Cosmetic surgeries and non-orientable surfaces

TL;DR

This paper demonstrates that certain Dehn surgeries on a specific 2-bridge knot produce non-homeomorphic manifolds by analyzing non-orientable surfaces, expanding understanding of cosmetic surgeries in knot theory.

Contribution

It introduces a novel approach using non-orientable surfaces to distinguish non-homeomorphic manifolds resulting from Dehn surgeries, specifically on the knot 9_{27}.

Findings

10/3- and -10/3-Dehn surgeries are not cosmetic.

The methods go beyond previous invariants like Casson invariant and Heegaard Floer homology.

Provides new tools for detecting non-homeomorphic manifolds in knot theory.

Abstract

By considering non-orientable surfaces in the surgered manifolds, we show that the 10/3- and -10/3-Dehn surgeries on the 2-bridge knot $9_{27} = S(49,19)$ are not cosmetic, i.e., they give mutually non-homeomorphic manifolds. The knot is unknown to have no cosmetic surgeries by previously known results; in particular, by using the Casson invariant and the Heegaard Floer homology.