More concordance homomorphisms from knot Floer homology
Irving Dai, Jennifer Hom, Matthew Stoffregen, Linh Truong
备注：
50 pages, 6 figures v2: Included a 2 page erratum pointing out an error in a lemma, along with a reference to a revised and correct version of the lemma. The main results of the paper are unaffected
We define an infinite family of linearly independent, integer-valued smooth concordance homomorphisms. Our homomorphisms are explicitly computable and rely on local equivalence classes of knot Floer complexes over the ring $\mathbb{F}[U, V]/(UV=0)$. We compare our invariants to other concordance homomorphisms coming from knot Floer homology, and discuss applications to topologically slice knots, concordance genus, and concordance unknotting number.
We review some recent results in knot concordance and homology cobordism. The proofs rely on various forms of Heegaard Floer homology. We also discuss related open problems.
We define a new smooth concordance homomorphism based on the knot Floer complex and an associated concordance invariant, epsilon. As an application, we show that an infinite family of topologically slice knots are independent in the smooth concordance group.
We prove that the map on knot Floer homology induced by a ribbon concordance is injective. As a consequence, we prove that the Seifert genus is monotonic under ribbon concordance. We also generalize a theorem of Gabai about the super-additivity of the Seifert genus under band connected sum. Our result gives evidence for a conjecture of Gordon that ribbon concordance is a partial order on the set of knots.
In this paper we construct a sequence of integer-valued concordance invariants $ν_n(K)$ that generalize the Ozsváth-Szabó $ν$-invariant and the Hom-Wu $ν^+$-invariant.
The knot Floer complex together with the associated concordance invariant epsilon can be used to define a filtration on the smooth concordance group. We show that the indexing set of this filtration contains the natural numbers cross the integers as an ordered subset.
In this survey article, we discuss several different knot concordance invariants coming from the Heegaard Floer homology package of Ozsvath and Szabo. Along the way, we prove that if two knots are concordant, then their knot Floer complexes satisfy a certain type of stable equivalence.
We modify the construction of knot Floer homology to produce a one-parameter family of homologies for knots in the three-sphere. These invariants can be used to give homomorphisms from the smooth concordance group to the integers, giving bounds on the four-ball genus and the concordance genus of knots. We give some applications of these homomorphisms.
It was recently proved by several authors that ribbon concordances induce injective maps in knot Floer homology, Khovanov homology, and the Heegaard Floer homology of the branched double cover. We give a simple proof of a similar statement in a more general setting, which includes knot Floer homology, Khovanov-Rozansky homologies, and all conic strong Khovanov-Floer theories. This gives a philosophical answer to the question of which aspects of a link TQFT make it injective under ribbon concordances.
We compute the effect of concordance surgery, a generalization of knot surgery defined using a self-concordance of a knot, on the Ozsváth-Szabó 4-manifold invariant. The formula involves the graded Lefschetz number of the concordance map on knot Floer homology. The proof uses the sutured Floer TQFT, and a version of sutured Floer homology perturbed by a 2-form.
We prove that twisted correction terms in Heegaard Floer homology provide lower bounds on the Thurston norm of certain cohomology classes determined by the strong concordance class of a 2-component link $L$ in $S^3$. We then specialise this procedure to knots in $S^2\times S^1$, and obtain a lower bound on their geometric winding number. Furthermore we produce an obstruction for a knot in $S^3$ to have untwisting number 1. We then provide an infinite family of null-homologous knots with increasing geometric winding number, on which the bound is sharp.
The knot Floer complex and the concordance invariant $\varepsilon$ can be used to define a filtration on the smooth concordance group. We exhibit an ordered subset of this filtration that is isomorphic to $\mathbb{N} \times \mathbb{N}$ and consists of topologically slice knots.
By considering a version of Khovanov homology incorporating both the Lee and $E(-1)$ differentials, we construct a $1$-parameter family of concordance homomorphisms similar to the Upsilon invariant from knot Floer homology. This invariant gives lower bounds on the slice genus and can be used to prove that certain infinite families of pretzel knots are linearly independent in the smooth concordance group.
We compute the knot Floer filtration induced by a cable of the meridian of a knot in the manifold obtained by large integer surgery along the knot. We give a formula in terms of the original knot Floer complex of the knot in the three-sphere. As an application, we show that a knot concordance invariant of Hom can equivalently be defined in terms of filtered maps on the Heegaard Floer homology groups induced by the two-handle attachment cobordism of surgery along a knot.
Kawauchi defined a group structure on the set of homology $S^1$$\times$$S^2$'s under an equivalence relation called $\widetilde{H}$-cobordism. This group receives a homomorphism from the knot concordance group, given by the operation of zero-surgery. It is natural to ask whether the zero-surgery homomorphism is injective. We show that this question has a negative answer in the smooth category. Indeed, using knot concordance invariants derived from knot Floer homology we show that the kernel of the zero-surgery homomorphism contains a $\mathbb{Z}^\infty$-subgroup.
We show that a decorated knot concordance $C$ from $K$ to $K'$ induces a homomorphism $F_C$ on knot Floer homology that preserves the Alexander and Maslov gradings. Furthermore, it induces a morphism of the spectral sequences to $\widehat{HF}(S^3) \cong \mathbb{Z}_2$ that agrees with $F_C$ on the $E^1$ page and is the identity on the $E^\infty$ page. It follows that $F_C$ is non-vanishing on $\widehat{HFK}_0(K, τ(K))$. We also obtain an invariant of slice disks in homology 4-balls bounding $S^3$. If $C$ is invertible, then $F_C$ is injective, hence $\dim \widehat{HFK}_j(K,i) \le \dim \widehat{…
Ozsvath-Stipsicz-Szabo recently defined a one-parameter family, upsilon of K at t, of concordance invariants associated to the knot Floer complex. We compare their invariant to the {-1, 0, 1}-valued concordance invariant epsilon, which is also associated to the knot Floer complex. In particular, we give an example of a knot K with upsilon uniformly equal to zero but epsilon non-zero.
Continuing our previous work, we effectively compute connected Heegaard Floer homologies of two families of Brieskorn spheres realized as the boundaries of almost simple linear graphs. Using Floer theoretic invariants of Dai, Hom, Stoffregen, and Truong, we show that these Brieskorn spheres also generate infinite rank summands in the homology cobordism group. Our computations also have applications to the concordance of classical knots and $0$-concordance of $2$-knots.
An immersed concordance between two links is a concordance with possible self-intersections. Given an immersed concordance we construct a smooth four-dimensional cobordism between surgeries on links. By applying $d$-invariant inequalities for this cobordism we obtain inequalities between the $H$-functions of links, which can be extracted from the link Floer homology package. As an application we show a Heegaard Floer theoretical criterion for bounding the splitting number of links. The criterion is especially effective for L-space links, and we present an infinite family of L-space links with …
We make use of link Floer homology to study cobordisms between links embedded in 4-dimensional ribbon homology cobordisms. Combining results of Daemi--Lidman--Vela-Vick--Wong and Zemke, we show that ribbon homology concordances induce split injections on $\mathcal{HFL}^-$. We also make use of the torsion submodule of $\text{HFL}^-$ to give restrictions on the number of critical points in ribbon homology concordances.
To a region $C$ of the plane satisfying a suitable convexity condition we associate a knot concordance invariant $Υ^C$. For appropriate choices of the domain this construction gives back some known knot Floer concordance invariants like Rasmussen's $h_i$ invariants, and the Ozsv\' ath-Stipsicz-Szab\' o upsilon invariant. Furthermore, to three such regions $C$, $C^+$ and $C^- $ we associate invariants $Υ_{C^\pm, C}$ generalising Kim-Livingston secondary invariant. We show how to compute these invariants for some interesting classes of knots (including alternating and torus knots), and we use th…
We define a concordance invariant, epsilon(K), associated to the knot Floer complex of K, and give a formula for the Ozsváth-Szabó 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.
Let $D(K)$ be the positively-clasped untwisted Whitehead double of a knot $K$, and $T_{p,q}$ be the $(p,q)$ torus knot. We show that $D(T_{2,2m+1})$ and $D^2(T_{2,2m+1})$ are linearly independent in the smooth knot concordance group $\mathcal{C}$ for each $m\geq 2$. Further, $D(T_{2,5})$ and $D^2(T_{2,5})$ generate a $\mathbb{Z}\oplus\mathbb{Z}$ summand in the subgroup of $\mathcal{C}$ generated by topologically slice knots. We use the concordance invariant $δ$ of Manolescu and Owens, using Heegaard Floer correction term. Interestingly, these results are not easily shown using other concordanc…
In an earlier work, we introduced a family of t-modified knot Floer homologies, defined by modifying the construction of knot Floer homology HFK-minus. The resulting groups were then used to define concordance homomorphisms indexed by t in [0,2]. In the present work we elaborate on the special case t=1, and call the corresponding modified knot Floer homology the unoriented knot Floer homology. Using elementary methods (based on grid diagrams and normal forms for surface cobordisms), we show that the resulting concordance homomorphism gives a lower bound for the smooth 4-dimensional crosscap nu…
In this paper we investigate the 0-concordance classes of 2-knots in $S^4$, an equivalence relation that is related to understanding smooth structures on 4-manifolds. Using Rochlin's invariant, and invariants arising from Heegaard-Floer homology, we will prove that there are infinitely many 0-concordance classes of 2-knots.
We show that a large class of satellite operators are rank-expanding; that is, they map some rank-one subgroup of the concordance group onto an infinite linearly independent set. Our work constitutes the first systematic study of this property in the literature and partially affirms a conjecture of the second author and Pinzón-Caicedo. More generally, we establish a Floer-theoretic condition for a family of companion knots to have infinite-rank image under satellites from this class. The methods we use are amenable to patterns which act trivially in topological concordance and are capable of h…
We define a filtration of the smooth concordance group based on the genus of representative knots. We use the Heegaard Floer epsilon and Upsilon invariants to prove the quotient groups with respect to this filtration are infinitely generated. Results are applied to three infinite families of topologically slice knots.
Using the covering involution on the double branched cover of the three-sphere branched along a knot, and adapting ideas of Hendricks-Manolescu and Hendricks-Hom-Lidman, we define new knot invariants and apply them to deduce novel linear independence results in the smooth concordance group of knots.
For all n > 0 there is a homomorphism from the smooth concordance group of knots in dimension 2n + 1 to an algebraically defined group called the rational algebraic concordance group. This algebraic concordance group splits as a direct sum of groups indexed by polynomials. For n > 1 the homomorphism is injective. This leads to what is called a primary decomposition theorem for knot concordance. In the classical dimension, the kernel of this homomorphism includes the smooth concordance group of topologically slice knots, and Jae Choon Cha has begun studying possible primary decompositions of th…
We define a nontrivial mod 2 valued additive concordance invariant defined on the torsion subgroup of the knot concordance group using involutive knot Floer package. For knots not contained in its kernel, we prove that their iterated $(\text{odd},1)$-cables have infinite order in the concordance group and, among them, infinitely many are linearly independent. Furthermore, by taking $(2,1)$-cables of the aforementioned knots, we present an infinite family of knots which are strongly rationally slice but not slice.
We define several concordance invariants using knot Floer homology which give improvements over known slice genus and clasp number bounds from Heegaard Floer homology. We also prove that the involutive correction terms of Hendricks and Manolescu give both a slice genus and a clasp number bound.
We prove a formula for the conjugation action on the knot Floer complex of the connected sum of two knots. Using the formula we construct a homomorphism from the smooth concordance group to an abelian group consisting of chain complexes with homotopy automorphisms, modulo an equivalence relation. Using our connected sum formula, we perform some example computations of Hendricks and Manolescu's involutive invariants on large surgeries of connected sums of knots.
We show that a decorated knot concordance $\mathcal{C}$ from $K_0$ to $K_1$ induces an $\mathbb{F}[U]$-module homomorphism \[G_{\mathcal{C}}: HFK^{-}(-S^3,K_0) \to HFK^{-}(-S^3,K_1)\] which preserves the Alexander and absolute $\mathbb{Z}_2$-Maslov gradings. Our construction generalizes the concordance maps induced on $\widehat{HFK}$ studied by Juhász and Marengon, but uses the description of $HFK^{-}$ as a direct limit of maps between sutured Floer homology groups discovered by Etnyre, Vela-Vick, and Zarev.
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.
The concordance genus of a knot K is the minimum Seifert genus of all knots smoothly concordant to K. Concordance genus is bounded below by the 4-ball genus and above by the Seifert genus. We give a lower bound for the concordance genus of K coming from the knot Floer complex of K. As an application, we prove that there are topologically slice knots with 4-ball genus equal to one and arbitrarily large concordance genus.
A knot in a solid torus defines a map on the set of (smooth or topological) concordance classes of knots in $S^3$. This set admits a group structure, but a conjecture of Hedden suggests that satellite maps never induce interesting homomorphisms: we give new evidence for this conjecture in both categories. First, we use Casson-Gordon signatures to give the first obstruction to a slice pattern inducing a homomorphism on the topological concordance group, constructing examples with every winding number besides $\pm 1$. We then provide subtle examples of satellite maps which map arbitrarily deep i…
In 2003, Ozsváth and Szabó defined the concordance invariant $τ$ for knots in oriented 3-manifolds as part of the Heegaard Floer homology package. In 2011, Sarkar gave a combinatorial definition of $τ$ for knots in $S^3$ and a combinatorial proof that $τ$ gives a lower bound for the slice genus of a knot. Recently, Harvey and O'Donnol defined a relatively bigraded combinatorial Heegaard Floer homology theory for transverse spatial graphs in $S^3$ which extends knot Floer homology. We define a $\mathbb{Z}$-filtered chain complex for balanced spatial graphs whose associated graded chain complex …
We study two homomorphisms to the rational homology sphere group. If $ψ$ denotes the inclusion homomorphism from the integral homology sphere group, then using work of Lisca we show that the image of $ψ$ intersects trivially with the subgroup of the rational homology sphere group generated by lens spaces. As corollaries this gives a new proof that the cokernel of $ψ$ is infinitely generated, and implies that a connected sum $K$ of 2-bridge knots is concordant to a knot with determinant 1 if and only if $K$ is smoothly slice. Furthermore, if $β$ denotes the homomorphism from the knot concordanc…
The existence of topologically slice knots that are of infinite order in the knot concordance group followed from Freedman's work on topological surgery and Donaldson's gauge theoretic approach to 4-manifolds. Here, as an application of Ozsvath and Szabo's Heegaard-Floer theory, we show the existence of an infinite subgroup of the smooth concordance group generated by topologically slice knots of concordance order two. In addition, no nontrivial element in this subgroup can be represented by a knot with Alexander polynomial one.
