Geometry & Topology

Knot concordance and Heegaard Floer homology invariants in branched covers

J Elisenda Grigsby, Daniel Ruberman, and Sašo Strle

Full-text: Open access


By studying the Heegaard Floer homology of the preimage of a knot KS3 inside its double branched cover, we develop simple obstructions to K having finite order in the classical smooth concordance group. As an application, we prove that all 2–bridge knots of crossing number at most 12 for which the smooth concordance order was previously unknown have infinite smooth concordance order.

Article information

Geom. Topol., Volume 12, Number 4 (2008), 2249-2275.

Received: 1 February 2007
Revised: 24 June 2008
Accepted: 13 June 2008
First available in Project Euclid: 20 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57R58: Floer homology 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}
Secondary: 57M12: Special coverings, e.g. branched 57M27: Invariants of knots and 3-manifolds

Smooth knot concordance Heegaard Floer homology branched covers Knot concordance branched cover $\tau$–invariant


Grigsby, J Elisenda; Ruberman, Daniel; Strle, Sašo. Knot concordance and Heegaard Floer homology invariants in branched covers. Geom. Topol. 12 (2008), no. 4, 2249--2275. doi:10.2140/gt.2008.12.2249.

Export citation


  • A J Casson, C M Gordon, Cobordism of classical knots, from: “À la recherche de la topologie perdue”, Progr. Math. 62, Birkhäuser, Boston (1986) 181–199
  • J C Cha, C Livingston, Unknown values in the table of knots (2005)
  • J C Cha, C Livingston, Knotinfo table of knot invariants (2006)\qua\char'176knotinfo/
  • T D Cochran, K E Orr, P Teichner, Knot concordance, Whitney towers and $L\sp 2$–signatures, Ann. of Math. $(2)$ 157 (2003) 433–519
  • T D Cochran, K E Orr, P Teichner, Structure in the classical knot concordance group, Comment. Math. Helv. 79 (2004) 105–123
  • S K Donaldson, An application of gauge theory to four-dimensional topology, J. Differential Geom. 18 (1983) 279–315
  • J E Grigsby, Combinatorial description of knot Floer homology of cyclic branched covers (2006)
  • J E Grigsby, Knot Floer homology in cyclic branched covers, Algebr. Geom. Topol. 6 (2006) 1355–1398
  • S Jabuka, S Naik, Order in the concordance group and Heegaard Floer homology, Geom. Topol. 11 (2007) 979–994
  • B J Jiang, A simple proof that the concordance group of algebraically slice knots is infinitely generated, Proc. Amer. Math. Soc. 83 (1981) 189–192
  • R C Kirby, Problems in low-dimensional topology, from: “Geometric topology (Athens, GA, 1993)”, AMS/IP Stud. Adv. Math. 2, Amer. Math. Soc., Providence, RI (1997) 35–473
  • D A Lee, R Lipshitz, Covering spaces and $\mathbb{Q}$ gradings on Heegaard Floer homology (2006)
  • A Levine, On knots with infinite smooth concordance order (2008)
  • P Lisca, Lens spaces, rational balls and the ribbon conjecture, Geom. Topol. 11 (2007) 429–472
  • P Lisca, Sums of lens spaces bounding rational balls, Algebr. Geom. Topol. 7 (2007) 2141–2164
  • C Livingston, S Naik, Obstructing four-torsion in the classical knot concordance group, J. Differential Geom. 51 (1999) 1–12
  • C Livingston, S Naik, Knot concordance and torsion, Asian J. Math. 5 (2001) 161–167
  • C Manolescu, B Owens, A concordance invariant from the Floer homology of double branched covers, Int. Math. Res. Not. IMRN (2007) Art. ID rnm077, 21
  • C Manolescu, P Ozsváth, S Sarkar, A Combinatorial Description of Knot Floer Homology (2006)
  • C Manolescu, P Ozsváth, Z Szabó, D Thurston, On combinatorial link Floer homology, Geom. Topol. 11 (2007) 2339–2412
  • B Owens, S Strle, Rational homology spheres and the four-ball genus of knots, Adv. Math. 200 (2006) 196–216
  • 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, 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, Holomorphic disks, link invariants and the multi-variable Alexander polynomial (2005)
  • P Ozsváth, Z Szabó, Holomorphic triangles and invariants for smooth four-manifolds, Adv. Math. 202 (2006) 326–400
  • J Rasmussen, Floer homology and knot complements, PhD thesis, Harvard University (2003)
  • J Rasmussen, Khovanov homology and the slice genus (2004) \qua To appear in Invent. Math.
  • J Rasmussen, Knot polynomials and knot homologies, from: “Geometry and topology of manifolds”, Fields Inst. Commun. 47, Amer. Math. Soc., Providence, RI (2005) 261–280
  • S Sarkar, J Wang, A Combinatorial Description of Some Heegaard Floer Homologies (2006)
  • The PARI Group, Bordeaux, PARI/GP, version 2.3.1 (2005)\qua Available from
  • V Turaev, Torsion invariants of ${\rm Spin}\sp c$–structures on $3$–manifolds, Math. Res. Lett. 4 (1997) 679–695