arXiv research

Study on knot concordance and homology cobordism using Heegaard Floer homology.

Khovanov homology shows (4,5) torus knot is a summand in its concordance class.

Study shows infinite-rank summand in homology concordance group of knots.

New subgroup found in knot homology concordance group.

Study ribbon homology concordances using link Floer homology.

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

New homomorphism from Khovanov homology for knot concordance.

New concordance invariant from spectral sequence on Khovanov homology.

New example shows figure eight knot not smoothly concordant but homology cobordant.

New homomorphism from Khovanov homology gives slice genus bounds.

Study of equivariant ribbon concordance using Khovanov homology.

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

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

Let $\widehat{\mathcal{C}}_{\mathbb{Z}}$ denote the group of knots in homology spheres that bound homology balls, modulo smooth concordance in homology cobordisms. Answering a question of Matsumoto, the second author previously showed that the natural map from the smooth knot concordance group $\mathcal{C}$ to $\wideha…

Defines knot concordance invariant using instanton homology and Donaldson invariants.

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…

In 2016 Levine showed that there exists a knot in a homology 3-sphere which is not smoothly concordant to any knot in the 3-sphere where one allows concordances in any smooth homology cobordism. Whether the same is true if one allows topological concordances is not known. One might hope that such an example might be de…

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

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 fr…

The following is a long-standing open question: "If the zero-framed surgeries on two knots in the 3-sphere are integral homology cobordant, are the knots themselves concordant?" We show that an obvious rational version of this question has a negative answer. Namely, we give examples of knots whose zero-framed surgeries…

New knot invariants from instanton homology.

New invariants from framed instanton homology for knot concordance.

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 app…

A formula for bordered Floer homology of concordances and satellites

We prove that the map on knot Floer homology induced by a strongly homotopy-ribbon concordance is injective.

New invariants help study satellite knots and their concordance.

Study of Khovanov homology invariants from $U(1) imes U(1)$-equivariant algebra.

The study limits the number of ribbon concordant fibered knots.

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 su…

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 discuss a concordance invariant constructed from Heegaard Floer homology "correction terms" and +/- 1 surgeries on knots in the three-sphere.

Study knot Floer homology to create concordance invariants and slice genus bounds.

Grid homology confirms the Upsilon invariant in knot theory.

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

Study invariants of null-homologous knots in thickened surfaces.

Any knot in $S^3$ may be reduced to a slice knot by crossing changes. Indeed, this slice knot can be taken to be the unknot. In this paper we study the question of when the same holds for knots in homology spheres. We show that a knot in a homology sphere is nullhomotopic in a smooth homology ball if and only if that k…

The paper describes and analyzes a knot concordance invariant ε using grid homology.

The Conway knot can't be smoothly tied to any other knot an infinite number of times.

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 fo…

By studying the Heegaard Floer homology of the preimage of a knot K in S^3 inside its double branched cover, we develop simple obstructions to K having finite order in the classical smooth concordance group. As an application, we prove that all 2-bridge knots of crossing number at most 12 for which the smooth concordan…

Combining known spectral sequences with a new spectral sequence relating reduced and unreduced sl(N)-homology yields a relationship between the Homflypt-homology of a knot and its sl(N)-concordance invariants. As an application, some of the sl(N)-concordance invariants are shown to be linearly independent.

A spectral sequence is established, from Bar-Natan's variant of Khovanov homology to a deformation of instanton homology for knots and links. This spectral sequence arises as a specialization of a spectral sequence from a characteristic-2 version of $F_5$ homology, in Khovanov'sclassification. Concordance invariants of…

Knots generating infinite subgroup bound rational homology balls.

The study examines obstructions to links being shake slice.

This paper shows that only finitely many knots can be ribbon concordant to any given knot.

In this note, we collect various properties of Seifert homology spheres from the viewpoint of Dehn surgery along a Seifert fiber. We expect that many of these are known to various experts, but include them in one place which we hope to be useful in the study of concordance and homology cobordism.

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…

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…