Deep and shallow slice knots in 4-manifolds
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- by Michael R. Klug and Benjamin M. Ruppik HTML | PDF
- Proc. Amer. Math. Soc. Ser. B 8 (2021), 204-218
Abstract:
We consider slice disks for knots in the boundary of a smooth compact 4-manifold $X^{4}$. We call a knot $K \subset \partial X$ deep slice in $X$ if there is a smooth properly embedded $2$-disk in $X$ with boundary $K$, but $K$ is not concordant to the unknot in a collar neighborhood $\partial X \times {I}$ of the boundary.
We point out how this concept relates to various well-known conjectures and give some criteria for the nonexistence of such deep slice knots. Then we show, using the Wall self-intersection invariant and a result of Rohlin, that every 4-manifold consisting of just one 0- and a nonzero number of 2-handles always has a deep slice knot in the boundary.
We end by considering 4-manifolds where every knot in the boundary bounds an embedded disk in the interior. A generalization of the Murasugi-Tristram inequality is used to show that there does not exist a compact, oriented $4$-manifold $V$ with spherical boundary such that every knot $K \subset {S}^{3} = \partial V$ is slice in $V$ via a null-homologous disk.
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Additional Information
- Michael R. Klug
- Affiliation: Department of Mathematics, University of California, Berkeley, Berkeley, California, 94720-3840
- MR Author ID: 1112294
- Email: michael.r.klug@gmail.com
- Benjamin M. Ruppik
- Affiliation: Max-Planck-Institut für Mathematik, Bonn, Germany
- ORCID: 0000-0001-9035-9217
- Email: bruppik@mpim-bonn.mpg.de
- Received by editor(s): October 3, 2020
- Received by editor(s) in revised form: January 11, 2021
- Published electronically: June 9, 2021
- Additional Notes: The authors were supported by the Max Planck Institute for Mathematics in Bonn.
- Communicated by: Shelly Harvey
- © Copyright 2021 by the authors under Creative Commons Attribution-Noncommercial 3.0 License (CC BY NC 3.0)
- Journal: Proc. Amer. Math. Soc. Ser. B 8 (2021), 204-218
- MSC (2020): Primary 57K40; Secondary 57K10
- DOI: https://doi.org/10.1090/bproc/89
- MathSciNet review: 4273166