## Geometry & Topology

### Order in the concordance group and Heegaard Floer homology

#### Abstract

We use the Heegaard–Floer homology correction terms defined by Ozsváth–Szabó to formulate a new obstruction for a knot to be of finite order in the smooth concordance group. This obstruction bears a formal resemblance to that of Casson and Gordon but is sensitive to the difference between the smooth versus topological category. As an application we obtain new lower bounds for the concordance order of small crossing knots.

#### Article information

Source
Geom. Topol., Volume 11, Number 2 (2007), 979-994.

Dates
Revised: 6 February 2007
Accepted: 30 January 2007
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513799866

Digital Object Identifier
doi:10.2140/gt.2007.11.979

Mathematical Reviews number (MathSciNet)
MR2326940

Zentralblatt MATH identifier
1132.57008

Subjects
Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}
Secondary: 57R58: Floer homology

#### Citation

Jabuka, Stanislav; Naik, Swatee. Order in the concordance group and Heegaard Floer homology. Geom. Topol. 11 (2007), no. 2, 979--994. doi:10.2140/gt.2007.11.979. https://projecteuclid.org/euclid.gt/1513799866

#### References

• A J Casson, C M Gordon, On slice knots in dimension three, from: “Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, CA, 1976), Part 2”, Proc. Sympos. Pure Math. XXXII, Amer. Math. Soc., Providence, R.I. (1978) 39–53
• 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
• 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
• T Cohcran, P Teichner, Knot concordance and von Neumann $\rho$–invariants (2004)
• D Coray, F Michel, Knot cobordism and amphicheirality, Comment. Math. Helv. 58 (1983) 601–616
• 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
• P Kirk, C Livingston, Twisted Alexander invariants, Reidemeister torsion, and Casson–Gordon invariants, Topology 38 (1999) 635–661
• J Levine, Invariants of knot cobordism, Invent. Math. 8 $(1969)$, 98–110; addendum, ibid. 8 (1969) 355
• J Levine, Knot cobordism groups in codimension two, Comment. Math. Helv. 44 (1969) 229–244
• C Livingston, Order 2 algebraically slice knots, from: “Proceedings of the Kirbyfest (Berkeley, CA, 1998)”, Geom. Topol. Monogr. 2, Geom. Topol. Publ., Coventry (1999) 335–342
• C Livingston, S Naik, Knot concordance and torsion, Asian J. Math. 5 (2001) 161–167
• B Owens, C Manolescu, A concordance invariant from the Floer homology of double branched covers (2005)
• 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 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, Knots with unknotting number one and Heegaard Floer homology, Topology 44 (2005) 705–745
• J Rasmussen, Khovanov homology and the slice genus (2004)
• A Tamulis, Knots of ten or fewer crossings of algebraic order 2, J. Knot Theory Ramifications 11 (2002) 211–222