arXiv research

The study examines obstructions to links being shake slice.

The study classifies $χ-$slice pretzel links and Seifert fiber spaces.

New invariant measures doubly slice links, disproving previous bounds.

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 paper shows some Montesinos links can't be doubly sliced strongly.

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.

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.

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-ma…

Defines concordance for spatial graphs and proves sliceness equivalence.

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…

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 $g_4(\ell_i)=i$.

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…

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 …

The paper defines a new invariant for links and uses it to show non-sliceness.

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 gener…

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 do…

Alternative proof for links not smoothly slice in $S^2 imes S^2$.

The paper shows links with 2 components are not smoothly slice in a specific 4-manifold.

New Milnor's invariant condition for topologically slice links.

The paper generalizes the $T$-genus to characterize slice knots and slice genus.

New findings show infinitely many knots cannot be smoothly round handle slices.

New exotic 4D spaces found using knot slicing techniques.

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 c…

Let p and q be distinct integers greater than one. We show that the 2-component pretzel link P(p,q,-p,-q) is not slice, even though it has a ribbon mutant, by using 3-fold branched covers and an obstruction based on Donaldson's diagonalization theorem. As a consequence, we prove the slice-ribbon conjecture for 4-strand…

The paper offers new methods to determine if certain 3D links can be formed by intersecting spheres in 4D space.

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 gen…

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 …

Study framed links in simply connected 4-manifolds, linking to exotic phenomena.

New bounds on HOMFLY polynomial for homogeneous links.

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 show that if K is any knot whose Ozsvath-Szabo concordance invariant tau(K) is positive, the all-positive Whitehead double of any iterated Bing double of K is topologically but not smoothly slice. We also show that the all-positive Whitehead double of any iterated Bing double of the Hopf link (e.g., the all-positive…

Determines surgeries on chain links bounding rational homology balls using lattice-theoretic methods.

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.…

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.

Study complex slices on real algebraic varieties and their properties.

Ribbonness proven for slice knots, solving an old question.

Links can be transformed into many others using a specific operation.

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, t…

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 t…

Invariants from surface Khovanov-Jacobsson classes help detect knots and slices.

Bing doubling is an operation which produces a 2-component boundary link B(K) from a knot K. If K is slice, then B(K) is easily seen to be boundary slice. In this paper, we investigate whether the converse holds. Our main result is that if B(K) is boundary slice, then K is algebraically slice. We also show that the Ras…

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 …

We investigate the properties of knots in S^3 which bound Klein bottles, such that a pushoff of the knot has zero linking number with the knot, i.e. has zero framing. This is motivated by the many results in the literature regarding slice knots of genus one, for example, the existence of homologically essential zero se…

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 towe…

It is known that the linking form on the 2-cover of slice knots has a metabolizer. We show that several weaker conditions, or some other conditions related to sliceness, do not imply the existence of a metabolizer. We then show how the Rudolph-Bennequin inequality can be used indirectly to prove that some knots are not…

New invariants refine link homology, showing large genus differences.

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: …

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.