Ozsváth–Szabó invariants of contact surgeries

Golla, Marco¹

¹ Department of Mathematics, University of Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy

Geometry & topology, Tome 19 (2015) no. 1, pp. 171-235.

Résumé

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.

Détail

DOI : 10.2140/gt.2015.19.171

Classification : 57R17, 57R57
Keywords: tight contact structures, Ozsváth–Szabó invariants

Bibliographie

[1] J A Baldwin, Capping off open books and the Ozsváth–Szabó contact invariant, J. Symplectic Geom. 11 (2013) 525

[2] 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

[3] J C Cha, C Livingston, KnotInfo: Table of knot invariants (2014)

[4] T D Cochran, B D Franklin, M Hedden, P D Horn, Knot concordance and homology cobordism, Proc. Amer. Math. Soc. 141 (2013) 2193

[5] F Ding, H Geiges, A Legendrian surgery presentation of contact 3–manifolds, Math. Proc. Cambridge Philos. Soc. 136 (2004) 583

[6] F Ding, H Geiges, Legendrian knots and links classified by classical invariants, Commun. Contemp. Math. 9 (2007) 135

[7] F Ding, H Geiges, A unique decomposition theorem for tight contact 3–manifolds, Enseign. Math. 53 (2007) 333

[8] Y Eliashberg, Classification of overtwisted contact structures on 3–manifolds, Invent. Math. 98 (1989) 623

[9] Y Eliashberg, Contact 3–manifolds twenty years since J Martinet’s work, Ann. Inst. Fourier (Grenoble) 42 (1992) 165

[10] J B Etnyre, Legendrian and transversal knots, from: "Handbook of knot theory" (editors W Menasco, M Thistlethwaite), Elsevier (2005) 105

[11] J B Etnyre, K Honda, Knots and contact geometry, I : Torus knots and the figure eight knot, J. Symplectic Geom. 1 (2001) 63

[12] J B Etnyre, K Honda, Cabling and transverse simplicity, Ann. of Math. 162 (2005) 1305

[13] J B Etnyre, D J Lafountain, B Tosun, Legendrian and transverse cables of positive torus knots, Geom. Topol. 16 (2012) 1639

[14] P Ghiggini, J Van Horn-Morris, Tight contact structures on the Brieskorn spheres −Σ(2,3,6n − 1) and contact invariants,

[15] M Golla, Comparing invariants of Legendrian knots, to appear in Quantum Topol.

[16] M Hedden, Knot Floer homology of Whitehead doubles, Geom. Topol. 11 (2007) 2277

[17] M Hedden, An Ozsváth–Szabó Floer homology invariant of knots in a contact manifold, Adv. Math. 219 (2008) 89

[18] J Hom, Heegaard Floer Invariants and Cabling, PhD thesis, University of Pennsylvania (2011)

[19] K Honda, On the classification of tight contact structures, I, Geom. Topol. 4 (2000) 309

[20] K Honda, W Kazez, G Matić, Contact structures, sutured Floer homology and TQFT,

[21] K Honda, W H Kazez, G Matić, The contact invariant in sutured Floer homology, Invent. Math. 176 (2009) 637

[22] A Juhász, Holomorphic discs and sutured manifolds, Algebr. Geom. Topol. 6 (2006) 1429

[23] A Juhász, D Thurston, Naturality and mapping class groups in Heegaard–Floer homology,

[24] K Kodama, M Sakuma, Symmetry groups of prime knots up to 10 crossings, from: "Knots 90" (editor A Kawauchi), de Gruyter (1992) 323

[25] R Lipshitz, A cylindrical reformulation of Heegaard–Floer homology, Geom. Topol. 10 (2006) 955

[26] R Lipshitz, P Ozsváth, D Thurston, Bordered Floer homology : Invariance and pairing,

[27] 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

[28] P Lisca, A I Stipsicz, Ozsváth–Szabó invariants and tight contact three-manifolds, I, Geom. Topol. 8 (2004) 925

[29] P Lisca, A I Stipsicz, Notes on the contact Ozsváth–Szabó invariants, Pacific J. Math. 228 (2006) 277

[30] P Lisca, A I Stipsicz, Contact surgery and transverse invariants, J. Topol. 4 (2011) 817

[31] L Ng, P Ozsváth, D Thurston, Transverse knots distinguished by knot Floer homology, J. Symplectic Geom. 6 (2008) 461

[32] B Ozbagci, An open book decomposition compatible with rational contact surgery, from: "Proceedings of Gökova Geometry–Topology Conference 2005" (editors S Akbulut, T Önder, R J Stern), GGT (2006) 175

[33] B Ozbagci, Contact handle decompositions, Topology Appl. 158 (2011) 718

[34] P Ozsváth, A I Stipsicz, Contact surgeries and the transverse invariant in knot Floer homology, J. Inst. Math. Jussieu 9 (2010) 601

[35] P Ozsváth, Z Szabó, Knot Floer homology and the four-ball genus, Geom. Topol. 7 (2003) 615

[36] P Ozsváth, Z Szabó, Holomorphic disks and three-manifold invariants : Properties and applications, Ann. of Math. 159 (2004) 1159

[37] P Ozsváth, Z Szabó, Holomorphic disks and topological invariants for closed three-manifolds, Ann. of Math. 159 (2004) 1027

[38] P Ozsváth, Z Szabó, Heegaard–Floer homology and contact structures, Duke Math. J. 129 (2005) 39

[39] P Ozsváth, Z Szabó, Holomorphic triangles and invariants for smooth four-manifolds, Adv. Math. 202 (2006) 326

[40] P Ozsváth, Z Szabó, Knot Floer homology and integer surgeries, Algebr. Geom. Topol. 8 (2008) 101

[41] P Ozsváth, Z Szabó, Knot Floer homology and rational surgeries, Algebr. Geom. Topol. 11 (2011) 1

[42] P Ozsváth, Z Szabó, D Thurston, Legendrian knots, transverse knots and combinatorial Floer homology, Geom. Topol. 12 (2008) 941

[43] O Plamenevskaya, Bounds for the Thurston–Bennequin number from Floer homology, Algebr. Geom. Topol. 4 (2004) 399

[44] J Rasmussen, Lens space surgeries and L–space homology spheres,

[45] J Rasmussen, Triangle counts and gluing maps, in preparation

[46] L Rudolph, An obstruction to sliceness via contact geometry and “classical” gauge theory, Invent. Math. 119 (1995) 155

[47] B Sahamie, Dehn twists in Heegaard–Floer homology, Algebr. Geom. Topol. 10 (2010) 465

[48] S Sarkar, Maslov index formulas for Whitney n–gons, J. Symplectic Geom. 9 (2011) 251

[49] A I Stipsicz, V Vértesi, On invariants for Legendrian knots, Pacific J. Math. 239 (2009) 157

[50] R Zarev, Bordered Floer homology for sutured manifolds,

[51] R Zarev, Equivalence of gluing maps for SFH, in preparation