Geometry & Topology

Ozsváth–Szabó invariants of contact surgeries

Marco Golla

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We give new tightness criteria for positive surgeries along knots in the 3–sphere, generalising results of Lisca and Stipsicz, and Sahamie. The main tools will be Honda, Kazez and Matić’s, and Ozsváth and Szabó’s Floer-theoretic contact invariants. We compute Ozsváth–Szabó contact invariant of positive contact surgeries along Legendrian knots in the 3–sphere in terms of the classical invariants of the knot. We also combine a Legendrian cabling construction with contact surgeries to get results about rational contact surgeries.

Article information

Geom. Topol., Volume 19, Number 1 (2015), 171-235.

Received: 16 January 2013
Revised: 19 April 2014
Accepted: 9 May 2014
First available in Project Euclid: 16 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57R17: Symplectic and contact topology
Secondary: 57R57: Applications of global analysis to structures on manifolds, Donaldson and Seiberg-Witten invariants [See also 58-XX]

tight contact structures Ozsváth–Szabó invariants


Golla, Marco. Ozsváth–Szabó invariants of contact surgeries. Geom. Topol. 19 (2015), no. 1, 171--235. doi:10.2140/gt.2015.19.171.

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  • J A Baldwin, Capping off open books and the Ozsváth–Szabó contact invariant, J. Symplectic Geom. 11 (2013) 525–561
  • J A Baldwin, D S Vela-Vick, V Vértesi, On the equivalence of Legendrian and transverse invariants in knot Floer homology, Geom. Topol. 17 (2013) 925–974
  • J C Cha, C Livingston, KnotInfo: Table of knot invariants (2014) Available at \setbox0\makeatletter\@url {\unhbox0
  • T D Cochran, B D Franklin, M Hedden, P D Horn, Knot concordance and homology cobordism, Proc. Amer. Math. Soc. 141 (2013) 2193–2208
  • F Ding, H Geiges, A Legendrian surgery presentation of contact $3$–manifolds, Math. Proc. Cambridge Philos. Soc. 136 (2004) 583–598
  • F Ding, H Geiges, Legendrian knots and links classified by classical invariants, Commun. Contemp. Math. 9 (2007) 135–162
  • F Ding, H Geiges, A unique decomposition theorem for tight contact $3$–manifolds, Enseign. Math. 53 (2007) 333–345
  • Y Eliashberg, Classification of overtwisted contact structures on $3$–manifolds, Invent. Math. 98 (1989) 623–637
  • Y Eliashberg, Contact $3$–manifolds twenty years since J Martinet's work, Ann. Inst. Fourier $($Grenoble$)$ 42 (1992) 165–192
  • J B Etnyre, Legendrian and transversal knots, from: “Handbook of knot theory”, (W Menasco, M Thistlethwaite, editors), Elsevier, Amsterdam (2005) 105–185
  • J B Etnyre, K Honda, Knots and contact geometry, I: Torus knots and the figure eight knot, J. Symplectic Geom. 1 (2001) 63–120
  • J B Etnyre, K Honda, Cabling and transverse simplicity, Ann. of Math. 162 (2005) 1305–1333
  • J B Etnyre, D J LaFountain, B Tosun, Legendrian and transverse cables of positive torus knots, Geom. Topol. 16 (2012) 1639–1689
  • P Ghiggini, J Van Horn-Morris, Tight contact structures on the Brieskorn spheres $-\Sigma(2,3,6n-1)$ and contact invariants
  • M Golla, Comparing invariants of Legendrian knots, to appear in Quantum Topol.
  • M Hedden, Knot Floer homology of Whitehead doubles, Geom. Topol. 11 (2007) 2277–2338
  • M Hedden, An Ozsváth–Szabó Floer homology invariant of knots in a contact manifold, Adv. Math. 219 (2008) 89–117
  • J Hom, Heegaard Floer Invariants and Cabling, PhD thesis, University of Pennsylvania (2011) Available at \setbox0\makeatletter\@url {\unhbox0
  • K Honda, On the classification of tight contact structures, I, Geom. Topol. 4 (2000) 309–368
  • K Honda, W Kazez, G Matić, Contact structures, sutured Floer homology and TQFT
  • K Honda, W H Kazez, G Matić, The contact invariant in sutured Floer homology, Invent. Math. 176 (2009) 637–676
  • A Juhász, Holomorphic discs and sutured manifolds, Algebr. Geom. Topol. 6 (2006) 1429–1457
  • A Juhász, D Thurston, Naturality and mapping class groups in Heegaard–Floer homology
  • K Kodama, M Sakuma, Symmetry groups of prime knots up to $10$ crossings, from: “Knots 90”, (A Kawauchi, editor), de Gruyter (1992) 323–340
  • R Lipshitz, A cylindrical reformulation of Heegaard–Floer homology, Geom. Topol. 10 (2006) 955–1097
  • R Lipshitz, P Ozsváth, D Thurston, Bordered Floer homology: Invariance and pairing
  • P Lisca, P Ozsváth, A I Stipsicz, Z Szabó, Heegaard Floer invariants of Legendrian knots in contact three-manifolds, J. Eur. Math. Soc. $($JEMS$)$ 11 (2009) 1307–1363
  • P Lisca, A I Stipsicz, Ozsváth–Szabó invariants and tight contact three-manifolds, I, Geom. Topol. 8 (2004) 925–945
  • P Lisca, A I Stipsicz, Notes on the contact Ozsváth–Szabó invariants, Pacific J. Math. 228 (2006) 277–295
  • P Lisca, A I Stipsicz, Contact surgery and transverse invariants, J. Topol. 4 (2011) 817–834
  • L Ng, P Ozsváth, D Thurston, Transverse knots distinguished by knot Floer homology, J. Symplectic Geom. 6 (2008) 461–490
  • B Ozbagci, An open book decomposition compatible with rational contact surgery, from: “Proceedings of Gökova Geometry–Topology Conference 2005”, (S Akbulut, T Önder, R J Stern, editors), GGT, Gökova (2006) 175–186
  • B Ozbagci, Contact handle decompositions, Topology Appl. 158 (2011) 718–727
  • P Ozsváth, A I Stipsicz, Contact surgeries and the transverse invariant in knot Floer homology, J. Inst. Math. Jussieu 9 (2010) 601–632
  • P Ozsváth, Z Szabó, Knot Floer homology and the four-ball genus, Geom. Topol. 7 (2003) 615–639
  • P Ozsváth, Z Szabó, Holomorphic disks and three-manifold invariants: Properties and applications, Ann. of Math. 159 (2004) 1159–1245
  • P Ozsváth, Z Szabó, Holomorphic disks and topological invariants for closed three-manifolds, Ann. of Math. 159 (2004) 1027–1158
  • P Ozsváth, Z Szabó, Heegaard–Floer homology and contact structures, Duke Math. J. 129 (2005) 39–61
  • P Ozsváth, Z Szabó, Holomorphic triangles and invariants for smooth four-manifolds, Adv. Math. 202 (2006) 326–400
  • P Ozsváth, Z Szabó, Knot Floer homology and integer surgeries, Algebr. Geom. Topol. 8 (2008) 101–153
  • P Ozsváth, Z Szabó, Knot Floer homology and rational surgeries, Algebr. Geom. Topol. 11 (2011) 1–68
  • P Ozsváth, Z Szabó, D Thurston, Legendrian knots, transverse knots and combinatorial Floer homology, Geom. Topol. 12 (2008) 941–980
  • O Plamenevskaya, Bounds for the Thurston–Bennequin number from Floer homology, Algebr. Geom. Topol. 4 (2004) 399–406
  • J Rasmussen, Lens space surgeries and $L$–space homology spheres
  • J Rasmussen, Triangle counts and gluing maps, in preparation
  • L Rudolph, An obstruction to sliceness via contact geometry and “classical” gauge theory, Invent. Math. 119 (1995) 155–163
  • B Sahamie, Dehn twists in Heegaard–Floer homology, Algebr. Geom. Topol. 10 (2010) 465–524
  • S Sarkar, Maslov index formulas for Whitney $n$–gons, J. Symplectic Geom. 9 (2011) 251–270
  • A I Stipsicz, V Vértesi, On invariants for Legendrian knots, Pacific J. Math. 239 (2009) 157–177
  • R Zarev, Bordered Floer homology for sutured manifolds
  • R Zarev, Equivalence of gluing maps for SFH, in preparation