Abstract
In the note we study Legendrian and transverse knots in rationally null-homologous knot types. In particular, we generalize the standard definitions of self-linking number, Thurston–Bennequin invariant, and rotation number. We then prove a version of Bennequin’s inequality for these knots and classify precisely when the Bennequin bound is sharp for fibered knot types. Finally, we study rational unknots and show that they are weakly Legendrian and transversely simple.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Kenneth L. Baker, John B. Etnyre, and Jeremy Van Horn-Morris. Cabling, contact structures, and mapping class monoids. J. Differ. Geom.to appear.
Kenneth L. Baker and J. Elisenda Grigsby. Grid diagrams and Legendrian lens space links. J. Symplectic. Geom. 7(4):415–448, 2009.
Kenneth L. Baker, J. Elisenda Grigsby, and Matthew Hedden. Grid diagrams for lens spaces and combinatorial knot Floer homology. Int. Math. Res. Not. IMRN, (10):Art. ID rnm024, 39, 2008.
Vladimir Chernov. Framed knots in 3-manifolds and affine self-linking numbers. J. Knot Theory Ramifications, 14(6):791–818, 2005.
Yakov Eliashberg. Filling by holomorphic discs and its applications. In Geometry of low-dimensional manifolds, 2 (Durham, 1989), volume 151 of London Math. Soc. Lecture Note Ser., pages 45–67. Cambridge Univ. Press, Cambridge, 1990.
Yakov Eliashberg. Legendrian and transversal knots in tight contact 3-manifolds. In Topological methods in modern mathematics (Stony Brook, NY, 1991), pages 171–193. Publish or Perish, Houston, TX, 1993.
John B. Etnyre. Introductory lectures on contact geometry. In Topology and geometry of manifolds (Athens, GA, 2001), volume 71 of Proc. Sympos. Pure Math., pages 81–107. Amer. Math. Soc., Providence, RI, 2003.
John B. Etnyre. Legendrian and transversal knots. In Handbook of knot theory, pages 105–185. Elsevier B. V., Amsterdam, 2005.
John B. Etnyre and Ko Honda. Knots and contact geometry. I. Torus knots and the figure eight knot. J. Symplectic Geom., 1(1):63–120, 2001.
John B. Etnyre and Jeremy Van Horn-Morris. Fibered transverse knots and the Bennequin bound. IMRN 2011:1483–1509, 2011.
Robert E. Gompf and András I. Stipsicz. 4-manifolds and Kirby calculus, volume 20 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 1999.
Ko Honda. On the classification of tight contact structures. I. Geom. Topol., 4:309–368 (electronic), 2000.
Ferit Öztürk. Generalised Thurston-Bennequin invariants for real algebraic surface singularities. Manuscripta Math., 117(3):273–298, 2005.
W. P. Thurston and H. E. Winkelnkemper. On the existence of contact forms. Proc. Amer. Math. Soc., 52:345–347, 1975.
Acknowledgments
The first author was partially supported by NSF Grant DMS-0239600. The second author was partially supported by NSF Grants DMS-0239600 and DMS-0804820.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer Science+Business Media, LLC
About this chapter
Cite this chapter
Baker, K., Etnyre, J. (2012). Rational Linking and Contact Geometry. In: Itenberg, I., Jöricke, B., Passare, M. (eds) Perspectives in Analysis, Geometry, and Topology. Progress in Mathematics, vol 296. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-8277-4_2
Download citation
DOI: https://doi.org/10.1007/978-0-8176-8277-4_2
Published:
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-0-8176-8276-7
Online ISBN: 978-0-8176-8277-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)