Filtered instanton homology and cosmetic surgery

TL;DR

This paper advances the understanding of the cosmetic surgery conjecture by refining the types of surgeries that could produce identical three-manifolds, using filtered instanton homology and new invariants.

Contribution

It introduces a Chern-Simons filtration on instanton homology to eliminate certain surgeries and defines new invariants and surgery relations, narrowing the conjecture to specific cases.

Findings

Eliminates $m{ extstyle rac{1}{n}}$-surgeries as cosmetic surgeries.

Reduces the conjecture to $m{ extstyle ext{±} 2}$-surgeries on genus 2 knots with trivial Alexander polynomial.

Establishes new invariants and surgery relations for Floer's instanton homology.

Abstract

The cosmetic surgery conjecture predicts that for a non-trivial knot in the three-sphere, performing two different Dehn surgeries results in distinct oriented three-manifolds. Hanselman reduced the problem to $\pm 2$ or $\pm 1/n$ surgeries being the only possible cosmetic surgeries. We remove the case of $\pm 1/n$-surgeries using the Chern-Simons filtration on Floer's original irreducible-only instanton homology, reducing the conjecture to the case of $\pm 2$ surgery on genus $2$ knots with trivial Alexander polynomial. We also prove some similar results for surgeries on knots in $S^2 \times S^1$. As key steps in establishing these results, we define invariants of the oriented homeomorphism type of three-manifolds derived from filtered instanton Floer homology and introduce a new surgery relationship for Floer's instanton homology.