On The Virtual Cosmetic Surgery Conjecture

TL;DR

This paper explores the conditions under which one 3-manifold obtained from Dehn surgery on a knot covers another, providing a complete classification for torus knots and partial results for hyperbolic knots.

Contribution

It offers a complete classification of covering relations for Dehn surgeries on torus knots and advances understanding of the conjecture for hyperbolic knots.

Findings

Torus knots have a complete classification of surgery covers.

Partial results support the conjecture that hyperbolic knot surgeries do not cover each other.

The work advances the understanding of the virtual cosmetic surgery conjecture.

Abstract

Let K be a knot in S^3, and M and M' be distinct Dehn surgeries along K. We investigate when M covers M'. When K is a torus knot, we provide a complete classification of such covers. When K is a hyperbolic knot, we provide partial results in the direction of the conjecture that M never covers M'.