Exotic Mazur manifolds and knot trace invariants

Hayden, Kyle; Mark, Thomas E.; Piccirillo, Lisa

doi:10.1016/j.aim.2021.107994

Cited by 7 publications

(11 citation statements)

References 37 publications

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“…By the connectivity of the space of positive scalar curvature metrics proven in [7], we can take h t so that h t is a positive scalar curvature metric for every t ∈ [0, 1]. Under these settings, we obtain a diffeomorphism between moduli spaces of the form (27) and the rest of proof is the same as that of Theorem 6.6.…”

Section: 2mentioning

confidence: 97%

“…Instead of assuming isometric property of diffemorphisms, we use the contractivity of the space of positive scalar curvature metrics proven in [7]. Let us explain how to take a fiberwise Riemann metric to obtain a diffeomorphism corresponding to (27). Because the family Seiberg-Witten invariant is an isotopy invariant, we can assume that f is product in a neighborhood N of Y preserving the level of N = [0, 1] × Y .…”

Section: 2mentioning

confidence: 99%

“…We shall focus on the case that the complements of the submanifolds are diffeomorphic to each other, since if two 3-manifolds are topologically isotopic but their complements are not diffeomorphic then there cannot exist any smooth isotopy between them. For example, let W and W be a pair of Mazur corks that are homeomorhic but not diffeomorphic [27]. Then their doubles D(W ) and D(W ) [25] are diffeomorphic to the standard S 4 [29], and thus we have two embedded 3manifolds Y = ∂W and Y = ∂W that are not smoothly isotopic in S 4 , since if they were smoothly isotopic, then by the isotopy extension theorem, their complements would be diffeomorphic as well.…”

Section: Introductionmentioning

confidence: 99%

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Diffeomorphisms of 4-manifolds with boundary and exotic embeddings

Iida¹,

Konno²,

Mukherjee³

et al. 2022

Preprint

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Section: 2mentioning

confidence: 97%

Section: 2mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Diffeomorphisms of 4-manifolds with boundary and exotic embeddings

Iida¹,

Konno²,

Mukherjee³

et al. 2022

Preprint

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“…In recent years, satellite knots with winding number ±1 have been instrumental in producing exotic structures on smooth 4-manifolds, see [Yas15,HMP19]. One of the more well-known winding number 1 patterns is the Mazur knot Q in Figure 1.…”

Section: Introductionmentioning

confidence: 99%

Twisted Mazur pattern satellite knots and bordered Floer theory

Petkova¹,

Wong²

2020

Preprint

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We use bordered Floer theory to study properties of twisted Mazur pattern satellite knots Q n (K). We prove that Q n (K) is not Floer homologically thin, with two exceptions. We calculate the 3-genus of Q n (K) in terms of the twisting parameter n and the 3-genus of the companion K, and we determine when Q n (K) is fibered. As an application to our results on Floer thickness and 3-genus, we verify the Cosmetic Surgery Conjecture for many of these satellite knots.

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“…Framed trace embedding lemma,[14, Lemma 3.3]). A framed knot (𝐾, 𝑘) in 𝑆3 is smoothly slice in 𝑊 if and only if −𝑋 𝑘 (𝐾) smoothly embeds in 𝑊.…”

mentioning

confidence: 99%

Trace embeddings from zero surgery homeomorphisms

Nakamura

2023

Journal of Topology

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Manolescu and Piccirillo (2023) recently initiated a program to construct an exotic or by using zero surgery homeomorphisms and Rasmussen's ‐invariant. They find five knots that if any were slice, one could construct an exotic and disprove the Smooth 4‐dimensional Poincaré conjecture. We rule out this exciting possibility and show that these knots are not slice. To do this, we use a zero surgery homeomorphism to relate slice properties of two knots stably after a connected sum with some 4‐manifold. Furthermore, we show that our techniques will extend to the entire infinite family of zero surgery homeomorphisms constructed by Manolescu and Piccirillo. However, our methods do not completely rule out the possibility of constructing an exotic or as Manolescu and Piccirillo proposed. We explain the limits of these methods hoping this will inform and invite new attempts to construct an exotic or . We also show that a family of homotopy spheres constructed by Manolescu and Piccirillo using annulus twists of a ribbon knot are all standard.

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Exotic Mazur manifolds and knot trace invariants

Cited by 7 publications

References 37 publications

Diffeomorphisms of 4-manifolds with boundary and exotic embeddings

Diffeomorphisms of 4-manifolds with boundary and exotic embeddings

Twisted Mazur pattern satellite knots and bordered Floer theory

Trace embeddings from zero surgery homeomorphisms

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