Integral Klein bottle surgeries and Heegaard Floer homology

TL;DR

This paper classifies certain three-manifolds obtained from Dehn surgery on knots in S^3 that contain Klein bottles, using Heegaard Floer invariants to identify specific manifolds and knots involved.

Contribution

It provides a classification of 8-surgeries on genus two knots resulting in manifolds with Klein bottles, identifying the specific dihedral manifold and the knot T(2,5).

Findings

X is an L-space.

X is the dihedral manifold (-1; 1/2, 1/2, 2/5).

The knot is T(2,5).

Abstract

We study which closed, connected, orientable three-manifolds $X$ containing a Klein bottle arise as integral Dehn surgery along a knot in $S^3$. Such $X$ are presentable as a gluing of the twisted $I$-bundle over the Klein bottle to a knot manifold, and we use a variety of Heegaard Floer type invariants to generate surgery obstructions. Suppose that $X$ is $8$-surgery along a genus two knot, and arises by gluing the twisted $I$-bundle over the Klein bottle to an $S^3$ knot complement. We show that $X$ is an L-space, it must be the dihedral manifold $\left(-1; \tfrac{1}{2}, \tfrac{1}{2}, \tfrac{2}{5}\right)$, and the surgery knot must be $K=T(2,5)$.