Hostname: page-component-745bb68f8f-kw2vx Total loading time: 0 Render date: 2025-01-12T07:39:20.087Z Has data issue: false hasContentIssue false
  • English
  • Français

$(1,1)$ L-space knots

Part of: Low-dimensional topology

Published online by Cambridge University Press:  20 March 2018

Joshua Evan Greene ,
Sam Lewallen  and
Faramarz Vafaee
Joshua Evan Greene
Affiliation:
Department of Mathematics, Boston College, Chestnut Hill, MA 02467, USA email [email protected]
Sam Lewallen
Affiliation:
Princeton Neuroscience Institute, Princeton University, Princeton, NJ 08544, USA email [email protected]
Faramarz Vafaee
Affiliation:
Mathematics Department, California Institute of Technology, Pasadena, CA 91125, USA email [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We characterize the $(1,1)$ knots in the 3-sphere and lens spaces that admit non-trivial L-space surgeries. As a corollary, 1-bridge braids in these manifolds admit non-trivial L-space surgeries. We also recover a characterization of the Berge manifold among 1-bridge braid exteriors.

Keywords

Floer homologyL-spaceDehn surgery

MSC classification

Primary: 57M27: Invariants of knots and 3-manifolds
Type
Research Article
Information
Compositio Mathematica , Volume 154 , Issue 5 , May 2018 , pp. 918 - 933
Copyright
© The Authors 2018 

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.)

References

Berge, J., Some knots with surgeries yielding lens spaces, Preprint (2018), arXiv:1802.09722.Google Scholar
Berge, J., The knots in D 2 × S 1 which have nontrivial Dehn surgeries that yield D 2 × S 1 , Topology Appl. 38 (1991), 1–19.CrossRefGoogle Scholar
Boileau, M., Boyer, S., Cebanu, R. and Walsh, G. S., Knot commensurability and the Berge conjecture , Geom. Topol. 16 (2012), 625–664.CrossRefGoogle Scholar
Culler, M., Gordon, C. McA., Luecke, J. and Shalen, P. B., Dehn surgery on knots , Ann. of Math. (2) 125 (1987), 237–300.CrossRefGoogle Scholar
Gabai, D., 1-bridge braids in solid tori , Topology Appl. 37 (1990), 221–235.CrossRefGoogle Scholar
Goda, H., Matsuda, H. and Morifuji, T., Knot Floer homology of (1, 1)-knots , Geom. Dedicata 112 (2005), 197–214.CrossRefGoogle Scholar
Hedden, M., On Floer homology and the Berge conjecture on knots admitting lens space surgeries , Trans. Amer. Math. Soc. 363 (2011), 949–968.CrossRefGoogle Scholar
Hom, J., Lidman, T. and Vafaee, F., Berge–Gabai knots and L-space satellite operations , Algebr. Geom. Topol. 14 (2014), 3745–3763.CrossRefGoogle Scholar
Juhász, A., A survey of Heegaard Floer homology , New Ideas in Low Dimensional Topology, vol. 56 (World Scientific, Hackensack, NJ, 2015), 237–296.Google Scholar
Menasco, W. and Zhang, X., Notes on tangles, 2-handle additions and exceptional Dehn fillings , Pacific J. Math. 198 (2001), 149–174.CrossRefGoogle Scholar
Ozsváth, P. and Szabó, Z., Holomorphic disks and knot invariants , Adv. Math. 186 (2004), 58–116.CrossRefGoogle Scholar
Ozsváth, P. and Szabó, Z., On knot Floer homology and lens space surgeries , Topology 44 (2005), 1281–1300.CrossRefGoogle Scholar
Rasmussen, J., Knot polynomials and knot homologies , Geometry and Topology of Manifolds, vol. 47 (American Mathematical Society, Providence, RI, 2005), 261–280.Google Scholar
Rasmussen, J. and Rasmussen, S. D., Floer simple manifolds and L-space intervals , Adv. Math. 322C (2017), 738–805.CrossRefGoogle Scholar
Vafaee, F., On the knot Floer homology of twisted torus knots , Int. Math. Res. Not. IMRN (2015), 6516–6537.CrossRefGoogle Scholar
Wu, Y.-Q., The classification of Dehn fillings on the outer torus of a 1-bridge braid exterior which produce solid tori , Math. Ann. 330 (2004), 1–15.CrossRefGoogle Scholar