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Doubly pointed trisection diagrams and surgery on 2-knots

Published online by Cambridge University Press:  11 August 2021

DAVID GAY  and
JEFFREY MEIER
DAVID GAY
Affiliation:
Euclid Lab, 160 Milledge Terrace, Athens, GA 30606, U.S.A.
JEFFREY MEIER
Affiliation:
Western Washington University, Department of Mathematics, 516 High Street, Bellingham, WA 98225 e-mail: meierj3@www.edu
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Abstract

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We study embedded spheres in 4–manifolds (2–knots) via doubly pointed trisection diagrams, showing that such descriptions are unique up to stabilisation and handleslides, and we describe how to obtain trisection diagrams for certain cut-and-paste operations along 2–knots directly from doubly pointed trisection diagrams. The operations described are classical surgery, Gluck surgery, blowdown, and (±4)–rational blowdown, and we illustrate our techniques and results with many examples.

Keywords

57K45 (Primary)57K40 (Secondary)
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 172 , Issue 1 , January 2022 , pp. 163 - 195
Creative Commons
Creative Common License - CCCreative Common License - BYCreative Common License - NCCreative Common License - SA
This is an Open Access article, distributed under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike licence (http://creativecommons.org/licenses/by-nc-sa/4.0/), which permits non-commercial re-use, distribution, and reproduction in any medium, provided the same Creative Commons licence is included and the original work is properly cited. The written permission of Cambridge University Press must be obtained for commercial re-use.
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

References

Bonahon, F.. DiffÉotopies des espaces lenticulaires. Topology 22 (1983), no. 3, 305–314.Google Scholar
Castro, N. A., Gay, D. T. and PinzÓn-Caicedo, J.. Diagrams for relative trisections. Pacific J. Math. 294 (2018), no. 2, 275–305.Google Scholar
Castro, N. A. and Ozbagci, B.. Trisections of 4–manifolds via Lefschetz fibrations. Proceedings of the National Academy of Sciences Oct 2018, 115 (2017), no. 43, 10861–10868. DOI: 10.1073/pnas.1717170115.CrossRefGoogle Scholar
Cochran, T.. Ribbon knots in S 4. J. London Math. Soc. (2) 28 (1983), no. 3, 563–576.Google Scholar
Fintushel, R. and Stern, R. J.. Rational blowdowns of smooth 4-manifolds. J. Differential Geom. 46 (1997), no. 2, 181–235.Google Scholar
Gay, D. and Kirby, R.. Trisecting 4-manifolds. Geom. Topol. 20 (2016), no. 6, 3097–3132.Google Scholar
Gluck, H.. The embedding of two-spheres in the four-sphere. Trans. Amer. Math. Soc. 104 (1962), 308–333.Google Scholar
H, Gluck. Tangled manifolds. Ann. of Math. (2) 76 (1962), 62–72.CrossRefGoogle Scholar
C. McA. Gordon. Knots in the 4-sphere. Comment. Math. Helv. 51 (1976), no. 4, 585–596.CrossRefGoogle Scholar
Gompf, R. E. and Stipsicz, A. I.. 4-manifolds and Kirby calculus, Graduate Studies in Math, vol. 20 (American Mathematical Society, Providence, RI, 1999).CrossRefGoogle Scholar
Johannson, K.. Topology and combinatorics of 3-manifolds . Lecture Notes in Math, vol. 1599 (Springer-Verlag, Berlin, 1995).Google Scholar
Kim, S. and Miller, M.. Trisections of surface complements and the Price twist. Algebr. Geom. Topol. 20 (2020), 343–373. DOI: 10.2140/agt.2020.20.343.CrossRefGoogle Scholar
Laudenbach, F. and PoÉnaru, V.. A note on 4-dimensional handlebodies. Bull. Soc. Math. France 100 (1972), 337–344.CrossRefGoogle Scholar
Meier, J.. Trisections and spun 4–manifolds. Math. Res. Lett. 25 (2018), no. 5, 1497–1524. DOI: 10.4310/MRL.2018.v25.n5.a7.CrossRefGoogle Scholar
Meier, J., Schirmer, T. and Alexander, Zupan. Classification of trisections and the generalized property R conjecture. Proc. Amer. Math. Soc. 144 (2016), no. 11, 4983–4997.Google Scholar
Meier, J. and Zupan, A.. Bridge trisections of knotted surfaces in 4–manifolds. Proceedings of the National Academy of Sciences Oct 2018, 115 (2017), no. 43, 10880–10886. DOI: 10.1073/pnas.1717171115.CrossRefGoogle Scholar
Meier, J. and Zupan, A.. Bridge trisections of knotted surfaces in S 4. Trans. Amer. Math. Soc. 369 (2017), no. 10, 7343–7386.Google Scholar
Meier, J. and Zupan, A.. Characterising Dehn surgeries on links via trisections. Proceedings of the National Academy of Sciences Oct 2018, 115 (2017), no. 43, 10887–10893. DOI: 10.1073/pnas.1717187115.CrossRefGoogle Scholar
Pao, P. S.. The topological structure of 4-manifolds with effective torus actions. I. Trans. Amer. Math. Soc. 227 (1977), 279–317.Google Scholar
Plotnick, S. P.. Equivariant intersection forms, knots in S 4, and rotations in 2-spheres. Trans. Amer. Math. Soc. 296 (1986), no. 2, 543–575.Google Scholar
Sunukjian, N. S.. Surfaces in 4-manifolds: concordance, isotopy, and surgery. Int. Math. Res. Not. IMRN (2015), no. 17, 7950–7978.Google Scholar