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A gauge theoretic invariant of embedded surfaces in $4$-manifolds and exotic $P^2$-knots

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arxiv 2312.02041 v1 pith:VNM2EBX3 submitted 2023-12-04 math.GT math.DG

Jin Miyazawa This is my paper · ORCID
classification math.GTmath.DG
keywords isotopicembeddedgaugeinvariantknotsmanifoldsotherseiberg--witten
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abstract

We give an infinite family of embeddings of $\mathbb{R} P^2$ to $S^4$ such that they are mutually topologically isotopic however are not smoothly isotopic to each other. Moreover, they are topologically isotopic to the standard $P^2$-knot. To prove that these $P^2$-knots are not smoothly isotopic to each other, we construct a gauge theoretic invariant of embedded surfaces in $4$-manifolds using a variant of the Seiberg--Witten theory, which is called the Real Seiberg--Witten theory.

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Cited by 4 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. Real sutured Heegaard Floer homology

    math.GT 2026-07 accept novelty 7.0 of 10

    Real sutured Heegaard Floer homology is defined, shown invariant and combinatorially computable via nice real diagrams, and proven to satisfy surface and arc decomposition formulas with new non-Künneth and vanishing p...

  3. An adjunction inequality for Real embedded surfaces

    math.GT 2025-07 conditional novelty 7.0 of 10

    Real embedded surfaces satisfy a new adjunction inequality when the Real Seiberg-Witten invariant is non-zero, and a class is Real-representable iff it lifts to Z_2-equivariant cohomology.

  4. More Exotic $\mathbb{RP}^2$-knots and Homotopy Spheres

    math.GT 2025-07 conditional novelty 7.0 of 10

    For every integer s, the branched double cover of the roll-spun Montesinos knot K(2,3,|6s+1|) is a homotopy 4-sphere, giving infinitely many exotic topologically unknotted RP^2-knots in S^4 with all odd Seiberg-Witten...

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