Knot Floer homology of Whitehead doubles

Hedden, Matthew¹

¹ Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge MA 02139-4307, USA

Geometry & topology, Tome 11 (2007) no. 4, pp. 2277-2338.

Résumé

In this paper we study the knot Floer homology invariants of the twisted and untwisted Whitehead doubles of an arbitrary knot, $K$. A formula is presented for the filtered chain homotopy type of $\widehat{HFK}\left(D_{\pm }\left(K,t\right)\right)$ in terms of the invariants for $K$, where $D_{\pm }\left(K,t\right)$ denotes the $t$–twisted positive (resp. negative)-clasped Whitehead double of $K$. In particular, the formula can be used iteratively and can be used to compute the Floer homology of manifolds obtained by surgery on Whitehead doubles. An immediate corollary is that $\tau \left(D_{+}\left(K,t\right)\right)=1$ if $t<2\tau \left(K\right)$ and zero otherwise, where $\tau$ is the Ozsváth–Szabó concordance invariant. It follows that the iterated untwisted Whitehead doubles of a knot satisfying $\tau \left(K\right)>0$ are not smoothly slice. Another corollary is a closed formula for the Floer homology of the three-manifold obtained by gluing the complement of an arbitrary knot, $K$, to the complement of the trefoil.

Détail

DOI : 10.2140/gt.2007.11.2277

Keywords: Whitehead double, Heegaard diagram, Floer homology

Bibliographie

[1] S Akbulut, R Matveyev, Exotic structures and adjunction inequality, Turkish J. Math. 21 (1997) 47

[2] S Akbulut, R Matveyev, A convex decomposition theorem for $4$–manifolds, Internat. Math. Res. Notices (1998) 371

[3] D Auckly, Surgery numbers of 3–manifolds: a hyperbolic example, from: "Geometric topology (Athens, GA, 1993)", AMS/IP Stud. Adv. Math. 2, Amer. Math. Soc. (1997) 21

[4] T D Cochran, R E Gompf, Applications of Donaldson's theorems to classical knot concordance, homology 3–spheres and property $P$, Topology 27 (1988) 495

[5] E Eftekhary, Filtration of Heegaard Floer homology and gluing formulas (2004)

[6] E Eftekhary, Longitude Floer homology and the Whitehead double, Algebr. Geom. Topol. 5 (2005) 1389

[7] M H Freedman, A geometric reformulation of $4$–dimensional surgery, Topology Appl. 24 (1986) 133

[8] M H Freedman, F Quinn, Topology of $4$–manifolds, Princeton Mathematical Series 39, Princeton University Press (1990)

[9] R E Gompf, A I Stipsicz, $4$–manifolds and Kirby calculus, Graduate Studies in Mathematics 20, American Mathematical Society (1999)

[10] M Hedden, On knot Floer homology and cabling, Algebr. Geom. Topol. 5 (2005) 1197

[11] M Hedden, On knot Floer homology and cabling, PhD thesis, Columbia University (2005)

[12] M Hedden, P Ording, The Ozsváth–Szabó and Rasmussen concordance invariants are not equal

[13] M Khovanov, A categorification of the Jones polynomial, Duke Math. J. 101 (2000) 359

[14] R Kirby, Problems in Low-Dimensional Topology (1995)

[15] W B R Lickorish, An introduction to knot theory, Graduate Texts in Mathematics 175, Springer (1997)

[16] R Lipshitz, A cylindrical reformulation of Heegaard Floer homology, Geom. Topol. 10 (2006) 955

[17] R Lipshitz, Heegaard–Floer invariants of bordered $3$–manifolds, PhD thesis, Stanford University (2006)

[18] C Livingston, Computations of the Ozsváth–Szabó knot concordance invariant, Geom. Topol. 8 (2004) 735

[19] C Livingston, S Naik, Ozsváth–Szabó and Rasmussen invariants of doubled knots, Algebr. Geom. Topol. 6 (2006) 651

[20] C Manolescu, B Owens, A concordance invariant from the Floer homology of double branched covers (2005)

[21] J Mccleary, User's guide to spectral sequences, Mathematics Lecture Series 12, Publish or Perish (1985)

[22] Y Ni, Sutured Heegaard diagrams for knots, Algebr. Geom. Topol. 6 (2006) 513

[23] P Ording, The knot Floer homology of satellite (1,1) knots, PhD thesis, Columbia University (2006)

[24] P S Ozsváth, Z Szabó, Holomorphic triangles and invariants for smooth four-manifolds (2001)

[25] P Ozsváth, Z Szabó, Heegaard Floer homology and alternating knots, Geom. Topol. 7 (2003) 225

[26] P Ozsváth, Z Szabó, Knot Floer homology and the four-ball genus, Geom. Topol. 7 (2003) 615

[27] P Ozsváth, Z Szabó, Holomorphic disks and knot invariants, Adv. Math. 186 (2004) 58

[28] P Ozsváth, Z Szabó, Holomorphic disks and three-manifold invariants: properties and applications, Ann. of Math. $(2)$ 159 (2004) 1159

[29] P Ozsváth, Z Szabó, Holomorphic disks and topological invariants for closed three-manifolds, Ann. of Math. $(2)$ 159 (2004) 1027

[30] P Ozsváth, Z Szabó, Knot Floer homology and integer surgeries (2005)

[31] P Ozsváth, Z Szabó, Knot Floer homology and rational surgeries (2005)

[32] J Rasmussen, Floer homology of surgeries on two-bridge knots, Algebr. Geom. Topol. 2 (2002) 757

[33] J Rasmussen, Floer homology and knot complements, PhD thesis, Harvard University (2003)

[34] J Rasmussen, Khovanov homology and the slice genus (2004)

[35] L Rudolph, Quasipositivity as an obstruction to sliceness, Bull. Amer. Math. Soc. $($N.S.$)$ 29 (1993) 51