The Michigan Mathematical Journal

Legendrian Lens Space Surgeries

Hansjörg Geiges and Sinem Onaran

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We show that every tight contact structure on any of the lens spaces L(ns2s+1,s2) with n2 and s1 can be obtained by a single Legendrian surgery along a suitable Legendrian realisation of the negative torus knot T(s,(sn1)) in the tight or an overtwisted contact structure on the 3-sphere.

Article information

Michigan Math. J., Volume 67, Issue 2 (2018), 405-422.

Received: 2 November 2016
Revised: 22 November 2016
First available in Project Euclid: 6 April 2018

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Zentralblatt MATH identifier

Primary: 53D35: Global theory of symplectic and contact manifolds [See also 57Rxx] 53D10: Contact manifolds, general 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45} 57R6


Geiges, Hansjörg; Onaran, Sinem. Legendrian Lens Space Surgeries. Michigan Math. J. 67 (2018), no. 2, 405--422. doi:10.1307/mmj/1522980162.

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  • [1] J. Bailey and D. Rolfsen, An unexpected surgery construction of a lens space, Pacific J. Math. 71 (1977), 295–298.
  • [2] M. Culler, C. M. Gordon, J. Luecke, and P. B. Shalen, Dehn surgery on knots, Ann. of Math. (2) 125 (1987), 237–300.
  • [3] F. Ding and H. Geiges, A Legendrian surgery presentation of contact $3$-manifolds, Math. Proc. Cambridge Philos. Soc. 136 (2004), 583–598.
  • [4] F. Ding, H. Geiges, and A. I. Stipsicz, Surgery diagrams for contact $3$-manifolds, Turkish J. Math. 28 (2004), 41–74.
  • [5] S. Durst and M. Kegel, Computing rotation and self-linking numbers in contact surgery diagrams, Acta Math. Hungar. 150 (2016), 524–540.
  • [6] J. B. Etnyre and K. Honda, Knots and contact geometry I: torus knots and the figure eight knot, J. Symplectic Geom. 1 (2001), 63–120.
  • [7] H. Geiges, An introduction to contact topology, Cambridge Stud. Adv. Math., 109, Cambridge University Press, Cambridge, 2008.
  • [8] H. Geiges and S. Onaran, Legendrian rational unknots in lens spaces, J. Symplectic Geom. 13 (2015), 17–50.
  • [9] E. Giroux, Structures de contact en dimension trois et bifurcations des feuilletages de surfaces, Invent. Math. 141 (2000), 615–689.
  • [10] R. E. Gompf, Handlebody construction of Stein surfaces, Ann. of Math. (2) 148 (1998), 619–693.
  • [11] R. E. Gompf and A. I. Stipsicz, $4$-Manifolds and Kirby calculus, Grad. Stud. Math., 20, American Mathematical Society, Providence, RI, 1999.
  • [12] M. Hedden, On Floer homology and the Berge conjecture on knots admitting lens space surgeries, Trans. Amer. Math. Soc. 363 (2011), 949–968.
  • [13] K. Honda, On the classification of tight contact structures I, Geom. Topol. 4 (2000), 309–368.
  • [14] P. Kronheimer, T. Mrowka, P. Ozsváth, and Z. Szabó, Monopoles and lens space surgeries, Ann. of Math. (2) 165 (2007), 457–546.
  • [15] P. Lisca and G. Matić, Tight contact structures and Seiberg–Witten invariants, Invent. Math. 129 (1997), 509–525.
  • [16] P. Lisca, P. Ozsváth, A. I. Stipsicz, and Z. Szabó, Heegaard Floer invariants of Legendrian knots in contact three-manifolds, J. Eur. Math. Soc. (JEMS) 11 (2009), 1307–1363.
  • [17] L. Moser, Elementary surgery along a torus knot, Pacific J. Math. 38 (1971), 737–745.
  • [18] P. Ozsváth and Z. Szabó, On knot Floer homology and lens space surgeries, Topology 44 (2005), 1281–1300.
  • [19] O. Plamenevskaya, On Legendrian surgeries between lens spaces, J. Symplectic Geom. 10 (2012), 165–181.
  • [20] J. Rasmussen, Lens space surgeries and $L$-space homology spheres, arXiv:0710.2531.
  • [21] M. Tange and Y. Yamada, Four-dimensional manifolds constructed by lens space surgeries along torus knots, J. Knot Theory Ramifications 21 (2012), 1250111.