Fintushel--Stern knot surgery in torus bundles

TL;DR

This paper characterizes when a torus bundle over a surface admits a symplectic structure after Fintushel--Stern knot surgery, showing it occurs precisely when the knot is fibered, using Seiberg--Witten theory and twisted Alexander polynomials.

Contribution

It establishes a necessary and sufficient condition for the existence of a symplectic structure after knot surgery on torus bundles, linking fibered knots to symplectic geometry.

Findings

$X_K$ admits a symplectic structure iff $K$ is fibered.

Uses Seiberg--Witten theory and twisted Alexander polynomials.

Provides a complete characterization of symplectic structures post-surgery.

Abstract

Suppose that $X$ is a torus bundle over a closed surface with homologically essential fibers. Let $X_K$ be the manifold obtained by Fintushel--Stern knot surgery on a fiber using a knot $K\subset S^3$. We prove that $X_K$ has a symplectic structure if and only if $K$ is a fibered knot. The proof uses Seiberg--Witten theory and a result of Friedl--Vidussi on twisted Alexander polynomials.