Two-bridge knots admit no purely cosmetic surgeries

TL;DR

This paper proves that certain classes of knots, specifically two-bridge and alternating fibered knots, cannot have different surgeries that produce the same 3-manifold, using advanced invariants and recent theoretical results.

Contribution

It establishes the non-existence of purely cosmetic surgeries for two-bridge and alternating fibered knots, expanding understanding of knot surgery uniqueness.

Findings

Two-bridge knots admit no purely cosmetic surgeries.

Alternating fibered knots admit no purely cosmetic surgeries.

Uses invariants like signature, finite type invariants, and the $SL(2, ext{C})$ Casson invariant.

Abstract

We show that two-bridge knots and alternating fibered knots admit no purely cosmetic surgeries, i.e., no pair of distinct Dehn surgeries on such a knot produce 3-manifolds that are homeomorphic as oriented manifolds. Our argument, based on a recent result by Hanselman, uses several invariants of knots or 3-manifolds; for knots, we study the signature and some finite type invariants, and for 3-manifolds, we deploy the $SL(2,\mathbb{C})$ Casson invariant.