Abstract
We give an explicit formula to compute the rotation number of a nullhomologous Legendrian knot in contact (1/n)-surgery diagrams along Legendrian links and obtain a corresponding result for the self-linking number of transverse knots. Moreover, we extend the formula by Ding–Geiges–Stipsicz for computing the d 3-invariant to (1/n)-surgeries.
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09 October 2017
There was a minor mistake in the formula for computing the Poincarédual of the Euler class of the contact structure in Theorem 5.1(1).
References
K. Baker and J. Etnyre, Rational linking and contact geometry, in Perspectives in Analysis, Geometry, and Topology, Progr. Math., 296 Birkhäuser Verlag (Basel, 2012), pp. 19–37.
Baker K., Grigsby J.: Grid diagrams and Legendrian lens space links. J. Symplectic Geom. 7, 415–448 (2009)
J. Conway, Transverse surgery on knots in contact 3-manifolds, arXiv:1409.7077.
Ding F., Geiges H.: A Legendrian surgery presentation of contact 3-manifolds. Math. Proc. Cambridge Philos. Soc. 136, 583–598 (2004)
F. Ding and H. Geiges, Symplectic fillability of tight contact structures on torus bundles, Algebr. Geom. Topol., 1 (2001), 153–172 (electronic).
Ding F., Geiges H., Stipsicz A.: Surgery diagrams for contact 3-manifolds. Turkish J. Math. 28, 41–74 (2004)
S. Durst, M. Kegel and M. Klukas, Computing the Thurston–Bennequin invariant in open books, Acta Math. Hungar., to appear.
J. Etnyre, Legendrian and transversal knots, in Handbook of knot theory, Elsevier B.V. (Amsterdam, 2005), pp. 105–185.
H. Geiges, An Introduction to Contact Topology, Cambridge University Press (Cambridge, 2008).
Geiges H., Onaran S.: Legendrian rational unknots in lens spaces. J. Symplectic Geom. 13, 17–50 (2015)
Gompf R.E.: Handlebody construction of Stein surfaces. Ann. of Math. (2(148), 619–693 (1998)
R. E. Gompf and A. Stipsicz, 4-Manifolds and Kirby Calculus, American Mathematical Society (Providence, 1999).
M. Kegel, The Legendrian knot complement problem, arXiv:1604.05196.
Lisca P., Ozsváth P., Stipsicz A., Szabó Z.: Heegaard Floer invariants of Legendrian knots in contact three-manifolds. J. Eur. Math. Soc. (JEMS) 11, 1307–1363 (2009)
P. Ozsváth, A. Stipsicz and and Z. Szabó, Grid Homology for Knots and Links, Mathematical Surveys and Monographs, American Mathematical Society (Providence, 2015).
D. Rolfsen, Knots and Links, AMS Chelsea Pub. (Providence, 2003).
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A correction to this article is available online at https://doi.org/10.1007/s10474-017-0759-6.
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Durst, S., Kegel, M. Computing rotation and self-linking numbers in contact surgery diagrams. Acta Math. Hungar. 150, 524–540 (2016). https://doi.org/10.1007/s10474-016-0660-8
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DOI: https://doi.org/10.1007/s10474-016-0660-8