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A note on cables and the involutive concordance invariants
Abstract
We prove a formula for the involutive concordance invariants of the cabled knots in terms of those of the companion knot and the pattern knot. As a consequence, we show that any iterated cable of a knot with parameters of the form (odd,1) is not smoothly slice as long as either of the involutive concordance invariants of the knot is nonzero. Our formula also gives new bounds for the unknotting number of a cabled knot, which are sometimes stronger than other known bounds coming from knot Floer homology.
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We prove that the (2,1)(2,1)-cable of the figure-eight knot is not smoothly slice by showing that its branched double cover bounds no equivariant homology ball.
We prove that the quotient of the integer homology cobordism group by the subgroup generated by the Seifert fibered spaces is infinitely generated.
We prove a connected sum formula for involutive Heegaard Floer homology, and use it to study the involutive correction terms of connected sums. In particular, we give an example of a three-manifold with d̲(Y)≠d(Y)≠d¯(Y). We also construct a homomorphism from the three-dimensional homology cobordism group to an algebraically defined Abelian group, consisting of certain complexes (equipped with a homotopy involution) modulo a notion of local equivalence.
Using the conjugation symmetry on Heegaard Floer complexes, we define a
three-manifold invariant called involutive Heegaard Floer homology, which is
meant to correspond to -equivariant Seiberg-Witten Floer
homology. Further, we obtain two new invariants of homology cobordism,
and , and two invariants of smooth knot concordance,
and . We also develop a formula for the
involutive Heegaard Floer homology of large integral surgeries on knots. We
give explicit calculations in the case of L-space knots and thin knots. In
particular, we show that detects the non-sliceness of the
figure-eight knot. Other applications include constraints on which large
surgeries on alternating knots can be homology cobordant to other large
surgeries on alternating knots.
Given an equivariant knot of order 2, we study the induced action of the symmetry on the knot Floer homology. We relate this action with the induced action of the symmetry on the Heegaard Floer homology of large surgeries on . This surgery formula can be thought of as an equivariant analog of the involutive large surgery formula proved by Hendricks and Manolescu. As a consequence, we obtain that for certain double branched covers of and corks, the induced action of the involution on Heegaard Floer homology can be identified with an action on the knot Floer homology. As an application, we calculate equivariant correction terms which are invariants of a generalized version of the spin rational homology cobordism group, and define two knot concordance invariants. We also compute the action of the symmetry on the knot Floer complex of for several equivariant knots.
We define several equivariant concordance invariants using knot Floer homology. We show that our invariants provide a lower bound for the equivariant slice genus and use this to give a family of strongly invertible slice knots whose equivariant slice genus grows arbitrarily large, answering a question of Boyle and Issa. We also apply our formalism to several seemingly nonequivariant questions. In particular, we show that knot Floer homology can be used to detect exotic pairs of slice disks, recovering an example due to Hayden, and extend a result due to Miller and Powell regarding stabilization distance. Our formalism suggests a possible route toward establishing the noncommutativity of the equivariant concordance group.
For certain kinds of 3-manifolds, the question whether such a manifold can be obtained by nontrivial Dehn surgery on a knot in S3 is reduced to the corresponding question for hyperbolic knots. Examples are, whether one can obtain S3, a fake S3, a fake S3 with nonzero Rohhn invariant, S1 × S2 a fake S1 × S2, S1 × S2 # M with M nonsimply-connected, or a fake lens space.
We prove a cabling formula for the concordance invariant , defined by
the author and Hom. This gives rise to a simple and effective 4-ball genus
bound for many cable knots.
Based on work of Rasmussen, we construct a concordance invariant associated
to the knot Floer complex, and exhibit examples in which this invariant gives
arbitrarily better bounds on the 4-ball genus than the Ozsvath-Szabo tau
invariant.
In [27], we introduced Floer homology theories HF - (Y,s), HF∞(Y,s), HF + (Y, t), HF(Y,s),and HF red (Y, s) associated to closed, oriented three-manifolds Y equipped with a Spiny structures s ∈ Spin c (Y). In the present paper, we give calculations and study the properties of these invariants. The calculations suggest a conjectured relationship with Seiberg-Witten theory. The properties include a relationship between the Euler characteristics of HF ± and Turaev's torsion, a relationship with the minimal genus problem (Thurston norm), and surgery exact sequences. We also include some applications of these techniques to three-manifold topology.
In Ozsváth and Szabó (Holomorphic triangles and invariants for smooth four-manifolds, math. SG/0110169, 2001), we introduced absolute gradings on the three-manifold invariants developed in Ozsváth and Szabó (Holomorphic disks and topological invariants for closed three-manifolds, math.SG/0101206, Ann. of Math. (2001), to appear). Coupled with the surgery long exact sequences, we obtain a number of three- and four-dimensional applications of this absolute grading including strengthenings of the “complexity bounds” derived in Ozsváth and Szabó (Holomorphic disks and three-manifold invariants: properties and applications, math.SG/0105202, Ann. of Math. (2001), to appear), restrictions on knots whose surgeries give rise to lens spaces, and calculations of HF+ for a variety of three-manifolds. Moreover, we show how the structure of HF+ constrains the exoticness of definite intersection forms for smooth four-manifolds which bound a given three-manifold. In addition to these new applications, the techniques also provide alternate proofs of Donaldson's diagonalizability theorem and the Thom conjecture for .
We define a concordance invariant, epsilon(K), associated to the knot Floer
complex of K, and give a formula for the Ozsv\'ath-Szab\'o concordance
invariant tau of K_{p,q}, the (p,q)-cable of a knot K, in terms of p, q,
tau(K), and epsilon(K). We also describe the behavior of epsilon under cabling,
allowing one to compute tau of iterated cables. Various properties and
applications of epsilon are also discussed.
Two Dehn surgeries on a knot are called {\it purely cosmetic}, if they yield
manifolds that are homeomorphic as oriented manifolds. Suppose there exist
purely cosmetic surgeries on a knot in , we show that the two surgery
slopes must be the opposite of each other. One ingredient of our proof is a
Dehn surgery formula for correction terms in Heegaard Floer homology.
We use bordered Floer homology to give a formula for the knot Floer homology
of any (p, pn+1)-cable of a thin knot K in terms of Delta_K(t), tau(K), p, and
n. We also give a formula for the Ozsvath-Szabo concordance invariant tau(K_{p,
pn+1}) in terms of tau(K), p, and n, and a formula for tau(K_{p,q}) for almost
all relatively prime p and q.
In this paper a classification of the manifolds obtained by a (p, q) surgery along an (r, s) torus knot is given. If |σ| = |rsp + q| ≠ 0, then the manifold is a Seifert manifold, singularly fibered by simple closed curves over the 2-sphere with singularities of types α1= s, α2 = r, and α3= |σ|. If M = 1, then there are only two singular fibers of types α1= s, α2 = r, and the manifold is a lens space L(|q|, ps2). If |σ| =0, then the manifold is not singularly fibered but is the connected sum of two lens spaces L(r, s)#L(s, r). It is also shown that the torus knots are the only knots whose comple¬ments can be singularly fibered.
Let K be a rationally null-homologous knot in a three-manifold Y. We construct a version of knot Floer homology in this context, including a description of the Floer homology of a three-manifold obtained as Morse surgery on the knot K. As an application, we express the Heegaard Floer homology of rational surgeries on Y along a null-homologous knot K in terms of the filtered homotopy type of the knot invariant for K. This has applications to Dehn surgery problems for knots in . In a different direction, we use the techniques developed here to calculate the Heegaard Floer homology of an arbitrary Seifert fibered three-manifold.
Let K be a null-homologous knot in a three-manifold Y. We give a description of the Heegaard Floer homology of integer surgeries on Y along K in terms of the filtered homotopy type of the knot invariant for K. As an illustration, we calculate the Heegaard Floer homology groups of non-trivial circle bundles over Riemann surfaces.
This paper is devoted to the study of the knot Floer homology groups HFK(S^3,K_{2,n}), where K_{2,n} denotes the (2,n) cable of an arbitrary knot, K. It is shown that for sufficiently large |n|, the Floer homology of the cabled knot depends only on the filtered chain homotopy type of CFK(K). A precise formula for this relationship is presented. In fact, the homology groups in the top 2 filtration dimensions for the cabled knot are isomorphic to the original knot's Floer homology group in the top filtration dimension. The results are extended to (p,pn+-1) cables. As an example we compute HFK((T_{2,2m+1})_{2,2n+1}) for all sufficiently large |n|, where T_{2,2m+1} denotes the (2,2m+1)-torus knot.
Using work of Ozsvath and Szabo, we show that if a nontrivial knot in S^3 admits a lens space surgery with slope p, then p <= 4g+3, where g is the genus of the knot. This is a close approximation to a bound conjectured by Goda and Teragaito.
The aim of this article is to introduce and study certain topological invariants for closed, oriented three-manifolds Y. These groups are relatively Z-graded Abelian groups associated to SpinC structures over Y. Given a genus g Heegaard splitting of Y, these theories are variants of the Lagrangian Floer homology for the g-fold symmetric product of the surface relative to certain totally real subspaces associated to the handlebodies. Comment: 118 pages, 10 figures, to appear in Annals of Mathematics. Reorganized both this paper and its sequel: the first paper now gives the definitions for closed, oriented three-manifolds. Properties and examples are given in the second paper
We continue our study of the knot Floer homology invariants of cable knots. For large |n|, we prove that many of the filtered subcomplexes in the knot Floer homology filtration associated to the (p,pn+1) cable of a knot, K, are isomorphic to those of K. This result allows us to obtain information about the behavior of the Ozsvath-Szabo concordance invariant under cabling, which has geometric consequences for the cabling operation. Applications considered include quasipositivity in the braid group, the knot theory of complex curves, smooth concordance, and lens space (or, more generally, L-space) surgeries.
Zemke New Heegaard Floer slice genus
- A Juhászandi