Skip to main content Accessibility help

Maximal Thurston–Bennequin number and reducible Legendrian surgery

  • Kouichi Yasui (a1)


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.



Hide All
[AM97] Akbulut, S. and Matveyev, R., Exotic structures and adjunction inequality , Turkish J. Math. 21 (1997), 4753.
[Eli90a] Eliashberg, Y., Topological characterization of Stein manifolds of dimension > 2 , Int. J. Math. 1 (1990), 2946.
[Eli90b] 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), 4567.
[Eli92] Eliashberg, Y., Contact 3-manifolds twenty years since J. Martinet’s work , Ann. Inst. Fourier (Grenoble) 42 (1992), 165192.
[FS95] Fintushel, R. and Stern, R. J., Immersed spheres in 4-manifolds and the immersed Thom conjecture , Turkish J. Math. 19 (1995), 145157.
[FT97] Fuchs, D. and Tabachnikov, S., Invariants of Legendrian and transverse knots in the standard contact space , Topology 36 (1997), 10251053.
[Gol15] Golla, M., Ozsváth–Szabó invariants of contact surgeries , Geom. Topol. 19 (2015), 171235.
[Gom98] Gompf, R. E., Handlebody construction of Stein surfaces , Ann. of Math. (2) 148 (1998), 619693.
[GS99] Gompf, R. E. and Stipsicz, A. I., 4-manifolds and Kirby calculus, Graduate Studies in Mathematics, vol. 20 (American Mathematical Society, Providence, RI, 1999).
[GS86] Gonzalez-Acuna, F. and Short, H., Knot surgery and primeness , Math. Proc. Cambridge Philos. Soc. 99 (1986), 89102.
[GL87] Gordon, C. McA. and Luecke, J., Only integral Dehn surgeries can yield reducible manifolds , Math. Proc. Cambridge Philos. Soc. 102 (1987), 97101.
[Gre15] Greene, J. E., L-space surgeries, genus bounds, and the cabling conjecture , J. Differential Geom. 100 (2015), 491506.
[KM94] Kronheimer, P. and Mrowka, T., The genus of embedded surfaces in the projective plane , Math. Res. Lett. 1 (1994), 797808.
[LS16] Lidman, T. and Sivek, S., Contact structures and reducible surgeries , Compos. Math. 152 (2016), 152186.
[LM97] Lisca, P. and Matić, G., Tight contact structures and Seiberg–Witten invariants , Invent. Math. 129 (1997), 509525.
[LM98] Lisca, P. and Matić, G., Stein 4-manifolds with boundary and contact structures , Topology Appl. 88 (1998), 5566.
[MST96] 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), 706788.
[Ng05] Ng, L., A Legendrian Thurston–Bennequin bound from Khovanov homology , Algebr. Geom. Topol. 5 (2005), 16371653.
[Ng12] Ng, L., On arc index and maximal Thurston–Bennequin number , J. Knot Theory Ramifications 21 (2012), 1250031, 11 pp.
[OS04] 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.
[OS00] Ozsváth, P. and Szabó, Z., The symplectic Thom conjecture , Ann. of Math. (2) 151 (2000), 93124.
[OS03] Ozsvath, P. and Szabo, Z., Knot Floer homology and the four-ball genus , Geom. Topol. 7 (2003), 615639.
[Pla04] Plamenevskaya, O., Bounds for the Thurston–Bennequin number from Floer homology , Algebr. Geom. Topol. 4 (2004), 399406.
[Pla06] Plamenevskaya, O., Transverse knots and Khovanov homology , Math. Res. Lett. 13 (2006), 571586.
[Ras10] Rasmussen, J., Khovanov homology and the slice genus , Invent. Math. 182 (2010), 419447.
[Shu07] Shumakovitch, A., Rasmussen invariant, slice-Bennequin inequality, and sliceness of knots , J. Knot Theory Ramifications 16 (2007), 14031412.
[Yas15] Yasui, K., Nonexistence of Stein structures on 4-manifolds and maximal Thurston–Bennequin numbers, J. Symplectic Geom., to appear. Preprint (2015), arXiv:1508.01491.
MathJax is a JavaScript display engine for mathematics. For more information see


MSC classification

Maximal Thurston–Bennequin number and reducible Legendrian surgery

  • Kouichi Yasui (a1)


Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed