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Maximal Thurston–Bennequin number and reducible Legendrian surgery

Part of: Low-dimensional topology Differential topology

Published online by Cambridge University Press:  30 June 2016

Kouichi Yasui
Kouichi Yasui*
Affiliation:
Department of Mathematics, Graduate School of Science, Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima, 739-8526, Japan email kyasui@hiroshima-u.ac.jp
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Abstract

We give a method for constructing a Legendrian representative of a knot in $S^{3}$ which realizes its maximal Thurston–Bennequin number under a certain condition. The method utilizes Stein handle decompositions of $D^{4}$ , and the resulting Legendrian representative is often very complicated (relative to the complexity of the topological knot type). As an application, we construct infinitely many knots in $S^{3}$ each of which yields a reducible 3-manifold by a Legendrian surgery in the standard tight contact structure. This disproves a conjecture of Lidman and Sivek.

Keywords

4-manifoldLegendrian knotStein manifoldDehn surgerycabling conjecture

MSC classification

Primary: 57R17: Symplectic and contact topology 57R65: Surgery and handlebodies 57M25: Knots and links in $S^3$

Information

Type
Research Article
Information
Compositio Mathematica , Volume 152 , Issue 9 , September 2016 , pp. 1899 - 1914
Copyright
© The Author 2016 

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References

Akbulut, S. and Matveyev, R., Exotic structures and adjunction inequality , Turkish J. Math. 21 (1997), 47–53.Google Scholar
Eliashberg, Y., Topological characterization of Stein manifolds of dimension > 2 , Int. J. Math. 1 (1990), 29–46.CrossRefGoogle Scholar
Eliashberg, Y., Filling by holomorphic discs and its applications , in Geometry of low-dimensional manifolds, 2 (Durham, 1989), London Mathematical Society Lecture Note Series, vol. 151 (Cambridge University Press, Cambridge, 1990), 45–67.Google Scholar
Eliashberg, Y., Contact 3-manifolds twenty years since J. Martinet’s work , Ann. Inst. Fourier (Grenoble) 42 (1992), 165–192.CrossRefGoogle Scholar
Fintushel, R. and Stern, R. J., Immersed spheres in 4-manifolds and the immersed Thom conjecture , Turkish J. Math. 19 (1995), 145–157.Google Scholar
Fuchs, D. and Tabachnikov, S., Invariants of Legendrian and transverse knots in the standard contact space , Topology 36 (1997), 1025–1053.Google Scholar
Golla, M., Ozsváth–Szabó invariants of contact surgeries , Geom. Topol. 19 (2015), 171–235.Google Scholar
Gompf, R. E., Handlebody construction of Stein surfaces , Ann. of Math. (2) 148 (1998), 619–693.CrossRefGoogle Scholar
Gompf, R. E. and Stipsicz, A. I., 4-manifolds and Kirby calculus, Graduate Studies in Mathematics, vol. 20 (American Mathematical Society, Providence, RI, 1999).Google Scholar
Gonzalez-Acuna, F. and Short, H., Knot surgery and primeness , Math. Proc. Cambridge Philos. Soc. 99 (1986), 89–102.Google Scholar
Gordon, C. McA. and Luecke, J., Only integral Dehn surgeries can yield reducible manifolds , Math. Proc. Cambridge Philos. Soc. 102 (1987), 97–101.CrossRefGoogle Scholar
Greene, J. E., L-space surgeries, genus bounds, and the cabling conjecture , J. Differential Geom. 100 (2015), 491–506.Google Scholar
Kronheimer, P. and Mrowka, T., The genus of embedded surfaces in the projective plane , Math. Res. Lett. 1 (1994), 797–808.Google Scholar
Lidman, T. and Sivek, S., Contact structures and reducible surgeries , Compos. Math. 152 (2016), 152–186.CrossRefGoogle Scholar
Lisca, P. and Matić, G., Tight contact structures and Seiberg–Witten invariants , Invent. Math. 129 (1997), 509–525.Google Scholar
Lisca, P. and Matić, G., Stein 4-manifolds with boundary and contact structures , Topology Appl. 88 (1998), 55–66.Google Scholar
Morgan, J., Szabó, Z. and Taubes, C., A product formula for the Seiberg–Witten invariants and the generalized Thom conjecture , J. Differential Geom. 44 (1996), 706–788.Google Scholar
Ng, L., A Legendrian Thurston–Bennequin bound from Khovanov homology , Algebr. Geom. Topol. 5 (2005), 1637–1653.Google Scholar
Ng, L., On arc index and maximal Thurston–Bennequin number , J. Knot Theory Ramifications 21 (2012), 1250031, 11 pp.Google Scholar
Ozbagci, B. and Stipsicz, A. I., Surgery on contact 3-manifolds and Stein surfaces, Bolyai Society Mathematical Studies, vol. 13 (Springer, Berlin, 2004), Janos Bolyai Mathematical Society, Budapest.CrossRefGoogle Scholar
Ozsváth, P. and Szabó, Z., The symplectic Thom conjecture , Ann. of Math. (2) 151 (2000), 93–124.CrossRefGoogle Scholar
Ozsvath, P. and Szabo, Z., Knot Floer homology and the four-ball genus , Geom. Topol. 7 (2003), 615–639.Google Scholar
Plamenevskaya, O., Bounds for the Thurston–Bennequin number from Floer homology , Algebr. Geom. Topol. 4 (2004), 399–406.Google Scholar
Plamenevskaya, O., Transverse knots and Khovanov homology , Math. Res. Lett. 13 (2006), 571–586.Google Scholar
Rasmussen, J., Khovanov homology and the slice genus , Invent. Math. 182 (2010), 419–447.Google Scholar
Shumakovitch, A., Rasmussen invariant, slice-Bennequin inequality, and sliceness of knots , J. Knot Theory Ramifications 16 (2007), 1403–1412.CrossRefGoogle Scholar
Yasui, K., Nonexistence of Stein structures on 4-manifolds and maximal Thurston–Bennequin numbers, J. Symplectic Geom., to appear. Preprint (2015), arXiv:1508.01491.Google Scholar