## Algebraic & Geometric Topology

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

#### Abstract

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.

#### Article information

Source
Algebr. Geom. Topol., Volume 8, Number 2 (2008), 615-692.

Dates
Revised: 9 November 2007
Accepted: 9 November 2007
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.agt/1513796839

Digital Object Identifier
doi:10.2140/agt.2008.8.615

Mathematical Reviews number (MathSciNet)
MR2443092

Zentralblatt MATH identifier
1144.57011

#### Citation

Ozsváth, Peter; Szabó, Zoltán. Holomorphic disks, link invariants and the multi-variable Alexander polynomial. Algebr. Geom. Topol. 8 (2008), no. 2, 615--692. doi:10.2140/agt.2008.8.615. https://projecteuclid.org/euclid.agt/1513796839

#### References

• F Bourgeois, Y Eliashberg, H Hofer, K Wysocki, E Zehnder, Compactness results in symplectic field theory, Geom. Topol. 7 (2003) 799–888
• N M Dunfield, S Gukov, J Rasmussen, The Superpolynomial for Knot Homologies, Experiment. Math 15 (2006) 129–160
• 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) 560–673
• A Floer, Morse theory for Lagrangian intersections, J. Differential Geom. 28 (1988) 513–547
• K Fukaya, Y G Oh, H Ohta, K Ono, Lagrangian intersection Floer theory-anomaly and obstruction, preprint 487 (2000) 488
• B Gornik, Note on Khovanov link cohomology (2004)
• M Gromov, Pseudoholomorphic curves in symplectic manifolds, Invent. Math. 82 (1985) 307–347
• E-N Ionel, T H Parker, Relative Gromov-Witten invariants, Ann. of Math. $(2)$ 157 (2003) 45–96
• M Khovanov, A categorification of the Jones polynomial, Duke Math. J. 101 (2000) 359–426
• M Khovanov, L Rozansky, Matrix factorizations and link homology II
• M Khovanov, L Rozansky, Matrix factorizations and link homology (2004)
• P B Kronheimer, T S Mrowka, Monopoles and contact structures, Invent. Math. 130 (1997) 209–255
• E S Lee, The support of the Khovanov's invariants for alternating knots (2002)
• A-M Li, Y Ruan, Symplectic surgery and Gromov-Witten invariants of Calabi-Yau 3-folds, Invent. Math. 145 (2001) 151–218
• R Lipshitz, A Cylindrical Reformulation of Heegaard Floer Homology (2005)
• D McDuff, D Salamon, $J$-holomorphic curves and quantum cohomology, University Lecture Series 6, American Mathematical Society, Providence, RI (1994)
• J Milnor, Whitehead torsion, Bull. Amer. Math. Soc. 72 (1966) 358–426
• P Ozsváth, Z Szabó, Heegaard Floer homology and alternating knots, Geom. Topol. 7 (2003) 225–254
• P Ozsváth, Z Szabó, Knot Floer homology and the four-ball genus, Geom. Topol. 7 (2003) 615–639
• P Ozsváth, Z Szabó, Holomorphic disks and knot invariants, Adv. Math. 186 (2004) 58–116
• P Ozsváth, Z Szabó, Holomorphic disks and topological invariants for closed three-manifolds, Ann. of Math. $(2)$ 159 (2004) 1027–1158
• P Ozsváth, Z Szabó, Link Floer homology and the Thurston norm (2006)
• J A Rasmussen, Floer homology of surgeries on two-bridge knots, Algebr. Geom. Topol. 2 (2002) 757–789
• J A Rasmussen, Floer homology and knot complements, PhD thesis, Harvard University (2003)
• J A Rasmussen, Khovanov homology and the slice genus (2004)
• D Rolfsen, Knots and links, Mathematics Lecture Series 7, Publish or Perish, Houston, TX (1990) Corrected reprint of the 1976 original