arXiv research

This work shows non-abelian quotient groups in string link concordance.

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 D…

We give infinitely many $2$-component links with unknotted components which are topologically concordant to the Hopf link, but not smoothly concordant to any $2$-component link with trivial Alexander polynomial. Our examples are pairwise non-concordant.

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…

Adding a braid closure to a fibered knot makes a link ribbon concordance minimal.

New obstructions show links with vanishing Milnor invariants may not be concordant to homology boundary links.

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 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, includ…

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 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.

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 def…

New equivalence relation for links using cut-diagrams.

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 ob…

We give new Casson-Gordon style obstructions for a two-component link to be topologically concordant to the Hopf link.

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…

Study ribbon homology concordances using link Floer homology.

The study examines obstructions to links being shake slice.

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…

Defines concordance for spatial graphs and proves sliceness equivalence.

A quandle coloring obstruction prevents a specific link from being ribbon concordant.

We generalize an algorithm of Rudolph to establish that every link is topologically concordant to a strongly quasipositive link.

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, whi…

Link concordance and Whitney towers linked to Milnor invariants.

Study of equivariant ribbon concordance using Khovanov homology.

We give new examples of 2-component links with linking number one and unknotted components that are topologically concordant to the positive Hopf link, but not smoothly so - in fact they are not smoothly concordant to the positive Hopf link with a knot tied in the first component. Such examples were previously construc…

Four-dimensional surgery is used to show that a two component link with Alexander polynomial one is topologically concordant to the Hopf link.

Proves non-solvability of concordance groups using Milnor invariants.

Extended signatures help distinguish non-concordant links.

The slope invariant is shown to be unchanged by concordance of colored links.

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 …

This paper solves the structure of link concordance groups, proving they are infinitely generated.

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 wel…

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 e…

The study establishes a link between fibered links and their concordance invariants.

We determine for which complex numbers on the unit circle the Levine-Tristram signature and the nullity give rise to link concordance invariants.

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 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).

Study shows how Khovanov homology behaves for split links and cobordisms.

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, a…

Prove strong ribbon concordance induces a partial order on links, certify minimality for a handful of knots, and find minimal ribbon minimal knots.

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 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-m…

Study shows knots in homology spheres can be topologically equivalent to those in $S^3$.

Extends knot concordance invariant to balanced spatial graphs using grid homology.

A theory of signatures for odd-dimensional links in rational homology spheres is studied via their generalized Seifert surfaces. The jump functions of signatures are shown invariant under appropriately generalized concordance and a special care is given to accommodate 1-dimensional links with mutual linking. Furthermor…

Defines a Rasmussen invariant for links in RP^3.

We show that a ribbon concordance between two links induces an injective map on Khovanov homology.