Geometry & Topology

Manifolds with small Heegaard Floer ranks

Matthew Hedden and Yi Ni

Full-text: Open access


We show that the only irreducible three-manifold with positive first Betti number and Heegaard Floer homology of rank two is homeomorphic to zero-framed surgery on the trefoil. We classify links whose branched double cover gives rise to this manifold. Together with a spectral sequence from Khovanov homology to the Floer homology of the branched double cover, our results show that Khovanov homology detects the unknot if and only if it detects the two component unlink.

Article information

Geom. Topol., Volume 14, Number 3 (2010), 1479-1501.

Received: 11 August 2009
Revised: 26 May 2010
Accepted: 24 February 2010
First available in Project Euclid: 20 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Heegaard Floer homology Khovanov homology two-component unlink torus bundle


Hedden, Matthew; Ni, Yi. Manifolds with small Heegaard Floer ranks. Geom. Topol. 14 (2010), no. 3, 1479--1501. doi:10.2140/gt.2010.14.1479.

Export citation


  • Y Ai, Y Ni, Two applications of twisted Floer homology, Int. Math. Res. Not. (2009) 3726–3746
  • Y Ai, T Peters, The twisted Floer homology of torus bundles, Algebr. Geom. Topol. 10 (2010) 679–695
  • D Bar-Natan, On Khovanov's categorification of the Jones polynomial, Algebr. Geom. Topol. 2 (2002) 337–370
  • M J Dunwoody, An equivariant sphere theorem, Bull. London Math. Soc. 17 (1985) 437–448
  • E Eftekhary, Floer homology and existence of incompressible tori in homology spheres
  • S Eliahou, L H Kauffman, M B Thistlethwaite, Infinite families of links with trivial Jones polynomial, Topology 42 (2003) 155–169
  • M Freedman, J Hass, P Scott, Least area incompressible surfaces in $3$–manifolds, Invent. Math. 71 (1983) 609–642
  • D Gabai, Foliations and the topology of $3$–manifolds, J. Differential Geom. 18 (1983) 445–503
  • P Ghiggini, Knot Floer homology detects genus-one fibred knots, Amer. J. Math. 130 (2008) 1151–1169
  • J E Grigsby, S Wehrli, On the colored Jones polynomial, sutured floer homology, and knot floer homology, Adv. Math. 223 (2010) 2114–2165
  • A Hatcher, Algebraic topology, Cambridge Univ. Press (2002)
  • M Hedden, Khovanov homology of the $2$–cable detects the unknot, Math. Res. Lett. 16 (2009) 991–994
  • M Hedden, L Watson, Does Khovanov homology detect the unknot?
  • U Hirsch, W D Neumann, On cyclic branched coverings of spheres, Math. Ann. 215 (1975) 289–291
  • S Jabuka, T E Mark, Product formulae for Ozsváth–Szabó $4$–manifold invariants, Geom. Topol. 12 (2008) 1557–1651
  • M Khovanov, A categorification of the Jones polynomial, Duke Math. J. 101 (2000) 359–426
  • M Khovanov, Patterns in knot cohomology. I, Experiment. Math. 12 (2003) 365–374
  • P Kronheimer, T Mrowka, Khovanov homology is an unknot-detector
  • E S Lee, An endomorphism of the Khovanov invariant, Adv. Math. 197 (2005) 554–586
  • W H Meeks, III, P Scott, Finite group actions on $3$–manifolds, Invent. Math. 86 (1986) 287–346
  • W H Meeks, III, L Simon, S T Yau, Embedded minimal surfaces, exotic spheres, and manifolds with positive Ricci curvature, Ann. of Math. $(2)$ 116 (1982) 621–659
  • Y Ni, Non-separating spheres and twisted Heegaard Floer homology
  • Y Ni, Knot Floer homology detects fibred knots, Invent. Math. 170 (2007) 577–608
  • Y Ni, Heegaard Floer homology and fibred $3$–manifolds, Amer. J. Math. 131 (2009) 1047–1063
  • P Ozsváth, Z Szabó, Absolutely graded Floer homologies and intersection forms for four-manifolds with boundary, Adv. Math. 173 (2003) 179–261
  • P Ozsváth, Z Szabó, Holomorphic disks and genus bounds, Geom. Topol. 8 (2004) 311–334
  • P Ozsváth, Z Szabó, Holomorphic disks and knot invariants, Adv. Math. 186 (2004) 58–116
  • P Ozsváth, Z Szabó, Holomorphic disks and three-manifold invariants: properties and applications, Ann. of Math. $(2)$ 159 (2004) 1159–1245
  • 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, Heegaard Floer homology and contact structures, Duke Math. J. 129 (2005) 39–61
  • P Ozsváth, Z Szabó, On Heegaard diagrams and holomorphic disks, from: “European Congress of Mathematics”, Eur. Math. Soc., Zürich (2005) 769–781
  • P Ozsváth, Z Szabó, On the Heegaard Floer homology of branched double-covers, Adv. Math. 194 (2005) 1–33
  • P Ozsváth, Z Szabó, Holomorphic triangles and invariants for smooth four-manifolds, Adv. Math. 202 (2006) 326–400
  • P Ozsváth, Z Szabó, An introduction to Heegaard Floer homology, from: “Floer homology, gauge theory, and low-dimensional topology”, (D A Ellwood, P Ozsváth, A I Stipsicz, Z Szabó, editors), Clay Math. Proc. 5, Amer. Math. Soc. (2006) 3–27
  • P Ozsváth, Z Szabó, Holomorphic disks, link invariants and the multi-variable Alexander polynomial, Algebr. Geom. Topol. 8 (2008) 615–692
  • P Ozsváth, Z Szabó, Link Floer homology and the Thurston norm, J. Amer. Math. Soc. 21 (2008) 671–709
  • J Rasmussen, Khovanov homology and the slice genus, to appear in Invent. Math.
  • A Shumakovitch, Torsion of the Khovanov homology
  • E H Spanier, Algebraic topology, McGraw-Hill, New York (1966)
  • M Thistlethwaite, Links with trivial Jones polynomial, J. Knot Theory Ramifications 10 (2001) 641–643
  • H Zieschang, Finite groups of mapping classes of surfaces, Lecture Notes in Math. 875, Springer, Berlin (1981)