arXiv research

Study exact Lagrangian cobordisms in cotangent bundles, proving bounds on sheaf interleaving distance and shadow distance.

Study shows surprising cobordism distances between certain torus knots.

Defines a measure of knot concordance using cobordism distance.

Study cobordism distances between 3-braid links and trefoil knots.

This article is about a natural distance function induced by smooth cobordisms between links. We show that the cobordism distance of torus links is determined by the profiles of their signature functions, up to a constant factor.

Study non-orientable link cobordisms using Floer homologies to prove inequalities.

Study on the cobordism distance between knots and their reverses.

We construct cobordisms of small genus between torus knots and use them to determine the cobordism distance between torus knots of small braid index. In fact, the cobordisms we construct arise as the intersection of a smooth algebraic curve in $\mathbb{C}^2$ with the unit 4-ball from which a 4-ball of smaller radius is…

Defines a new Upsilon torsion function for knot Floer homology.

Given a connected cobordism between two knots in the 3-sphere, our main result is an inequality involving torsion orders of the knot Floer homology of the knots, and the number of local maxima and the genus of the cobordism. This has several topological applications: The torsion order gives lower bounds on the bridge i…

Solves asymptotic $A_n$-realization problem for curves.

We give asymptotically sharp upper bounds for the Khovanov width and the dealternation number of positive braid links, in terms of their crossing number. The same braid-theoretic technique, combined with Ozsváth, Stipsicz, and Szabó's Upsilon invariant, allows us to determine the exact cobordism distance between torus …

Signals are geometric submanifolds with specific properties.

The paper calculates a knot invariant for 3-braid knots.

The study extends cobordism theory to complex sections, defining and calculating cobordism groups.

Study cobordisms of nested manifolds and their invariants.

The paper introduces a new invariant for cobordism classes of manifolds and extends cobordism groups.

Defines cobordism maps connecting Khovanov and instanton homologies.

The study explores conditions for $h$-cobordisms between smooth 4-manifolds.

Study shows Khovanov homology's relation to decomposable Lagrangian cobordisms.

The article examines how many stabilizations are needed to transform 5D s-cobordisms into product cobordisms.

Extends cobordism groups of immersions to projections with new results.

We give complete geometric invariants of cobordisms of fold maps with oriented singular set and cobordisms of even codimensional fold maps. These invariants are given in terms of cobordisms of stably framed manifolds and cobordisms of immersions with prescribed normal bundles defined by the author in his earlier works.

New results on algebraic knots with Brieskorn polynomials.

Study $h$-cobordisms of complexity 2 in 5D, finding obstructions and examples.

Study finds conditions for existence of specific pseudo-Riemannian cobordisms.

We study the problem of defining maps on link Floer homology induced by unoriented link cobordisms. We provide a natural notion of link cobordism, disoriented link cobordism, which tracks the motion of index zero and index three critical points. Then we construct a map on unoriented link Floer homology associated to a …

We study necessary and sufficient conditions for the existence of Lorentzian and weak Lorentzian cobordisms between closed smooth manifolds of arbitrary dimension such that the structure group of the frame bundle of the cobordism is $\Spin(1, n)_0$. This extends a result of Gibbons-Hawking on $\Sl(2, \C)$-Lorentzian co…

Using methods inspired from algebraic $K$-theory, we give a new proof of the Genauer fibration sequence, relating the cobordism categories of closed manifolds with cobordism categories of manifolds with boundaries, and of the Bökstedt-Madsen delooping of the cobordism category. Unlike the existing proofs, this approach…

We show that the theory of stable complex $G$-cobordisms, for a torus $G$, is embedded into the theory of stable complex $G$-cobordisms of not necessarily compact manifolds equipped with proper abstract moment maps. Thus the introduction of such non-compact cobordisms in the stable complex $G$-cobordism theory does not…

We study cobordisms and cobordisms rel boundary of PL locally-flat disk knots $D^{n-2}\into D^n$. Cobordisms of disk knots that do not fix the boundary sphere knots are easily classified by the cobordism properties of these boundaries, and any two even-dimensional disk knots with isotopic boundary knots are cobordant r…

Researchers develop methods to construct Lagrangian cobordisms between Legendrian knots.

Periodic Floer homology (PFH) is a Gromov-Floer type invariant for fibered three-manifolds with Hamiltonian structures. The cobordism maps on periodic Floer homology induced by symplectic cobordisms are currently only defined indirectly by using Seiberg-Witten theory. In this paper, we investigate the cobordism maps in…

Paper extends cobordism maps in Khovanov and instanton homologies.

New cobordisms found between certain quasipositive knots.

We introduce a relation of cobordism for knots in thickened surfaces and study cobordism invariants of such knots.

Paper presents new cobordism sequences for singular maps.

The h-cobordism theorem is a noted theorem in differential and PL topology. A generalization of the h-cobordism theorem for possibly non simply connected manifolds is the so called s-cobordism theorem. In this paper, we prove semialgebraic and Nash versions of these theorems. That is, starting with semialgebraic or Nas…

We introduce an equivalence relation, called cobordism, for words and study cobordism invariants of words inspired by methods of low-dimensional topology.

3-manifolds with specific homology are cobordant if homeomorphic.

Foam cobordism groups linked to interval exchange automorphisms.

It has been a central open problem in Heegaard Floer theory whether cobordisms of links induce homomorphisms on the associated link Floer homology groups. We provide an affirmative answer by introducing a natural notion of cobordism between sutured manifolds, and showing that such a cobordism induces a map on sutured F…

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

Study shows mapping class groups differ for $h$-cobordant manifolds.

In this article, we reformulate the cobordism map of embedded contact homology, which is induced by exact symplectic cobordism and defined as direct limit of homomorphisms called filtered ECH cobordism map. The filtered ECH cobordism map is defined by counting embedded holomorphic curves with zero ECH index and we prov…

Paper shows string cobordism at 24 dims can be determined by elliptic genus.

Cobordism of virtual string links on $n$ strands is a combinatorial generalization of link cobordism. There exists a bijection between virtual string links up to cobordisms and elements of the group $\mathbb{Z}^{n(n-1)}$. This paper also shows that virtual string links up to unwelded equivalence are classified by those…

We consider the set of connected surfaces in the 4-ball with boundary a fixed knot in the 3-sphere. We define the stabilization distance between two surfaces as the minimal $g$ such that we can get from one to the other using stabilizations and destabilizations through surfaces of genus at most $g$. Similarly, we consi…