English

Embedding spheres in knot traces

Geometric Topology 2023-04-12 v2

Abstract

The trace of nn-framed surgery on a knot in S3S^3 is a 4-manifold homotopy equivalent to the 2-sphere. We characterise when a generator of the second homotopy group of such a manifold can be realised by a locally flat embedded 2-sphere whose complement has abelian fundamental group. Our characterisation is in terms of classical and computable 3-dimensional knot invariants. For each nn, this provides conditions that imply a knot is topologically nn-shake slice, directly analogous to the result of Freedman and Quinn that a knot with trivial Alexander polynomial is topologically slice.

Keywords

Cite

@article{arxiv.2004.04204,
  title  = {Embedding spheres in knot traces},
  author = {Peter Feller and Allison N. Miller and Matthias Nagel and Patrick Orson and Mark Powell and Arunima Ray},
  journal= {arXiv preprint arXiv:2004.04204},
  year   = {2023}
}

Comments

37 pages, 2 figures. v2 some typos fixed

R2 v1 2026-06-23T14:44:45.758Z