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A note on the four-dimensional clasp number of knots

Part of: Topological manifolds

Published online by Cambridge University Press:  16 July 2021

PETER FELLER  and
JUNGHWAN PARK
PETER FELLER
Affiliation:
ETH Zurich, Rämistrasse 101, 8092 Zurich, Switzerland e-mail: [email protected]
JUNGHWAN PARK
Affiliation:
KAIST, 291 Daehak-ro, Daejeon, 34141, South Korea e-mail: [email protected]
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Abstract

Among the knots that are the connected sum of two torus knots with cobordism distance 1, we characterise those that have 4-dimensional clasp number at least 2, and we show that their n-fold connected self-sum has 4-dimensional clasp number at least 2n. Our proof works in the topological category. To contrast this, we build a family of topologically slice knots for which the n-fold connected self-sum has 4-ball genus n and 4-dimensional clasp number at least 2n.

MSC classification

Primary: 57N70: Cobordism and concordance
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 173 , Issue 1 , July 2022 , pp. 213 - 226
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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References

Bodnár, J., Celoria, D., and Golla, M.. A note on cobordisms of algebraic knots. Algebr. Geom. Topol., 17(4):2543–2564, 2017.CrossRefGoogle Scholar
Borodzik, M. and Livingston, C.. Semigroups, d-invariants and deformations of cuspidal singular points of plane curves. J. Lond. Math. Soc. (2), 93(2):439–463, 2016.CrossRefGoogle Scholar
Brieskorn, E.. Beispiele zur Differentialtopologie von Singularitäten. Invent. Math., 2 (1966), 1–14.CrossRefGoogle Scholar
Conway, A., Nagel, M., and Toffoli, E.. Multivariable signatures, genus bounds, and $0.5$ -solvable cobordisms. Michigan Math. J., 69(2) (2020), 381–427.CrossRefGoogle Scholar
Daemi, A. and Scaduto, C.. Chern-simons functional, singular instantons, and the four-dimensional clasp number. arXiv:2007.13160, (2020).Google Scholar
Daemi, A. and Scaduto, C.. Equivariant singular instanton homology, virtual talk in nearly carbon neutral geometry and topology conference. Available at http://ncngt.org/2020/05/16/floer-theory-and-low-dimensional-topology-gauge-theory/, (June 2020).Google Scholar
Feller, P.. Gordian adjacency for torus knots. Algebr. Geom. Topol., 14(2) (2014), 769–793.CrossRefGoogle Scholar
Friedl, S., Livingston, C., and Zentner, R.. Knot concordances and alternating knots. Michigan Math. J., 66(2) (2017), 421–432.CrossRefGoogle Scholar
Feller, P. and Park, J.. Genus one cobordisms between torus knots. arXiv:1910.01672 to appear: Int. Math. Res. Not. IMRN, (2019).CrossRefGoogle Scholar
Freedman, M. H.. The topology of four-dimensional manifolds. J. Differential Geom., 17(3) (1982), 357–453.CrossRefGoogle Scholar
Gambaudo, J.-M. and Ghys, É.. Braids and signatures. Bull. Soc. Math. France, 133(4) (2005), 541–579.CrossRefGoogle Scholar
Hedden, M.. On knot Floer homology and cabling. II. Int. Math. Res. Not. IMRN, (12) (2009), 2248–2274.CrossRefGoogle Scholar
Hedden, M., Kim, S.-G., and Livingston, C.. Topologically slice knots of smooth concordance order two. J. Differential Geom., 102(3) (2016), 353–393.CrossRefGoogle Scholar
Hom, J.. A survey on Heegaard Floer homology and concordance. J. Knot Theory Ramifications, 26(2) (2017), 1740015, 24.CrossRefGoogle Scholar
Hom, J. and Wu, Z.. Four-ball genus bounds and a refinement of the Ozváth-Szabó tau invariant. J. Symplectic Geom., 14(1) (2016), 305–323.CrossRefGoogle Scholar
Juhász, A. and Zemke, I.. Concordance invariants with applications to the 4-dimensional clasp number. arXiv:2007.07106, (2020).Google Scholar
Kim, M. H., Krcatovich, D., and Park, J.. Links with non-trivial Alexander polynomial which are topologically concordant to the Hopf link. Trans. Amer. Math. Soc., 371(8) (2019), 5379–5400.CrossRefGoogle Scholar
Kronheimer, P. B. and Mrowka, T. S.. Instantons and some concordance invariants of knots. arXiv:1910.11129, (2019).Google Scholar
Kim, M. H. and Park, K.. An infinite-rank summand of knots with trivial Alexander polynomial. J. Symplectic Geom., 16(6) (2018), 1749–1771.CrossRefGoogle Scholar
Kronheimer, P. B.. Genus versus double-points for immersed surfaces, virtual talk in the regensburg low-dimensional geometry and topology seminar. Available at https://www-app.uni-regensburg.de/Fakultaeten/MAT/sfb-higher-invariants/index.php/Regensb-urg_low-dimensional_geometry_and_topology_seminar, (May 2020).Google Scholar
Kauffman, L. H. and Taylor, L. R.. Signature of links. Trans. Amer. Math. Soc., 216 (1976), 351–365.CrossRefGoogle Scholar
Levine, J.. Knot cobordism groups in codimension two. Comment. Math. Helv., 44 1969, 229–244.CrossRefGoogle Scholar
Litherland, R. A.. Signatures of iterated torus knots. In Topology of low-dimensional manifolds (Proc. Second Sussex Conf., Chelwood Gate, 1977), Lecture Notes in Math. vol. 722 (Springer, Berlin, 1979), pages 71–84.CrossRefGoogle Scholar
Livingston, C.. Computations of the Ozsváth–Szabó knot concordance invariant. Geom. Topol., 8 (2004), 735–742.CrossRefGoogle Scholar
Livingston, C.. Notes on the knot concordance invariant upsilon. Algebr. Geom. Topol., 17(1) (2017), 111–130.CrossRefGoogle Scholar
Livingston, C.. Signature functions of knots. Proc. Amer. Math. Soc., 146(10) (2018), 4513–4520.CrossRefGoogle Scholar
Owens, B. and Strle, S.. Immersed disks, slicing numbers and concordance unknotting numbers. Comm. Anal. Geom., 24(5) (2016), 1107–1138.CrossRefGoogle Scholar
Ozsváth, P., Stipsicz, A. I., and Szabó, Z.. Concordance homomorphisms from knot Floer homology. Adv. Math., 315 (2017), 366–426.CrossRefGoogle Scholar
Powell, M.. The four-genus of a link, Levine–Tristram signatures and satellites. J. Knot Theory Ramifications, 26(2) (2017), 1740008, 28.CrossRefGoogle Scholar
Tristram, A. G.. Some cobordism invariants for links. Proc. Camb. Philo. Soc., 66 (1969), 251–264.CrossRefGoogle Scholar
Wang, S.. Semigroups of L-space knots and nonalgebraic iterated torus knots. Math. Res. Lett., 25(1) (2018), 335–346.CrossRefGoogle Scholar