The A-B slice problem is a reformulation of the topological 4-dimensional surgery conjecture in terms of decompositions of the 4-ball and link homotopy. We show that link groups, a recently developed invariant of 4-manifolds, provide an obstruction for the class of model decompositions, introduced by M. Freedman and X.-S. Lin. This unifies and extends the previously known partial obstructions in the A-B slice program. As a consequence, link groups satisfy Alexander duality when restricted to the class of model decompositions, but not for general submanifolds of the 4-ball.
Shake slice generalizes the notion of a slice link, naturally extending the notion of shake slice knots to links. There is also a relative version, shake concordance, that generalizes link concordance. We show that if two links are shake concordant, then their zero surgery manifolds are homology cobordant. Then we give several obstructions to a link being shake slice; for instance, the Arf invariants vanish for both the link and each component. Finally we show that a shake slice link bounds disjoint disks in a homology 4-ball and hence each component is algebraically slice.
We construct an infinite family of topologically slice 2--component boundary links $\ell_i$, none of which is smoothly concordant to a split link, such that $g_4(\ell_i)=i$.
A 2-component oriented link in $S^3$ is called weakly doubly slice if it is a cross-section of an unknotted sphere in $S^4$, and strongly doubly slice if it is a cross-section of a 2-component trivial spherical link in $S^4$. We give the first example of 2-component boundary links which are weakly doubly slice but not strongly doubly slice. We also introduce a new invariant $g_{st}$ of homotopically trivial links that measures the failure of a link from being strongly doubly slice and that bounds the doubly slice genus $g_{ds}$ from below. Our examples have arbitrarily large doubly slice genus…
We show that every good boundary link with a pair of derivative links on a Seifert surface satisfying a homotopically trivial plus assumption is freely slice. This subsumes all previously known methods for freely slicing good boundary links with two or more components, and provides new freely slice links.
This paper compares notions of double sliceness for links. The main result is to show that a large family of 2-component Montesinos links are not strongly doubly slice despite being weakly doubly slice and having doubly slice components. Our principal obstruction to strong double slicing comes by considering branched double covers. To this end we prove a result classifying Seifert fibered spaces which admit a smooth embeddings into integer homology $S^1 \times S^3$s by maps inducing surjections on the first homology group. A number of other results and examples pertaining to doubly slice links…
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 define Casson-Gordon sigma-invariants for links and give a lower bound of the slice genus of a link in terms of these invariants. We study as an example a family of two component links of genus h and show that their slice genus is h, whereas the Murasugi-Tristram inequality does not obstruct this link from bounding an annulus in the 4-ball.
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…
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.
The bipolar filtration introduced by T. Cochran, S. Harvey, and P. Horn is a framework for the study of smooth concordance of topologically slice knots and links. It is known that there are topologically slice 1-bipolar knots which are not 2-bipolar. For knots, this is the highest known level at which the filtration does not stabilize. For the case of links with two or more components, we prove that the filtration does not stabilize at any level: for any n, there are topologically slice links which are n-bipolar but not (n+1)-bipolar. In the proof we describe an explicit geometric construction…
We prove that, for any ordinary sense slice 1-link $L$, we can define the Arf invariant and Arf(L)=0. We prove that, for any m-component 1-link L_1, there exists a 3m-component ordinary sense slaice 1-link L_2 of which L_1 is a sublink.
We give an example of a 3-component smoothly slice boundary link, each of whose components has a genus one Seifert surface, such that any metaboliser of the boundary link Seifert form is represented by 3 curves on the Seifert surfaces that form a link with nonvanishing Milnor triple linking number. We also give a generalisation to m-component links and higher Milnor invariants. We prove that our examples are ribbon and that all ribbon links are boundary slice.
We provide obstructions to a link in $S^3$ arising as the cross section of any number of unlinked spheres in $S^4$. Our obstructions arise from the multivariable signature, the Blanchfield form and generalised Seifert matrices. We also obtain obstructions in the case of surfaces of higher genera, leading to a lower bound on the doubly slice genus of links.
A link in the 3-sphere is called (smoothly) slice if its components bound disjoint smoothly embedded disks in the 4-ball. More generally, given a 4-manifold M with a distinguished circle in its boundary, a link in the 3-sphere is called M-slice if its components bound in the 4-ball disjoint embedded copies of M. A 4-manifold M is constructed such that the Borromean rings are not M-slice but the Hopf link is. This contrasts the classical link-slice setting where the Hopf link may be thought of "the most non-slice" link. Further examples and an obstruction for a family of decompositions of the 4…
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…
It is well-known that all 2-knots are slice. Are all 2-links slice? This is an outstanding open question. In this paper we prove the following: For any 2-component 2-link (J,K)in the 4-sphere which bounds the 5-ball B^5, there is an embedded disc 2-disc D^2_J (respectively, D^2_K) in B^5 with the following properties: J (respectively K) bounds D^2_J (respectively, D^2_K). D^2_J and D^2_K intersect transversely. the intersection of D^2_J and D^2_K in D^2_J (respectively, D^2_K) is a trivial 1-knot.
We prove that a Casson tower of height 4 contains a flat embedded disc bounded by the attaching circle, and we prove disc embedding results for height 2 and 3 Casson towers which are embedded into a 4-manifold, with some additional fundamental group assumptions. In the proofs we create a capped grope from a Casson tower and use a refined height raising argument to establish the existence of a symmetric grope which has two layers of caps, data which is sufficient for a topological disc to exist, with the desired boundary. As applications, we present new slice knots and links by giving direct ge…
We use techniques of Freedman and Teichner to prove that, under certain circumstances, the multi-infection of a slice link is again slice (not necessarily smoothly slice). We provide a general context for proving links are slice that includes many of the previously known results.
We give a new geometric obstruction to the iterated Bing double of a knot being a slice link: for n>1 the (n+1)-st iterated Bing double of a knot is rationally slice if and only if the n-th iterated Bing double of the knot is rationally slice. The main technique of the proof is a covering link construction simplifying a given link. We prove certain similar geometric obstructions for n <= 1 as well. Our results are sharp enough to conclude, when combined with algebraic invariants, that if the n-th iterated Bing double of a knot is slice for some n, then the knot is algebraically slice. Also our…
We study the effect of Nielsen moves and their geometric counterparts, handle slides, on good boundary links. A collection of links, universal for 4-dimensional surgery, is shown to admit Seifert surfaces with trivial Lagrangian. They are good boundary links, with Seifert matrices of a more general form than in known constructions of slice links. We show that a certain more restrictive condition on Seifert matrices is sufficient for proving the links are slice. We also give a correction of a Kirby calculus identity in \cite{FK2}, useful for constructing surgery kernels associated to link-slice…
We define the stabilizing number $\operatorname{sn}(K)$ of a knot $K \subset S^3$ as the minimal number $n$ of $S^2 \times S^2$ connected summands required for $K$ to bound a nullhomotopic locally flat disc in $D^4 \# n S^2 \times S^2$. This quantity is defined when the Arf invariant of $K$ is zero. We show that $\operatorname{sn}(K)$ is bounded below by signatures and Casson-Gordon invariants and bounded above by the topological $4$-genus $g_4^{\operatorname{top}}(K)$. We provide an infinite family of examples with $\operatorname{sn}(K)<g_4^{\operatorname{top}}(K)$.
If there are any 2-component counterexamples to the Generalized Property R Conjecture, a least genus component of all such counterexamples cannot be a fibered knot. Furthermore, the monodromy of a fibered component of any such counterexample has unexpected restrictions. The simplest plausible counterexample to the Generalized Property R Conjecture could be a 2-component link containing the square knot. We characterize all two-component links that contain the square knot and which surger to (S^1 x S^2) # (S^1 x S^2). We exhibit a family of such links that are probably counterexamples to General…
Let C_T be the subgroup of the smooth knot concordance group generated by topologically slice knots and let C_D be the subgroup generated by knots with trivial Alexander polynomial. We prove the quotient C_T/C_D is infinitely generated, and uncover similar structure in the 3-dimensional rational spin bordism group. Our methods also lead to the construction of links that are topologically, but not smoothly, concordant to boundary links.
We show that the subgroup of the knot concordance group generated by links of isolated complex singularities intersects the subgroup of algebraically slice knots in an infinite rank subgroup.
We show that perturbing the definition of sl(n) Khovanov-Rozansky link homology gives a lower bound on the slice genus of a knot. As a corollary this yields another proof of Milnor's conjecture on the slice genus of torus knots.
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 Blanchfield form to obtain a lower bound on the equivariant slice genus of a strongly invertible knot. For our main application, let $K$ be a genus one strongly invertible slice knot with nontrivial Alexander polynomial. We show that the equivariant slice genus of an equivariant connected sum $\#^n K$ is at least $n/4$. We also formulate an equivariant algebraic concordance group, and show that the kernel of the forgetful map to the classical algebraic concordance group is infinite rank.
As a corollary of work of Ozsvath and Szabo [math.GT/0301149], it is shown that the classical concordance group of algebraically slice knots has an infinite cyclic summand and in particular is not a divisible group.
Let $Γ$ be a 3-dimensional Kleinian punctured torus group with ccidental parabolic transformations. The deformation space of $Γ$ in the group of Möbius transformations on the 2-sphere is well-known as the Maskit slice of punctured torus groups. In this paper, we study deformations $Γ'$ of $Γ$ in the group of Möbius transformations on the 3-sphere such that $Γ'$ does not contain screw parabolic transformations. We will show that the space of the deformations is realized as a domain of 3-space $\mathbb{R}^3$, which contains the Maskit slice of punctured torus groups as a slice through a plane. F…
The A-B slice problem, a reformulation of the 4-dimensional topological surgery conjecture for free groups, is shown to admit a link-homotopy+ solution. The proof relies on geometric applications of the group-theoretic 2-Engel relation. Implications for the surgery conjecture are discussed.
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.
We define an obstruction for a knot to be Z[Z]-homology ribbon, and use this to provide restrictions on the integers that can occur as the triple linking numbers of derivative links of knots that are either homotopy ribbon or doubly slice. Our main application finds new non-doubly slice knots. In particular this gives new information on the doubly solvable filtration of Taehee Kim: doubly algebraically slice ribbon knots need not be doubly (1)-solvable, and doubly algebraically slice knots need not be (0.5,1)-solvable. We also discuss potential connections to unsolved conjectures in knot conco…
We develop a theory of chain complex double-cobordism for chain complexes equipped with Poincaré duality. The resulting double-cobordism groups are a refinement of Ranicki's torsion algebraic $L$-groups for localisations of a commutative ring with involution. The refinement is analogous to the difference between metabolic and hyperbolic linking forms. We apply the double $L$-groups in high-dimensional knot theory to define an invariant for doubly-slice $n$-knots. We prove that the "stably doubly-slice implies doubly-slice" property holds (algebraically) for Blanchfield forms, Seifert forms and…
The concordance group of algebraically slice knots is the subgroup of the classical knot concordance group formed by algebraically slice knots. Results of Casson and Gordon and of Jiang showed that this group contains in infinitely generated free (abelian) subgroup. Here it is shown that the concordance group of algebraically slice knots also contain elements of finite order; in fact it contains an infinite subgroup generated by elements of order 2.
We introduce a new link invariant called the algebraic genus, which gives an upper bound for the topological slice genus of links. In fact, the algebraic genus is an upper bound for another version of the slice genus proposed here: the minimal genus of a surface in the four-ball whose complement has infinite cyclic fundamental group. We characterize the algebraic genus in terms of cobordisms in three-space, and explore the connections to other knot invariants related to the Seifert form, the Blanchfield form, knot genera and unknotting. Employing Casson-Gordon invariants, we discuss the algebr…
We study the set $\widehat{\mathcal S}_M$ of framed smoothly slice links which lie on the boundary of the complement of a 1-handlebody in a closed, simply connected, smooth 4-manifold $M$. We show that $\widehat{\mathcal S}_M$ is well-defined and describe how it relates to exotic phenomena in dimension four. In particular, in the case when $X$ is smooth, with a handle decompositions with no 1-handles and homeomorphic to but not smoothly embeddable in $D^4$, our results tell us that $X$ is exotic if and only if there is a link $L\hookrightarrow S^3$ which is smoothly slice in $X$, but not in $D…
We address primary decomposition conjectures for knot concordance groups, which predict direct sum decompositions into primary parts. We show that the smooth concordance group of topologically slice knots has a large subgroup for which the conjectures are true and there are infinitely many primary parts each of which has infinite rank. This supports the conjectures for topologically slice knots. We also prove analogues for the associated graded groups of the bipolar filtration of topologically slice knots. Among ingredients of the proof, we use amenable $L^2$-signatures, Ozsváth-Szabó $d$-inva…
A long-standing conjecture due to Michael Freedman asserts that the 4-dimensional topological surgery conjecture fails for non-abelian free groups, or equivalently that a family of canonical examples of links (the generalized Borromean rings) are not A-B slice. A stronger version of the conjecture, that the Borromean rings are not even weakly A-B slice, where one drops the equivariant aspect of the problem, has been the main focus in search for an obstruction to surgery. We show that the Borromean rings, and more generally all links with trivial linking numbers, are in fact weakly A-B slice. T…
We construct an infinite family of topologically slice knots that are not smoothly concordant to their reverses. More precisely, if T denotes the concordance group of topologically slice knots and R is the involution of T induced by string reversal, then T/Fix(R) contains an infinitely generated free subgroup. The result remains true modulo the subgroup of T generated by knots with trivial Alexander polynomial.
The linear slice of quasi-Fuchsian once-punctured torus groups is defined by fixing the complex length of some simple closed curve to be a fixed positive real number. It is known that the linear slice is a union of disks, and it always has one standard component containing Fuchsian groups. Komori and Yamashita proved that there exist non-standard components if the length is sufficiently large. We give two other proofs of their theorem, one is based on some properties of length functions, and the other is based on the theory of complex projective structures and complex earthquakes. From the lat…
The difference between slice and doubly-slice knots is reflected in algebra by the difference between metabolic and hyperbolic Blanchfield linking forms. We exploit this algebraic distinction to refine the classical Witt group of linking forms by defining a `double Witt group' of linking forms. We calculate the double Witt group for Dedekind domains and precisely determine its relationship to the classical Witt group. Finally, we prove that the double Witt group of Seifert forms is isomorphic to the double Witt group of Blanchfield forms.
In the early 1980's Mike Freedman showed that all knots with trivial Alexander polynomial are topologically slice (with fundamental group Z). This paper contains the first new examples of topologically slice knots. In fact, we give a sufficient homological condition under which a knot is slice with fundamental group Z semi-direct product Z[1/2]. These two fundamental groups are known to be the only solvable ribbon groups. Our homological condition implies that the Alexander polynomial equals (t-2)(t^{-1}-2) but also contains information about the metabelian cover of the knot complement (since …
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.
Using the symplectic geometry of certain manifolds which appear naturally in Lie theory, we define an invariant which assigns a graded abelian group to an oriented link. The relevant manifolds are transverse slices to certain nilpotent orbits inside sl_{2m}, and intersections of those with regular semisimple orbits. The invariant is conjectured to be equal to Khovanov's combinatorially defined homology theory (with the bigrading of that theory collapsed in a certain way).
Let S(D) be the surface produced by applying Seifert's algorithm to the oriented link diagram D. I prove that if D has no negative crossings then S(D) is a quasipositive Seifert surface, that is, S(D) embeds incompressibly on a fiber surface plumbed from positive Hopf annuli. This result, combined with the truth of the `local Thom Conjecture', has various interesting consequences; for instance, it yields an easily-computed estimate for the slice euler characteristic of the link L(D) (where D is arbitrary) that extends and often improves the `slice-Bennequin inequality' for closed-braid diagram…
