Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-10T14:59:04.427Z Has data issue: false hasContentIssue false
  • English
  • Français

Contact surgeries and the transverse invariant in knot Floer homology

Part of: Differential topology Low-dimensional topology

Published online by Cambridge University Press:  11 May 2010

Peter Ozsváth  and
András I. Stipsicz
Peter Ozsváth
Affiliation:
Department of Mathematics, Columbia University, New York, NY 10027, USA (petero@math.columbia.edu)
András I. Stipsicz
Affiliation:
Alfréd Rényi Institute of Mathematics, PO Box 127, H-1364 Budapest, Hungary and Department of Mathematics, Columbia University, New York, NY 10027, USA (stipsicz@renyi.hu)
Get access Rights & Permissions [Opens in a new window]

Abstract

We study naturality properties of the transverse invariant in knot Floer homology under contact (+1)-surgery. This can be used as a calculational tool for the transverse invariant. As a consequence, we show that the Eliashberg–Chekanov twist knots En are not transversely simple for n odd and n > 3.

Keywords

Legendrian knotstransverse knotsHeegaard Floer homology

MSC classification

Secondary: 57M27: Invariants of knots and 3-manifolds 57R58: Floer homology
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 9 , Issue 3 , July 2010 , pp. 601 - 632
Copyright
Copyright © Cambridge University Press 2010

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

1.Akbulut, S. and Ozbagci, B., Lefschetz fibrations on compact Stein surfaces, Geom. Topol. 5 (2001), 319–334 (electronic).CrossRefGoogle Scholar
2.Arikan, F., On the support genus of a contact structure, J. Gökova Geom. Topol. 1 (2007), 92–115.Google Scholar
3.Birman, J. S. and Menasco, W. M., Stabilization in the braid groups, II, Transversal simplicity of knots, Geom. Topol. 10 (2006), 1425–1452 (electronic).CrossRefGoogle Scholar
4.Chekanov, Y., Differential algebra of Legendrian links, Invent. Math. 150(3) (2002), 441–483.CrossRefGoogle Scholar
5.Epstein, J., Fuchs, D. and Meyer, M., Chekanov–Eliashberg invariants and transverse approximations of Legendrian knots, Pac. J. Math. 201(1) (2001), 89–106.CrossRefGoogle Scholar
6.Etnyre, J. B. and Honda, K., Cabling and transverse simplicity, Annals Math. (2) 162(3) (2005), 1305–1333.CrossRefGoogle Scholar
7.Etnyre, J., Ng, L. and Vértesi, V., Legendrian and transverse twist knots, preprint (arXiv:1002.2400; 2010).Google Scholar
8.Hatcher, A. and Thurston, W., Incompressible surfaces in 2-bridge knot complements, Invent. Math. 79 (1985), 225–246.CrossRefGoogle Scholar
9.Honda, K., Kazez, W. and Matić, G., The contact invariant in sutured Floer homology, Invent. Math. 176 (2009), 637–676.CrossRefGoogle Scholar
10.Honda, K., Kazez, W. and Matić, G., On the contact class in Heegaard Floer homology, J. Diff. Geom. 83 (2009), 289–311.Google Scholar
11.Lipshitz, R., A cylindrical reformulation of Heegaard Floer homology, Geom. Topol. 10 (2006), 955–1097 (electronic).CrossRefGoogle Scholar
12.Lisca, P., Ozsváth, P. S., Stipsicz, A. and Szabó, Z., Heegaard Floer invariants of Legendrian knots in contact three-manifolds, J. Eur. Math. Soc. 11 (2009), 1307–1363.Google Scholar
13.Manolescu, C., Ozsváth, P. S., Szabó, Z. and Thurston, D., On combinatorial link Floer homology, Geom. Topol. 11 (2007), 2339–2412.CrossRefGoogle Scholar
14.Manolescu, C., Ozsváth, P. S. and Sarkar, S., A combinatorial description of knot Floer homology, Annals Math. 169 (2009), 633–660.CrossRefGoogle Scholar
15.Ng, L. and Traynor, L., Legendrian solid-torus links, J. Symplectic Geom. 2 (2004), 411–443.CrossRefGoogle Scholar
16.Ng, L., Ozsváth, P. S. and Thurston, D. P., Transverse knots distinguished by knot Floer homology, J. Symplectic Geom. 6 (2008), 461–490.CrossRefGoogle Scholar
17.Ni, Y., Knot Floer homology detects the Thurston norm, Invent. Math. 170 (2007), 577–608.CrossRefGoogle Scholar
18.Ozsváth, P. S. and Szabó, Z., Absolutely graded Floer homologies and intersection forms for four–manifolds with boundary, Adv. Math. 173 (2003), 179–261.CrossRefGoogle Scholar
19.Ozsváth, P. S. and Szabó, Z., Heegaard Floer homology and alternating knots, Geom. Topol. 7 (2003), 225–254 (electronic).CrossRefGoogle Scholar
20.Ozsváth, P. S. and Szabó, Z., Holomorphic disks and knot invariants, Adv. Math. 186(1) (2004), 58–116.CrossRefGoogle Scholar
21.Ozsváth, P. S. and Szabó, Z., Holomorphic disks and three-manifold invariants: properties and applications, Annals Math. (2) 159(3) (2004), 1159–1245.CrossRefGoogle Scholar
22.Ozsváth, P. S. and Szabó, Z., Holomorphic disks and topological invariants for closed three-manifolds, Annals Math. (2) 159(3) (2004), 1027–1158.CrossRefGoogle Scholar
23.Ozsváth, P. S. and Szabó, Z., Heegaard Floer homology and contact structures, Duke Math. J. 129(1) (2005), 39–61.CrossRefGoogle Scholar
24.Ozsváth, P. S. and Szabó, Z., Holomorphic triangles and invariants for smooth four-manifolds, Adv. Math. 202(2) (2006), 326–400.CrossRefGoogle Scholar
25.Ozsváth, P., Szabó, Z. and Thurston, D., Legendrian knots, transverse knots and combinatorial Floer homology, Geom. Topol. 12 (2008), 941–980.CrossRefGoogle Scholar
26.Rasmussen, J. A., Floer homology of surgeries on 2-bridge knots, Alg. Geom. Topol. 2 (2002), 757–789 (electronic).CrossRefGoogle Scholar
27.Rasmussen, J. A., Floer homology and knot complements, PhD thesis, Harvard University (2003).Google Scholar
28.Reid, A. and Walsh, G., Commesurability classes of 2-bridge knot complements, Alg. Geom. Topol. 8 (2008), 1031–1057.CrossRefGoogle Scholar
29.Sahamie, B., Dehn twists in Heegaard Floer homology, Alg. Geom. Topol. 10 (2010), 465–524.CrossRefGoogle Scholar
30.Vértesi, V., Transversely non simple knots, Alg. Geom. Topol. 8 (2008), 1481–1498.CrossRefGoogle Scholar