arXiv research

The theory of signature invariants of links in rational homology spheres is applied to covering links of homology boundary links. From patterns and Seifert matrices of homology boundary links, an explicit formula is derived to compute signature invariants of their covering links. Using the formula, we produce fused bou…

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

New invariants derived from link homology for 4-manifolds.

The paper evaluates homology for links in a solid torus with special boundary conditions.

The abstract conjectures a link between knot homologies and quiver partition functions.

Khovanov homology extended to 3-manifolds, linking tangles.

The paper introduces a new linking form for 3-manifolds in $S^4$.

A celebrated theorem of Kirby identifies the set of closed oriented connected 3-manifolds with the set of framed links in $S^3$ modulo two moves. We give a similar description for the set of knots (and more generally, boundary links) in homology 3-spheres. As an application, we define a noncommutative version of the Al…

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

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…

Geometric trick simplifies link homotopy and concordance.

We give a combinatorial description of the Legendrian contact homology algebra associated to a Legendrian link in $S^1\times S^2$ or any connected sum $\#^k(S^1\times S^2)$, viewed as the contact boundary of the Weinstein manifold obtained by attaching 1-handles to the 4-ball. In view of the surgery formula for symplec…

We determine the algebraic structure underlying the geometric complex associated to a link in Bar-Natan's geometric formalism of Khovanov's link homology theory (n=2). We find an isomorphism of complexes which reduces the complex to one in a simpler category. This reduction enables us to specify exactly the amount of i…

Extends Rasmussen's invariant to new surfaces in four-manifolds.

We introduce augmented biracks and define a (co)homology theory associated to augmented biracks. The new homology theory extends the previously studied Yang-Baxter homology with a combinatorial formulation for the boundary map and specializes to $N$-reduced rack homology when the birack is a rack. We introduce augmente…

In this paper, we focus on L-spaces for which the boundary maps of the Heegaard Floer chain complexes vanish. We collect such manifolds systematically by using the smoothing order on links.

The paper develops a bordered $\mathit{HF}^-$ theory using link surgery formula.

New modules derived from Khovanov homology for links.

The study explores Legendrian invariants and half Giroux torsion in contact structures.

Develops persistent Khovanov homology for tangles.

Generalizes Seifert algorithm to integral homology spheres.

Extends Khovanov homology to surfaces with singularities.

We give an alternative presentation of Khovanov homology of links with strict functoriality result over integers. The construction uses an oriented $sl(2)$ state model allowing a natural definition of the boundary operator as twisted action of morphisms belonging to a TQFT for trivalent graphs and surfaces.

In these notes, I will sketch a new approach to Khovanov homology of knots and links based on counting the solutions of certain elliptic partial differential equations in four and five dimensions. The equations are formulated on four and five-dimensional manifolds with boundary, with a rather subtle boundary condition …

Third paper in series defines monopole Floer homology and gluing theorem for 3-manifolds.

For a 3-manifold with torus boundary admitting an appropriate involution, we show that Khovanov homology provides obstructions to certain exceptional Dehn fillings. For example, given a strongly invertible knot in S^3, we give obstructions to lens space surgeries, as well as obstructions to surgeries with finite fundam…

Protocorks link exotic 4-manifolds, with applications to Floer homology.

The paper uses Floer homology to study twist coefficients and their behavior after capping off.

This paper improves a result on homology concordance in contractible manifolds and two bridge links.

We compute the Heegaard Floer homology of any rational homology 3-sphere with an open book decomposition of the form (T,φ), where T is a genus one surface with one boundary component. In addition, we compute the Heegaard Floer homology of any T^2-bundle over S^1 with first Betti number equal to one, and we compare our …

We focus on L-spaces for which the boundary maps of the Heegaard Floer chain complexes vanish. In previous paper \cite{Usui}, we collect such manifolds systematically by using the smoothing order on links. In this paper, we classify such L-spaces under appropreate constraint.

Develops Floer cohomology for 4-manifolds with involutions and links.

We compute the Khovanov lasagna module of S²×S², confirming a conjecture.

We show that a null-homologous transverse knot K in the complement of an overtwisted disk in a contact 3-manifold is the boundary of a Legendrian ribbon if and only if it possesses a Seifert surface S such that the self-linking number of K with respect to S satisfies $\sel(K,S)=-χ(S)$. In particular, every null-homolog…

Sharp inequalities link manifold's systole to curvature of boundary.

The paper introduces new inequalities for knots in 4D cobordisms.

Study uses twisted Alexander polynomials to link fibered classes in 3-manifolds.

Detects knots in thickened surfaces using instanton homology.

We use a special kind of 2-dimensional extended Topological Quantum Field Theories (TQFTs), so-called open-closed TQFTs, in order to extend Khovanov homology from links to arbitrary tangles, not necessarily even. For every plane diagram of an oriented tangle, we construct a chain complex whose homology is invariant und…

The paper establishes criteria for symplectic surfaces in 4-manifolds.

In this paper, we characterize non-hyperbolic 3-component links in the 3-sphere whose exteriors contain essential 3-punctured spheres with non-integral boundary slopes. We also show the existence of embeddings of some multibranched surfaces in the 3-sphere which satisfy some homological conditions to be embedded in the…

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…

The paper introduces new invariants for genus one knots and surfaces.

Plateau's problem is to find a surface with minimal area spanning a given boundary. In 1960, Reifenberg and Adams developed a definition for "span" using Čech homology, and variants of this definition have been used ever sense. However, limitations of Čech homology resulted in the lack of a natural definition for a bou…

Matveev introduced Borromean surgery on 3-manifolds, and proved that the equivalence relation on closed, oriented 3-manifolds generated by Borromean surgeries is characterized by the first homology group and the torsion linking pairing. Massuyeau generalized this result to closed, spin 3-manifolds, and the second autho…

New link detection results using knot and link Floer homology.

The paper calculates homology and intersection pairing of branched covers using disoriented homology.

Study links with annuli using sutured Floer homology.