Hostname: page-component-cb9f654ff-5jtmz Total loading time: 0 Render date: 2025-08-28T22:49:53.479Z Has data issue: false hasContentIssue false
  • English
  • Français

Cross-sections of unknotted ribbon disks and algebraic curves

Part of: Complex manifolds Low-dimensional topology

Published online by Cambridge University Press:  11 February 2019

Kyle Hayden
Kyle Hayden*
Affiliation:
Department of Mathematics, Columbia University, New York, NY 10027, USA email hayden@math.columbia.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We resolve parts (A) and (B) of Problem 1.100 from Kirby’s list [Problems in low-dimensional topology, in Geometric topology, AMS/IP Studies in Advanced Mathematics, vol. 2 (American Mathematical Society, Providence, RI, 1997), 35–473] by showing that many nontrivial links arise as cross-sections of unknotted holomorphic disks in the four-ball. The techniques can be used to produce unknotted ribbon surfaces with prescribed cross-sections, including unknotted Lagrangian disks with nontrivial cross-sections.

Keywords

ribbon surfacesalgebraic curvesLagrangian surfacesquasipositive links

MSC classification

Primary: 57M25: Knots and links in $S^3$
Secondary: 32Q55: Topological aspects of complex manifolds

Information

Type
Research Article
Information
Compositio Mathematica , Volume 155 , Issue 2 , February 2019 , pp. 413 - 423
Copyright
© The Author 2019 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Birman, J., Ko, K. H. and Lee, S. J., A new approach to the word and conjugacy problems in the braid groups , Adv. Math. 139 (1998), 322–353.Google Scholar
Boileau, M. and Fourrier, L., Knot theory and plane algebraic curves , Chaos Solitons Fractals 9 (1998), 779–792.Google Scholar
Boileau, M. and Orevkov, S., Quasi-positivité d’une courbe analytique dans une boule pseudo-convexe , C. R. Acad. Sci. Paris Sér. I Math. 332 (2001), 825–830.Google Scholar
Borodzik, M., Morse theory for plane algebraic curves , J. Topol. 5 (2012), 341–365.Google Scholar
Bourgeois, F., Sabloff, J. M. and Traynor, L., Lagrangian cobordisms via generating families: construction and geography , Algebr. Geom. Topol. 15 (2015), 2439–2477.Google Scholar
Chantraine, B., On Lagrangian concordance of Legendrian knots , Algebr. Geom. Topol. 10 (2010), 63–85.Google Scholar
Chantraine, B., Some non-collarable slices of Lagrangian surfaces , Bull. Lond. Math. Soc. 44 (2012), 981–987.Google Scholar
Cornwell, C., Ng, L. and Sivek, S., Obstructions to Lagrangian concordance , Algebr. Geom. Topol. 16 (2016), 797–824.Google Scholar
Dimitroglou Rizell, G. , Legendrian ambient surgery and Legendrian contact homology , J. Symplectic Geom. 14 (2016), 811–901.Google Scholar
Ekholm, T., Honda, K. and Kálmán, T., Legendrian knots and exact Lagrangian cobordisms , J. Eur. Math. Soc. (JEMS) 18 (2016), 2627–2689.Google Scholar
Eliashberg, Ya., Topology of 2-knots in R 4 and symplectic geometry , in The Floer memorial volume, Progress in Mathematics, vol. 133 (Birkhäuser, Basel, 1995), 335–353.Google Scholar
Eliashberg, Ya. and Polterovich, L., Local Lagrangian 2-knots are trivial , Ann. of Math. (2) 144 (1996), 61–76.Google Scholar
Etnyre, J. B., Introductory lectures on contact geometry , in Topology and geometry of manifolds, Athens, GA, 2001, Proceedings of Symposia in Pure Mathematics, vol. 71 (American Mathematical Society, Providence, RI, 2003), 81–107.Google Scholar
Etnyre, J., Legendrian and transversal knots , in Handbook of knot theory (Elsevier, Amsterdam, 2005), 105–185.Google Scholar
Feller, P., Optimal cobordisms between torus knots , Comm. Anal. Geom. 24 (2016), 993–1025.Google Scholar
Fiedler, T., Complex plane curves in the ball , Invent. Math. 95 (1989), 479–506.Google Scholar
Gabai, D., The Murasugi sum is a natural geometric operation , in Low-dimensional topology, San Francisco, CA, 1981, Contemporary Mathematics, vol. 20 (American Mathematical Society, Providence, RI, 1983), 131–143.Google Scholar
Gordon, C. McA., Ribbon concordance of knots in the 3-sphere , Math. Ann. 257 (1981), 157–170.Google Scholar
Hass, J., The geometry of the slice-ribbon problem , Math. Proc. Cambridge Philos. Soc. 94 (1983), 101–108.Google Scholar
Hayden, K., Quasipositive links and Stein surfaces, Preprint (2017), arXiv:1703.10150.Google Scholar
Hayden, K. and Sabloff, J. M., Positive knots and Lagrangian fillability , Proc. Amer. Math. Soc. 143 (2015), 1813–1821.Google Scholar
Hedden, M., Notions of positivity and the Ozsváth–Szabó concordance invariant , J. Knot Theory Ramifications 19 (2010), 617–629.Google Scholar
Hirasawa, M. and Shimokawa, K., Dehn surgeries on strongly invertible knots which yield lens spaces , Proc. Amer. Math. Soc. 128 (2000), 3445–3451.Google Scholar
Kirby, R., Problems in low-dimensional topology , in Geometric topology, AMS/IP Studies in Advanced Mathematics, vol. 2, ed. Kazez, W. H. (American Mathematical Society, Providence, RI, 1997), 35–473.Google Scholar
Kronheimer, P. B. and Mrowka, T. S., The genus of embedded surfaces in the projective plane , Math. Res. Lett. 1 (1994), 797–808.Google Scholar
Orevkov, S. Yu., Rudolph diagrams and analytic realization of the Vitushkin covering , Mat. Zametki 60 (1996), 206–224, 319.Google Scholar
Rudolph, L., Braided surfaces and Seifert ribbons for closed braids , Comment. Math. Helv. 58 (1983), 1–37.Google Scholar
Rudolph, L., Algebraic functions and closed braids , Topology 22 (1983), 191–202.Google Scholar
Rudolph, L., A congruence between link polynomials , Math. Proc. Cambridge Philos. Soc. 107 (1990), 319–327.Google Scholar
Rudolph, L., Quasipositivity as an obstruction to sliceness , Bull. Amer. Math. Soc. (N.S.) 29 (1993), 51–59.Google Scholar
Scharlemann, M. and Thompson, A., Link genus and the Conway moves , Comment. Math. Helv. 64 (1989), 527–535.Google Scholar