What question did this study set out to answer?

The study aims to identify which three-manifolds arise as integral Dehn surgery along a knot in S^3, specifically those containing a Klein bottle.

July 5, 2026Quantum Topology0 citationsOpen Access

Integral Klein bottle surgeries and Heegaard Floer homology

RDRobert DeYesoUniversity of Tennessee at Martin

Key Points

The study aims to identify which three-manifolds arise as integral Dehn surgery along a knot in S^3, specifically those containing a Klein bottle.
Examined closed, connected, orientable three-manifolds resulting from integral Dehn surgery.
Utilized Heegaard Floer invariants to establish surgery obstructions.
Analyzed specific cases, including 8-surgery along a genus two knot.
Demonstrated that specific surgeries lead to L-space manifolds.
Identified that the dihedral manifold (-1; 1/2, 1/2, 2/5) results from the gluing process.
Established that the surgery knot must be K=T(2,5).

Abstract

In this paper, 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 (-1; 12, 12, 25), and the surgery knot must be K=T (2, 5).

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