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Smoothly knotted surfaces that remain distinct after many internal stabilizations

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arxiv 2307.16266 v1 pith:RWXMCT65 submitted 2023-07-30 math.GT

David Auckly This is my paper · ORCID
classification math.GT
keywords internalsmoothlysurfacesknottedmanifoldmanystabilizationsisotopic
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abstract

Internal stabilization adds a trivial handle to an embedded surface in a coordinate chart. It is known that any pair of smoothly knotted surfaces in a simply-connected $4$-manifold become smoothly isotopic after sufficiently many internal stabilizations. In this paper, we show that there is no upper bound on the number of internal stabilizations required. In fact, this behavior is fairly generic. The definition of a subtly smoothly knotted pair of surfaces is given and it is shown that many surfaces may be modified to obtain subtly knotted surfaces with large internal stabilization distance. Furthermore, it is shown that after stabilizing any $4$-manifold with contain topologically isotopic, smoothly related, non-isotopic copies of any $3$-manifold having a positive first betti number.

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Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Exotic knottings and symmetries of surfaces in 4-manifolds

    math.GT 2026-07 conditional novelty 8.0 of 10

    Iterated rim surgery produces topologically isotopic genus-g surfaces whose smoothly extendable mapping classes lose one projective homology symmetry per step, ending in projective rigidity.

  2. Families of diffeomorphisms, embeddings, and positive scalar curvature metrics via Seiberg-Witten theory

    math.GT 2025-01 accept novelty 7.0 of 10

    A new gluing theorem for parameterized Seiberg-Witten invariants gives infinite rank Z^∞ summands in higher homotopy and homology of diffeomorphism groups of 4-manifolds that are topologically trivial.

  3. Exotic families of embeddings

    math.GT 2025-01 conditional novelty 6.0 of 10

    Smooth embeddings of 3-manifolds in 4-manifolds that are topologically trivial but smoothly exotic, both as individual embeddings and in families parameterized by spheres, are constructed and detected.

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