Holomorphic disks, link invariants and the multi-variable Alexander polynomial

Ozsváth, Peter¹ ; Szabó, Zoltán²

¹Department of Mathematics, Columbia University, New York, NY 10027, USA
²Department of Mathematics, Princeton University, New Jersey 08544, USA

Algebraic and Geometric Topology, Tome 8 (2008) no. 2, pp. 615-692

Résumé

The knot Floer homology is an invariant of knots in S3 whose Euler characteristic is the Alexander polynomial of the knot. In this paper we generalize this to links in S3 giving an invariant whose Euler characteristic is the multi-variable Alexander polynomial. We study basic properties of this invariant, and give some calculations.

Détail

DOI : 10.2140/agt.2008.8.615

Keywords: Floer homology, links, link invariant, multi-variable Alexander polynomial

Bibliographie

[1] F Bourgeois, Y Eliashberg, H Hofer, K Wysocki, E Zehnder, Compactness results in symplectic field theory, Geom. Topol. 7 (2003) 799

[2] N Dunfield, S Gukov, J Rasmussen, The Superpolynomial for Knot Homologies, Experiment. Math 15 (2006) 129

[3] Y Eliashberg, A Givental, H Hofer, Introduction to symplectic field theory, from: "GAFA 2000 Visions in Mathematics – Towards 2000", Geom. Funct. Anal. Special Issue (2000) 560

[4] A Floer, Morse theory for Lagrangian intersections, J. Differential Geom. 28 (1988) 513

[5] K Fukaya, Y Oh, H Ohta, K Ono, Lagrangian intersection Floer theory-anomaly and obstruction, preprint 487 (2000) 488

[6] B Gornik, Note on Khovanov link cohomology (2004)

[7] M Gromov, Pseudoholomorphic curves in symplectic manifolds, Invent. Math. 82 (1985) 307

[8] E N Ionel, T H Parker, Relative Gromov-Witten invariants, Ann. of Math. $(2)$ 157 (2003) 45

[9] M Khovanov, A categorification of the Jones polynomial, Duke Math. J. 101 (2000) 359

[10] M Khovanov, L Rozansky, Matrix factorizations and link homology II

[11] M Khovanov, L Rozansky, Matrix factorizations and link homology (2004)

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

[13] E Lee, The support of the Khovanov's invariants for alternating knots (2002)

[14] A M Li, Y Ruan, Symplectic surgery and Gromov-Witten invariants of Calabi-Yau 3-folds, Invent. Math. 145 (2001) 151

[15] R Lipshitz, A Cylindrical Reformulation of Heegaard Floer Homology (2005)

[16] D Mcduff, D Salamon, $J$-holomorphic curves and quantum cohomology, University Lecture Series 6, American Mathematical Society (1994)

[17] J Milnor, Whitehead torsion, Bull. Amer. Math. Soc. 72 (1966) 358

[18] P Ozsváth, Z Szabó, Heegaard Floer homology and alternating knots, Geom. Topol. 7 (2003) 225

[19] P Ozsváth, Z Szabó, Knot Floer homology and the four-ball genus, Geom. Topol. 7 (2003) 615

[20] P Ozsváth, Z Szabó, Holomorphic disks and knot invariants, Adv. Math. 186 (2004) 58

[21] P Ozsváth, Z Szabó, Holomorphic disks and topological invariants for closed three-manifolds, Ann. of Math. $(2)$ 159 (2004) 1027

[22] P Ozsváth, Z Szabó, Link Floer homology and the Thurston norm (2006)

[23] J A Rasmussen, Floer homology of surgeries on two-bridge knots, Algebr. Geom. Topol. 2 (2002) 757

[24] J Rasmussen, Floer homology and knot complements, PhD thesis, Harvard University (2003)

[25] J Rasmussen, Khovanov homology and the slice genus (2004)

[26] D Rolfsen, Knots and links, Mathematics Lecture Series 7, Publish or Perish (1990)

[27] M Thistlethwaite, Link table

[28] V Turaev, Torsions of 3-manifolds, from: "Invariants of knots and 3-manifolds (Kyoto, 2001)" (editors T Ohtsuki, e al), Geom. Topol. Monogr. 4 (2002) 295