Monopole Floer homology and Legendrian knots

Sivek, Steven¹

¹ Department of Mathematics, Harvard University, Cambridge, MA 02138, USA

Geometry & topology, Tome 16 (2012) no. 2, pp. 751-779.

Résumé

We use monopole Floer homology for sutured manifolds to construct invariants of unoriented Legendrian knots in a contact $3$–manifold. These invariants assign to a knot $K\subset Y$ elements of the monopole knot homology $KHM\left(-Y,K\right)$, and they strongly resemble the knot Floer homology invariants of Lisca, Ozsváth, Stipsicz, and Szabó. We prove several vanishing results, investigate their behavior under contact surgeries, and use this to construct many examples of nonloose knots in overtwisted $3$–manifolds. We also show that these invariants are functorial with respect to Lagrangian concordance.

Détail

DOI : 10.2140/gt.2012.16.751

Classification : 57M27, 57R58, 57R17
Keywords: Legendrian knot, monopole Floer homology

Bibliographie

[1] J M Bloom, T Mrowka, P Ozsváth, The Künneth principle in Floer homology, in preparation

[2] S Boyer, D Lines, Conway potential functions for links in $\mathbf{Q}$–homology $3$–spheres, Proc. Edinburgh Math. Soc. 35 (1992) 53

[3] J C Cha, C Livingston, KnotInfo: Table of knot invariants, website (2010)

[4] B Chantraine, Lagrangian concordance of Legendrian knots, Algebr. Geom. Topol. 10 (2010) 63

[5] Y Chekanov, Differential algebra of Legendrian links, Invent. Math. 150 (2002) 441

[6] W Chongchitmate, L Ng, An atlas of Legendrian knots, to appear in Exp. Math.

[7] M Culler, Gridlink: a tool for knot theorists

[8] F Ding, H Geiges, Symplectic fillability of tight contact structures on torus bundles, Algebr. Geom. Topol. 1 (2001) 153

[9] F Ding, H Geiges, Handle moves in contact surgery diagrams, J. Topol. 2 (2009) 105

[10] F Ding, H Geiges, A I Stipsicz, Surgery diagrams for contact $3$–manifolds, Turkish J. Math. 28 (2004) 41

[11] Y Eliashberg, Invariants in contact topology, from: "Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998)" (1998) 327

[12] Y Eliashberg, M Fraser, Topologically trivial Legendrian knots, J. Symplectic Geom. 7 (2009) 77

[13] J B Etnyre, On contact surgery, Proc. Amer. Math. Soc. 136 (2008) 3355

[14] J B Etnyre, K Honda, Knots and contact geometry I: Torus knots and the figure eight knot, J. Symplectic Geom. 1 (2001) 63

[15] J B Etnyre, K Honda, On the nonexistence of tight contact structures, Ann. of Math. 153 (2001) 749

[16] J B Etnyre, D S Vela-Vick, Torsion and open book decompositions, Int. Math. Res. Not. 2010 (2010) 4385

[17] D Fuchs, Chekanov–Eliashberg invariant of Legendrian knots: existence of augmentations, J. Geom. Phys. 47 (2003) 43

[18] P Ghiggini, Knot Floer homology detects genus-one fibred knots, Amer. J. Math. 130 (2008) 1151

[19] E Giroux, Convexité en topologie de contact, Comment. Math. Helv. 66 (1991) 637

[20] K Honda, On the classification of tight contact structures I, Geom. Topol. 4 (2000) 309

[21] K Honda, W H Kazez, G Matić, Tight contact structures on fibered hyperbolic $3$–manifolds, J. Differential Geom. 64 (2003) 305

[22] A Juhász, Holomorphic discs and sutured manifolds, Algebr. Geom. Topol. 6 (2006) 1429

[23] T Kálmán, Contact homology and one parameter families of Legendrian knots, Geom. Topol. 9 (2005) 2013

[24] Y Kanda, On the Thurston–Bennequin invariant of Legendrian knots and nonexactness of Bennequin's inequality, Invent. Math. 133 (1998) 227

[25] P B Kronheimer, T S Mrowka, Monopoles and contact structures, Invent. Math. 130 (1997) 209

[26] P Kronheimer, T Mrowka, Monopoles and three-manifolds, New Math. Monogr. 10, Cambridge Univ. Press (2007)

[27] P Kronheimer, T Mrowka, Knots, sutures, and excision, J. Differential Geom. 84 (2010) 301

[28] P Kronheimer, T Mrowka, P Ozsváth, Z Szabó, Monopoles and lens space surgeries, Ann. of Math. 165 (2007) 457

[29] Y Lekili, Heegaard Floer homology of broken fibrations over the circle

[30] P Lisca, P Ozsváth, A I Stipsicz, Z Szabó, Heegaard Floer invariants of Legendrian knots in contact three-manifolds, J. Eur. Math. Soc. 11 (2009) 1307

[31] P Lisca, A I Stipsicz, Ozsváth–Szabó invariants and tight contact three-manifolds I, Geom. Topol. 8 (2004) 925

[32] T Mrowka, Y Rollin, Legendrian knots and monopoles, Algebr. Geom. Topol. 6 (2006) 1

[33] T Mrowka, Y Rollin, Contact invariants and monopole Floer homology, in preparation

[34] K Niederkrüger, C Wendl, Weak symplectic fillings and holomorphic curves, Ann. Sci. Éc. Norm. Supér. (4) 44 (2011) 801

[35] P Ozsváth, A I Stipsicz, Contact surgeries and the transverse invariant in knot Floer homology, J. Inst. Math. Jussieu 9 (2010) 601

[36] P Ozsváth, Z Szabó, D Thurston, Legendrian knots, transverse knots and combinatorial Floer homology, Geom. Topol. 12 (2008) 941

[37] B Sahamie, Dehn twists in Heegaard Floer homology, Algebr. Geom. Topol. 10 (2010) 465

[38] A I Stipsicz, V Vértesi, On invariants for Legendrian knots, Pacific J. Math. 239 (2009) 157

[39] C H Taubes, Embedded contact homology and Seiberg–Witten Floer cohomology V, Geom. Topol. 14 (2010) 2961

[40] C Wendl, A hierarchy of local symplectic filling obstructions for contact $3$–manifolds