Geometry & Topology

Knot Floer homology and the four-ball genus

Peter Ozsváth and Zoltán Szabó

Full-text: Open access


We use the knot filtration on the Heegaard Floer complex CF̂ to define an integer invariant τ(K) for knots. Like the classical signature, this invariant gives a homomorphism from the knot concordance group to . As such, it gives lower bounds for the slice genus (and hence also the unknotting number) of a knot; but unlike the signature, τ gives sharp bounds on the four-ball genera of torus knots. As another illustration, we calculate the invariant for several ten-crossing knots.

Article information

Geom. Topol., Volume 7, Number 2 (2003), 615-639.

Received: 16 January 2003
Revised: 17 October 2003
Accepted: 21 September 2003
First available in Project Euclid: 21 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57R58: Floer homology
Secondary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45} 57M27: Invariants of knots and 3-manifolds

Floer homology knot concordance signature 4–ball genus


Ozsváth, Peter; Szabó, Zoltán. Knot Floer homology and the four-ball genus. Geom. Topol. 7 (2003), no. 2, 615--639. doi:10.2140/gt.2003.7.615.

Export citation


  • J O Berge, Some knots with surgeries giving lens spaces, unpublished manuscript
  • M Boileau, C Weber, Le problème de J. Milnor sur le nombre gordien des noeuds algébriques, Enseign. Math. 30 (1984) 173–222
  • S K Donaldson, P B Kronheimer, The Geometry of Four-Manifolds, Oxford Mathematical Monographs, Oxford University Press (1990)
  • C A Giller, A family of links and the Conway calculus, Trans. Amer. Math. Soc. 270 (1982) 75–109
  • C McA Gordon, R A Litherland, K Murasugi, Signatures of covering links, Canad. J. Math. 33 (1981) 381–394
  • L H Kauffman, Formal knot theory, Mathematical Notes 30, Princeton University Press (1983)
  • T Kawamura, The unknotting numbers of $10\sb {139}$ and $10\sb {152}$ are $4$, Osaka J. Math. 35 (1998) 539–546
  • P B Kronheimer, T S Mrowka, Gauge theory for embedded surfaces. I, Topology 32 (1993) 773–826
  • J W Milnor, Singular points of complex hypersurfaces, Annals of Mathematics Studies 61, Princeton University Press (1968)
  • B Owens, S Strle, Rational homology spheres and four-ball genus (2003), preprint
  • P S Ozsváth, Z Szabó, Holomorphic triangles and invariants for smooth four-manifolds.
  • P S Ozsváth, Z Szabó, On knot Floer homology and lens space surgeries.
  • P S Ozsváth, Z Szabó, The symplectic Thom conjecture, Ann. of Math. 151 (2000) 93–124
  • P S Ozsváth, Z Szabó, Holomorphic disks and three-manifold invariants: properties and applications (2001), to appear in Annals of Math. arXiv:math.SG/0105202
  • P S Ozsváth, Z Szabó, Holomorphic disks and topological invariants for closed three-manifolds (2001). To appear in Annals of Math.
  • P S Ozsváth, Z Szabó, Holomorphic disk invariants for symplectic four-manifolds (2002).
  • P S Ozsváth, Z Szabó, Holomorphic disks and knot invariants (2002), to appear in Adv. in Math.arXiv:math.GT/0209056
  • P S Ozsváth, Z Szabó, Absolutely graded Floer homologies and intersection forms for four-manifolds with boundary, Adv. Math. 173 (2003) 179–261
  • P S Ozsváth, Z Szabó, Heegaard Floer homology and alternating knots, Geom. Topol. 7 (2003) 225–254.
  • P S Ozsváth, Z Szabó, Knot Floer homology, genus bounds, and mutation (2003).
  • J A Rasmussen, Floer homologies of surgeries on two-bridge knots, Algebr. Geom. Topol. 2 (2002) 757–589
  • J A Rasmussen, Floer homology and knot complements, Ph.D. thesis, Harvard University (2003)
  • D Rolfsen, Knots and links, Mathematics Lecture Series 7, Publish or Perish, Inc. (1976)
  • L Rudolph, Qusipositivity as an obstruction to sliceness, Bull. Amer. Math. Soc. (N.S.) 29 (1993) 51–59
  • T Tanaka, Unknotting numbers of quasipositive knots, Topology Appl. 88 (1998) 239–246