Link concordance and generalized doubling operators
Tim Cochran, Shelly Harvey, Constance Leidy
备注：
45 pages. Final version. Changed figures 1.3 and 4.2. Expanded Remark 5.4. Fixed typos and made other minor changes. Some of the results are renumbered. Updates references. Note: All results except Cor. 4.8, Ex. 4.4, Ex. 4.6, Lemmas 6.4, 6.5 appeared previously in 0705.3987 under different title: Knot concordance and Blanchfield duality
We introduce a technique for showing classical knots and links are not slice. As one application we show that the iterated Bing doubles of many algebraically slice knots are not topologically slice. Some of the proofs do not use the existence of the Cheeger-Gromov bound, a deep analytical tool used by Cochran-Teichner. We define generalized doubling operators, of which Bing doubling is an instance, and prove our nontriviality results in this more general context. Our main examples are boundary links that cannot be detected in the algebraic boundary link concordance group.
Various obstructions to knot concordance have been found using Casson-Gordon invariants, higher-order Alexander polynomials, as well as von-Neumann rho-invariants. Examples have been produced using (iterated) doubling operations K=R(c,J), and considering these as parametrized by invariants of the base knot J and doubling operator R. In this paper, we introduce a new mew method to obstruct concordance. We show that infinitely many distinct concordance classes may be constructed by varying the infecting curve c in S^3-R. Distinct concordance classes are found even while fixing the base knot, the…
We use moduli spaces of instantons and Chern-Simons invariants of flat connections to prove that the Whitehead doubles of (2,2^n-1) torus knots are independent in the smooth knot concordance group; that is, they freely generate a subgroup of infinite rank.
We study the eta-invariants of links and show that in many cases they form link concordance invariants, in particular that many eta-invariants vanish for slice links. This result contains and generalizes previous invariants by Smolinsky and Cha--Ko. We give a formula for the eta-invariant for boundary links. In several intersting cases this allows us to show that a given link is not slice. We show that even more eta-invariants have to vanish for boundary slice links. We give an example of a boundary link $L$ that is not boundary slice but where all the known link concordance invariants compute…
We show that the twisted signature invariants of boundary link concordance derived from unitary representations of the free group are actually ordinary link concordance invariants. We also show how the discontinuity locus of this signature function is determined by Seifert matrices of the link.
We prove the analogue of the Concordance Implies Isotopy in Codimension $\ge 3$ Theorem for link maps, together with some other its singular analogues. In the case of spherical link maps, a stronger result was independently obtained by P. Teichner (by different methods).
We define a family of link concordance invariants $\left\{ s_n \right\}_{n=2,3, \cdots}$. These link concordance invariants give lower bounds on the slice genus of a link $L$. We compute the slice genus of positive links. Moreover, these invariants give lower bounds on the link splitting number of a link. Especially, this new lower bound determines the splitting number of positive torus links. This is a generalization of Lobb's knot concordance invariants $\left\{ s_n \right\}$, obtained from Gornik's spectral sequence.
This paper is a generalization of the author's previous work on link homotopy to link concordance. We show that the only real-valued finite type link concordance invariants are the linking numbers of the components.
We define a notion of concordance based on Euler characteristic, and show that it gives rise to a concordance group of links in the three-sphere, which has the concordance group of knots as a direct summand with infinitely generated complement. We consider variants of this using oriented and nonoriented surfaces as well as smooth and locally flat embeddings.
We investigate the concordance properties of `parallel links' P(K), given by the (2,0) cable of a knot K. We focus on the question: if P(K) is concordant to a split link, is K necessarily slice? We show that if P(K) is smoothly concordant to a split link, then many smooth concordance invariants of K must vanish, including the tau and s-invariants, and suitably normalized d-invariants of surgeries on K. We also investigate the (2,2m) cables P_m(K), and find obstructions to smooth concordance to the sum of the (2,2m) torus link and a split link.
Satellite constructions on a knot can be thought of as taking some strands of a knot and then tying in another knot. Using satellite constructions one can construct many distinct isotopy classes of knots. Pushing this further one can construct distinct concordance classes of knots which preserve some algebraic invariants. Infection is a generalization of satellite operations, which has been previously studies. An infection by a string link can be thought of as grabbing a knot at multiple locations and then tying in a link. Cochran, Friedl and Techner showed that any algebraically slice knot is…
J. Davis showed that the topological concordance class of a link in the 3-sphere is uniquely determined by its Alexander polynomial for 2-component links with Alexander polynomial one. A similar result for knots with Alexander polynomial one was shown earlier by M. Freedman. We prove that these two cases are the only exceptional cases, by showing that the link concordance class is not determined by the Alexander invariants in any other case.
Fixing two concordant links in $3$--space, we study the set of all embedded concordances between them, as knotted annuli in $4$--space. When regarded up to surface-concordance or link-homotopy, the set $\mathcal{C}(L)$ of concordances from a link $L$ to itself forms a group. In order to investigate these groups, we define Milnor-type invariants of $\mathcal{C}(L)$, which are integers defined modulo a certain indeterminacy given by Milnor invariants of $L$. We show in particular that, for a slice link $L$, these invariants classify $\mathcal{C}(L)$ up to link-homotopy.
We construct links of arbitrarily many components each component of which is slice and yet are not concordant to any link with even one unknotted component. The only tool we use comes from the Alexander modules.
We introduce a generalization of the Ozsváth-Szabó $τ$-invariant to links by studying a filtered version of link grid homology. We prove that this invariant remains unchanged under strong concordance and we show that it produces a lower bound for the slice genus of a link. We show that this bound is sharp for torus links and we also give an application to Legendrian link invariants in the standard contact 3-sphere.
It was shown by Jim Davis that a 2-component link with Alexander polynomial one is topologically concordant to the Hopf link. In this paper, we show that there is a 2-component link with Alexander polynomial one that has unknotted components and is not smoothly concordant to the Hopf link, answering a question of Jim Davis. We construct infinitely many concordance classes of such links, and show that they have the stronger property of not being smoothly concordant to the Hopf link with knots tied in the components.
Knots and links in 3-manifolds are studied by applying intersection invariants to singular concordances. The resulting link invariants generalize the Arf invariant, the mod 2 Sato-Levine invariants, and Milnor's triple linking numbers. Besides fitting into a general theory of Whitney towers, these invariants provide obstructions to the existence of a singular concordance which can be homotoped to an embedding after stabilization by connected sums with $S^2\times S^2$. Results include classifications of stably slice links in orientable 3-manifolds, stable knot concordance in products of an orie…
We can construct a 4-manifold by attaching 2-handles to a 4-ball with framing r along the components of a link in the boundary of the 4-ball. We define a link as r-shake slice if there exists embedded spheres that represent the generators of the second homology of the 4-manifold. This naturally extends r-shake slice, a generalization of slice that has previously only been studied for knots, to links of more than one component. We also define a relative notion of shake r-concordance for links and versions with stricter conditions on the embedded spheres that we call strongly r-shake slice and s…
It is known that each of the successive quotient groups of the grope and solvable filtrations of the knot concordance group has an infinite rank subgroup. The generating knots of these subgroups are constructed using iterated doubling operators. In this paper, for each of the successive quotients of the filtrations we give a new infinite rank subgroup which trivially intersects the previously known infinite rank subgroups. Instead of iterated doubling operators, the generating knots of these new subgroups are constructed using the notion of algebraic $n$-solutions, which was introduced by Coch…
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 show that there are links whose individual components are concordant to the unknot, but which are not concordant to any link with unknotted components. We give examples in the topological category, and examples in the smooth category which are topologically slice. We also give generalizations regarding components of prescribed Alexander polynomials. The main tools are covering link calculus, algebraic invariants of rational knot concordance theory, and the correction term of Heegaard Floer homology.
A fibered concordance of knots, introduced by Harer, is a concordance between fibered knots that is well-behaved with respect to the fibrations. We consider semi-fibered concordance of two component ordered links $L=J \sqcup K$ with $J$ fibered. These are concordances that restrict to fibered concordances on the first component. Motivated by some examples of Gompf-Scharlemann-Thompson, we further limit our attention to those links $L$ where $K$ is "close to" a fiber of $J$. Such $L$ are studied with virtual covers, where a virtual knot $\upsilon$ is associated to $L$. We show that the concorda…
We study the effect of mutation on link concordance and 3-manifolds. We show that the set of links concordant to sublinks of homology boundary links is not closed under positive mutation. We show that mutation does not preserve homology cobordism classes of 3-manifolds. A significant consequence is that there exist 3-manifolds which have the same quantum SU(2)-invariants but are not homology cobordant. These results are obtained by investigating the effect of mutation on the Milnor invariants, or equivalently the Massey products.
We employ Hirzebruch-type invariants obtained from iterated p-covers to investigate concordance of links and string links. We show that the invariants naturally give various group homomorphisms of the string link concordance group into L-groups over number fields. We also obtain homomorphisms of successive quotients of the Cochran-Orr-Teichner filtration. As an application we show that the kernel of Harvey's $ρ_n$-invariant is large enough to contain a subgroup with infinite rank abelianization, modulo local knots. As another application, we show that recently discovered nontrivial 2-torsion e…
We use the knot homology of Khovanov and Lee to construct link concordance invariants generalizing the Rasmussen $s$-invariant of knots. The relevant invariant for a link is a filtration on a vector space of dimension $2^{|L|}$. The basic properties of the $s$-invariant all extend to the case of links; in particular, any orientable cobordism $Σ$ between links induces a map between their corresponding vector spaces which is filtered of degree $χ(Σ)$. A corollary of this construction is that any component preserving orientable cobordism from a $\Kh$-thin link to a link split into $k$ components …
In the study of homology cobordisms, knot concordance and link concordance, the following technical problem arises frequently: let $π$ be a group and let $M \to N$ be a homomorphism between projective $\Z[π]$-modules such that $\Z_p \otimes_{\Z[π]} M\to \Z_p \otimes_{\Z[π]} N$ is injective; for which other right $\Z[π]$-modules $V$ is the induced map $V \otimes_{\Z[π]} M\to V\otimes_{\Z[π]}N$ also injective? Our main theorem gives a new criterion which combines and generalizes many previous results.
We introduce a new class of links for which we give a lower bound for the slice genus $g_*$, using the generalized Rasmussen invariant. We show that this bound, in some cases, allows one to compute $g_*$ exactly; in particular, we compute $g_*$ for torus links. We also study another link invariant: the strong slice genus $g_*^*$. Studying the behaviour of a specific type of cobordisms in Lee homology, a lower bound for $g_*^*$ is also given.
Link-homotopy and self Delta-equivalence are equivalence relations on links. It was shown by J. Milnor (resp. the last author) that Milnor invariants determine whether or not a link is link-homotopic (resp. self Delta-equivalent) to a trivial link. We study link-homotopy and self Delta-equivalence on a certain component of a link with fixing the rest components, in other words, homotopy and Delta-equivalence of knots in the complement of a certain link. We show that Milnor invariants determine whether a knot in the complement of a trivial link is null-homotopic, and give a sufficient condition…
In the present paper we extend the definition of slice-torus invariant to links. We prove a few properties of the newly-defined slice-torus link invariants: the behaviour under crossing change, a slice genus bound, an obstruction to strong sliceness, and a combinatorial bound. Furthermore, we provide an application to the computation of the splitting number. Finally, we use the slice-torus link invariants, and the Whitehead doubling to define new strong concordance invariants for links, which are proven to be independent from the corresponding slice-torus link invariant.
We establish several new results about both the (n)-solvable filtration, F_n^m, of the set of link concordance classes and the (n)-solvable filtration of the string link concordance group. We first establish a relationship between Milnor's invariants and links, L, with certain restrictions on the 4-manifold bounded by M_L. Using this relationship, we can relate (n)-solvability of a link (or string link) with its Milnor's invariants. Specifically, we show that if a link is (n)-solvable, then its Milnor's invariants vanish for lengths up to 2^{n+2}-1. Previously, there were no known results abou…
The twisted torsion of a 3-manifold is well-known to be zero whenever the corresponding twisted Alexander module is non-torsion. Under mild extra assumptions we introduce a new twisted torsion invariant which is always non-zero. We show how this torsion invariant relates to the twisted intersection form of a bounding 4-manifold, generalizing a theorem of Milnor. Using this result, we give new obstructions to 3-manifolds being homology cobordant and to links being concordant. These obstructions are sufficiently strong to detect that the Bing double of the figure eight knot is not slice.
We obtain new invariants of topological link concordance and homology cobordism of 3-manifolds from Hirzebruch-type intersection form defects of towers of iterated p-covers. Our invariants can extract geometric information from an arbitrary depth of the derived series of the fundamental group, and can detect torsion which is invisible via signature invariants. Applications illustrating these features include the following: (1) There are infinitely many homology equivalent rational 3-spheres which are indistinguishable via multisignatures, eta-invariants, and L2-signatures but have distinct hom…
We show that if a link J in the 3-sphere is homotopy ribbon concordant to a link L then the Alexander polynomial of L divides the Alexander polynomial of J.
We introduce a new technique for showing classical knots and links are not slice. As one application we resolve a long-standing question as to whether certain natural families of knots contain topologically slice knots. We also present a simpler proof of the result of Cochran-Teichner that the successive quotients of the integral terms of the Cochran-Orr-Teichner filtration of the knot concordance group have rank 1. For links we have similar results. We show that the iterated Bing doubles of many algebraically slice knots are not topologically slice. Some of the proofs do not use the existence…
We use the Bar-Natan Zh-correspondence to identify the generalized Alexander polynomial of a virtual knot with the Alexander polynomial of a two component welded link. We show that the Zh-map is functorial under concordance, and also that Satoh's Tube map (from welded links to ribbon knotted tori in $S^4$) is functorial under concordance. In addition, we extend classical results of Chen, Milnor, and Hillman on the lower central series of link groups to links in thickened surfaces. Our main result is that the generalized Alexander polynomial vanishes on any knot in a thickened surface which is …
A cobordism between links in thickened surfaces consists of a surface $ S $ and a $3$-manifold $M $, with $ S $ properly embedded in $ M \times I $. We show that there exist links in thickened surfaces such that if $(S,M) $ is a cobordism between them in which $ S $ is simple, then $ M $ must be complex. That is, there are cases in which low complexity of the surface does not imply low complexity of the $3$-manifold. Specifically, we show that there exist concordant links in thickened surfaces between which a concordance can only be realised by passing through thickenings of higher genus surfa…
This paper describes grope and Whitney tower filtrations on the set of concordance classes of classical links in terms of class and order respectively. Using the tree-valued intersection theory of Whitney towers, the associated graded quotients are shown to be finitely generated abelian groups under a (surprisingly) well-defined connected sum operation. Twisted Whitney towers are also introduced, along with a corresponding quadratic enhancement of the intersection theory for framed Whitney towers that measures Whitney-disk framing obstructions. The obstruction theory in the framed setting is s…
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 …
Using the Gordon-Litherland pairing, one can define invariants (signature, nullity, determinant) for ${\mathbb Z}/2$ null-homologous links in thickened surfaces. In this paper, we study the concordance properties of these invariants. For example, if $K \subset Σ\times I$ is ${\mathbb Z}/2$ null-homologous and slice, we show that its signatures vanish and its determinants are perfect squares. These statements are derived from a cobordism result for closed unoriented surfaces in certain 4-manifolds. The Brown invariants are defined for ${\mathbb Z}/2$ null-homologous links in thickened surfaces.…
We consider the Grope filtration of the classical knot concordance group that was introduced in a paper of Cochran, Orr and Teichner. Our main result is that successive quotients at each stage in this filtration have infinite rank. We also establish the analogous result for the Grope filtration of the concordance group of string links consisting of more than one component.
We establish several results about two short exact sequences involving lower terms of the $n$-solvable filtration, $\{\mathcal{F}^m_n\}$ of the string link concordance group $\mathcal{C}^m$. We utilize the Thom-Pontryagin construction to show that the Sato-Levine invariants $\barμ_{(iijj)}$ must vanish for 0.5-solvable links. Using this result, we show that the short exact sequence $0\rightarrow \mathcal{F}^m_0/\mathcal{F}^m_{0.5} \rightarrow \mathcal{F}^m_{-0.5}/\mathcal{F}^m_{0.5} \rightarrow \mathcal{F}^m_{-0.5}/\mathcal{F}^m_0 \rightarrow 0$ does not split for links of two or more componen…
This paper computes Whitney tower filtrations of classical links. Whitney towers consist of iterated stages of Whitney disks and allow a tree-valued intersection theory, showing that the associated graded quotients of the filtration are finitely generated abelian groups. Twisted Whitney towers are studied and a new quadratic refinement of the intersection theory is introduced, measuring Whitney disk framing obstructions. It is shown that the filtrations are completely classified by Milnor invariants together with new higher-order Sato-Levine and higher-order Arf invariants, which are obstructi…
The set SL(n) of n-string links has a monoid structure, given by the stacking product. When considered up to concordance, SL(n) becomes a group, which is known to be abelian only if n=1. In this paper, we consider two families of equivalence relations which endow SL(n) with a group structure, namely the C_k-equivalence introduced by Habiro in connection with finite type invariants theory, and the C_k-concordance, which is generated by C_k-equivalence and concordance. We investigate under which condition these groups are abelian, and give applications to finite type invariants.
Milnor's $\barμ$-invariants of links in the $3$-sphere $S^3$ vanish on any link concordant to a boundary link. In particular, they are trivial on any knot in $S^3$. Here we consider knots in thickened surfaces $Σ\times [0,1]$, where $Σ$ is closed and oriented. We construct new concordance invariants by adapting the Chen-Milnor theory of links in $S^3$ to an extension of the group of a virtual knot. A key ingredient is the Bar-Natan $\textit{Zh}$ map, which allows for a geometric interpretation of the group extension. The group extension itself was originally defined by Silver-Williams. Our ext…
