Average four-genus of two-bridge knots
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- by S. Baader, A. Kjuchukova, L. Lewark, F. Misev and A. Ray;
- Proc. Amer. Math. Soc. 153 (2025), 357-362
- DOI: https://doi.org/10.1090/proc/14784
- Published electronically: November 25, 2024
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Abstract:
We prove that the expected value of the ratio between the smooth four-genus and the Seifert genus of two-bridge knots tends to zero as the crossing number tends to infinity.References
- F. Bonahon and L. C. Siebenmann, New geometric splittings of classical knots and the classification and symmetries of arborescent knots, preprint available at https://dornsife.usc.edu/francis-bonahon/preprints-available/.
- J. H. Conway, An enumeration of knots and links, and some of their algebraic properties, Computational Problems in Abstract Algebra (Proc. Conf., Oxford, 1967) Pergamon, Oxford-New York-Toronto, Ont., 1970, pp. 329–358. MR 258014
- David Gabai, The Murasugi sum is a natural geometric operation, Low-dimensional topology (San Francisco, Calif., 1981) Contemp. Math., vol. 20, Amer. Math. Soc., Providence, RI, 1983, pp. 131–143. MR 718138, DOI 10.1090/conm/020/718138
- Joshua Greene and Stanislav Jabuka, The slice-ribbon conjecture for 3-stranded pretzel knots, Amer. J. Math. 133 (2011), no. 3, 555–580. MR 2808326, DOI 10.1353/ajm.2011.0022
- Akio Kawauchi, A survey of knot theory, Birkhäuser Verlag, Basel, 1996. Translated and revised from the 1990 Japanese original by the author. MR 1417494
- P. B. Kronheimer and T. S. Mrowka, The genus of embedded surfaces in the projective plane, Math. Res. Lett. 1 (1994), no. 6, 797–808. MR 1306022, DOI 10.4310/MRL.1994.v1.n6.a14
- Ana G. Lecuona, On the slice-ribbon conjecture for Montesinos knots, Trans. Amer. Math. Soc. 364 (2012), no. 1, 233–285. MR 2833583, DOI 10.1090/S0002-9947-2011-05385-7
- Paolo Lisca, Lens spaces, rational balls and the ribbon conjecture, Geom. Topol. 11 (2007), 429–472. MR 2302495, DOI 10.2140/gt.2007.11.429
- Jacob Rasmussen, Khovanov homology and the slice genus, Invent. Math. 182 (2010), no. 2, 419–447. MR 2729272, DOI 10.1007/s00222-010-0275-6
- Kurt Reidemeister, Homotopieringe und Linsenräume, Abh. Math. Sem. Univ. Hamburg 11 (1935), no. 1, 102–109 (German). MR 3069647, DOI 10.1007/BF02940717
- Lee Rudolph, Quasipositivity as an obstruction to sliceness, Bull. Amer. Math. Soc. (N.S.) 29 (1993), no. 1, 51–59. MR 1193540, DOI 10.1090/S0273-0979-1993-00397-5
- Horst Schubert, Knoten mit zwei Brücken, Math. Z. 65 (1956), 133–170 (German). MR 82104, DOI 10.1007/BF01473875
- Yakov G. Sinai, Probability theory, Springer Textbook, Springer-Verlag, Berlin, 1992. An introductory course; Translated from the Russian and with a preface by D. Haughton. MR 1215487, DOI 10.1007/978-3-662-02845-2
Bibliographic Information
- S. Baader
- Affiliation: Mathematisches Institut, Sidlerstr. 5, 3012 Bern, Switzerland
- MR Author ID: 757518
- Email: sebastian.baader@math.unibe.ch
- A. Kjuchukova
- Affiliation: Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany
- MR Author ID: 1272582
- Email: sashka@mpim-bonn.mpg.de
- L. Lewark
- Affiliation: Mathematisches Institut, Sidlerstr. 5, 3012 Bern, Switzerland
- MR Author ID: 1064492
- ORCID: 0000-0002-1569-9219
- Email: lukas@lewark.de
- F. Misev
- Affiliation: Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany
- MR Author ID: 1254979
- Email: fmisev@mpim-bonn.mpg.de
- A. Ray
- Affiliation: Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany
- MR Author ID: 1039665
- ORCID: 0000-0002-5620-4139
- Email: aruray@mpim-bonn.mpg.de
- Received by editor(s): April 10, 2019
- Received by editor(s) in revised form: July 3, 2019
- Published electronically: November 25, 2024
- Additional Notes: The second author was partially supported by NSF grant DMS-1821257
- Communicated by: David Futer
- © Copyright 2024 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 153 (2025), 357-362
- MSC (2020): Primary 57K10
- DOI: https://doi.org/10.1090/proc/14784
- MathSciNet review: 4840282