## Geometry & Topology

### Knot Floer homology of Whitehead doubles

Matthew Hedden

#### Abstract

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 $HFK̂(D±(K,t))$ in terms of the invariants for $K$, where $D±(K,t)$ 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 $τ(D+(K,t))=1$ if $t<2τ(K)$ and zero otherwise, where $τ$ is the Ozsváth–Szabó concordance invariant. It follows that the iterated untwisted Whitehead doubles of a knot satisfying $τ(K)>0$ 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.

#### Article information

Source
Geom. Topol., Volume 11, Number 4 (2007), 2277-2338.

Dates
Accepted: 20 August 2007
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513799983

Digital Object Identifier
doi:10.2140/gt.2007.11.2277

Mathematical Reviews number (MathSciNet)
MR2372849

Zentralblatt MATH identifier
1187.57015

Subjects
Primary: 57M27: Invariants of knots and 3-manifolds
Secondary: 57R58: Floer homology

#### Citation

Hedden, Matthew. Knot Floer homology of Whitehead doubles. Geom. Topol. 11 (2007), no. 4, 2277--2338. doi:10.2140/gt.2007.11.2277. https://projecteuclid.org/euclid.gt/1513799983

#### References

• S Akbulut, R Matveyev, Exotic structures and adjunction inequality, Turkish J. Math. 21 (1997) 47–53
• S Akbulut, R Matveyev, A convex decomposition theorem for $4$–manifolds, Internat. Math. Res. Notices (1998) 371–381
• 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., Providence, RI (1997) 21–34
• T D Cochran, R E Gompf, Applications of Donaldson's theorems to classical knot concordance, homology 3–spheres and property $P$, Topology 27 (1988) 495–512
• E Eftekhary, Filtration of Heegaard Floer homology and gluing formulas (2004)
• E Eftekhary, Longitude Floer homology and the Whitehead double, Algebr. Geom. Topol. 5 (2005) 1389–1418
• M H Freedman, A geometric reformulation of $4$–dimensional surgery, Topology Appl. 24 (1986) 133–141
• M H Freedman, F Quinn, Topology of $4$–manifolds, Princeton Mathematical Series 39, Princeton University Press, Princeton, NJ (1990)
• R E Gompf, A I Stipsicz, $4$–manifolds and Kirby calculus, Graduate Studies in Mathematics 20, American Mathematical Society, Providence, RI (1999)
• M Hedden, On knot Floer homology and cabling, Algebr. Geom. Topol. 5 (2005) 1197–1222
• M Hedden, On knot Floer homology and cabling, PhD thesis, Columbia University (2005) Available at http://math.mit.edu/\char'176mhedden/thesis.ps
• M Hedden, P Ording, The Ozsváth–Szabó and Rasmussen concordance invariants are not equal to appear, Amer. J. Math.
• M Khovanov, A categorification of the Jones polynomial, Duke Math. J. 101 (2000) 359–426
• R Kirby, Problems in Low-Dimensional Topology (1995)
• W B R Lickorish, An introduction to knot theory, Graduate Texts in Mathematics 175, Springer, New York (1997)
• R Lipshitz, A cylindrical reformulation of Heegaard Floer homology, Geom. Topol. 10 (2006) 955–1097
• R Lipshitz, Heegaard–Floer invariants of bordered $3$–manifolds, PhD thesis, Stanford University (2006)
• C Livingston, Computations of the Ozsváth–Szabó knot concordance invariant, Geom. Topol. 8 (2004) 735–742
• C Livingston, S Naik, Ozsváth–Szabó and Rasmussen invariants of doubled knots, Algebr. Geom. Topol. 6 (2006) 651–657
• C Manolescu, B Owens, A concordance invariant from the Floer homology of double branched covers (2005)
• J McCleary, User's guide to spectral sequences, Mathematics Lecture Series 12, Publish or Perish, Wilmington, DE (1985)
• Y Ni, Sutured Heegaard diagrams for knots, Algebr. Geom. Topol. 6 (2006) 513–537
• P Ording, The knot Floer homology of satellite (1,1) knots, PhD thesis, Columbia University (2006)
• P S Ozsváth, Z Szabó, Holomorphic triangles and invariants for smooth four-manifolds (2001)
• P Ozsváth, Z Szabó, Heegaard Floer homology and alternating knots, Geom. Topol. 7 (2003) 225–254
• 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 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, Knot Floer homology and integer surgeries (2005)
• P Ozsváth, Z Szabó, Knot Floer homology and rational surgeries (2005)
• J Rasmussen, Floer homology of surgeries on two-bridge knots, Algebr. Geom. Topol. 2 (2002) 757–789
• J Rasmussen, Floer homology and knot complements, PhD thesis, Harvard University (2003)
• J Rasmussen, Khovanov homology and the slice genus (2004)
• L Rudolph, Quasipositivity as an obstruction to sliceness, Bull. Amer. Math. Soc. $($N.S.$)$ 29 (1993) 51–59