits at most two pairs of integral purely cosmetic surgeries. We also present constraints for null-homologous knots in certain -manifolds with the first Betti number one or two to admit purely cosmetic surgeries. As another application, we show that, for a null-homologous knot in some -manifolds, including , there are at most two knots which are inequivalent to the given one, but whose exteriors are orientation-preservingly homeomorphic to that of the given one.
Paper Structure (11 sections, 20 theorems, 73 equations, 2 figures)

This paper contains 11 sections, 20 theorems, 73 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Purely cosmetic surgeries on knots in rational homology spheres
  3. Surgery formula of $\lambda$
  4. Proof of Theorem \ref{['thm1']}
  5. Knot complement problem in a rational homology sphere
  6. Constraints on knots in a 3-manifold with the Betti number 1 or 2
  7. Proof of Theorem \ref{['thm3']}
  8. Proof of Theorem \ref{['thm4']}
  9. Proof of Theorem \ref{['thm5']}
  10. Examples
  11. Confirming several formulae

Key Result

Theorem 1.1

A null-homologous knot in a rational homology sphere admits at most two pairs of integral purely cosmetic surgeries.

Figures (2)

  • Figure 1: Whitehead link $W$ and the link $L_m$. The box labeled $m$ indicates $m$ times full-twists.
  • Figure 2: Identifying two $2$-spheres, we have a knot in $S^2 \times S^1$, which admits no purely cosmetic surgeries.

Theorems & Definitions (34)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Remark
  • Lemma 2.1
  • proof
  • ...and 24 more