Exceptional Cosmetic surgeries on $S^3$

TL;DR

This paper investigates the conditions under which truly cosmetic surgeries on hyperbolic knots in S^3 are exceptional, providing new constraints and properties related to such surgeries.

Contribution

It proves that exceptional truly cosmetic surgeries on hyperbolic knots in S^3 must have slope ±1 and be toroidal but not Seifert fibred, and identifies cases with no such surgeries.

Findings

Slope of exceptional truly cosmetic surgery must be ±1

Such surgeries are toroidal but not Seifert fibred

No exceptional truly cosmetic surgeries on certain hyperbolic knots

Abstract

This paper concerns the truly or purely cosmetic surgery conjecture. We give a survey on exceptional surgeries and cosmetic surgeries. We prove that the slope of an exceptional truly cosmetic surgery on a hyperbolic knot in $S^3$ must be $\pm 1$ and the surgery must be toroidal but not Seifert fibred. As consequence we show that there are no exceptional truly cosmetic surgeries on certain types of hyperbolic knot in $S^3$. We also give some properties of Heegaard Floer correction terms and torsion invariants for exceptional cosmetic surgeries on $S^3$.