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…
Determine pairs of torus knots with genus one cobordisms, with exceptions.
problem Identify pairs of torus knots with genus one cobordisms.
method Combine obstructions from Heegaard Floer knot complex with explicit constructions.
result Determine pairs of torus knots with genus one cobordisms, with exceptions.
New results on algebraic knots with Brieskorn polynomials.
problem Understanding cobordisms of algebraic knots defined by Brieskorn polynomials.
method Analyzing Fox--Milnor type relations, decomposing algebraic cobordism classes, and studying cyclic suspensions.
result Spherical algebraic knots associated with Brieskorn polynomials have infinite order in the knot cobordism group.
Study shows surprising cobordism distances between certain torus knots.
problem Determining cobordism distances between thin and thick torus knots.
method Analyzes locally flat cobordisms between torus knots with small and large braid indices.
result Surprising fact about torus knots as cross-sections of almost minimal cobordisms.
New example shows figure eight knot not smoothly concordant but homology cobordant.
problem Smooth concordance vs homology cobordism of knots.
method Construction of knots with specific properties.
result Figure eight knot not smoothly concordant but homology cobordant.
We introduce a relation of cobordism for knots in thickened surfaces and study cobordism invariants of such knots.
Study on knot concordance and homology cobordism using Heegaard Floer homology.
problem Knot concordance and homology cobordism.
method Heegaard Floer homology.
result Recent results in knot concordance and homology cobordism.
New cobordisms found between certain quasipositive knots.
problem Existence of complex cobordisms between quasipositive knots.
method Construction of specific knots and cobordisms of specified genus.
result No complex cobordism exists between certain quasipositive knots.
New cobordism maps for instanton knot homology help compute surgeries on knots.
problem Computing surgeries on knots using instanton knot homology.
method Construct cobordism maps for instanton knot homology.
result Compute the framed instanton Floer homology of certain knots.
We investigate cobordisms of free knots. Free knots and links are also called homotopy classes of Gauss words and phrases. We define a new strong invariant of free knots which allows to detect free knots not cobordant to the trivial one.
Defines a measure of knot concordance using cobordism distance.
problem Measuring how close knots are to being linearly dependent.
method Cobordism distance on cyclic subgroups of knot concordance group.
result Projective space of knot concordance group with integer-valued metric.
New obstructions found for Lagrangian cobordisms in knot Floer homology.
problem Existence of Lagrangian cobordisms in knot Floer homology.
method Functorial behavior of Legendrian invariants with respect to decomposable Lagrangian cobordisms.
result New computable obstructions to Lagrangian cobordisms.
Classifies knots that bound equivariant surfaces with free symmetries.
problem Classifying knots that bound equivariant surfaces with free symmetries.
method Homology cobordism classification of lens spaces using d-invariants.
result Numerical condition determining free periods for torus knots.
In [D.A. Fedoseev, V.O. Manturov, A sliceness criterion for odd free knots,arXiv:1707.04923], the authors proved a sliceness criterion for odd free knots: free knots with odd chords. In the present paper we give a similar criterion for stably odd free knots. Some additional results on knot sliceness and cobordism are g…
Researchers develop methods to construct Lagrangian cobordisms between Legendrian knots.
problem Understanding the relationship between Legendrian knots through Lagrangian cobordisms.
method Combinatorial and geometric methods, including Heegaard Floer Homology and contact surgery.
result Construction of nondecomposable Lagrangian cobordisms between Legendrian knots.
In this paper we study the cobordism of algebraic knots associated with weighted homogeneous polynomials, and in particular Brieskorn polynomials. Under some assumptions we prove that the associated algebraic knots are cobordant if and only if the Brieskorn polynomials have the same exponents.
Squeezed knots are slices of minimal cobordisms; obstructions come from quantum knot invariants.
problem Characterizing and obstructing squeezed knots.
method Analysis of cobordisms, quantum knot invariants, and stable cohomology operations.
result Effective obstructions to squeezedness come from quantum knot invariants, notably Rasmussen invariant refinements.
This paper explores how many positive integer surgeries on a knot produce a manifold rational homology cobordant to a lens space.
problem How many positive integer surgeries on a knot produce a manifold rational homology cobordant to a lens space?
method Uses Greene and McCoy's changemaker lattices from Heegaard Floer d-invariants and Aceto-Celoria-Park's rational cobordisms and integral homology.
result For a given knot, there are at most two positive integer surgeries that produce a manifold rational homology cobordant to a lens space.
Study shows torsion order bounds band-unlinking number for knot cobordisms.
problem Understanding bounds on knot cobordisms using torsion orders.
method Established an inequality involving local maxima, genus, and torsion orders of Khovanov homology.
result Torsion order gives a lower bound for the band-unlinking number.
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 C2 with the unit 4-ball from which a 4-ball of smaller radius is…
Python tool calculates cobordism maps in Khovanov homology.
problem Computing cobordism maps on Khovanov homology.
method Developed a Python module to calculate these maps.
result Computed cobordism maps for all incompressible Seifert surfaces of prime knots up to 10 crossings.
Study Lagrangian zigzag cobordisms for Legendrian knots, comparing to smooth concordance.
problem Understanding Legendrian knots through Lagrangian cobordisms.
method Defined an equivalence relation on Legendrian knots using interpolating zigzag Lagrangian cobordisms, studied metric monoid, and compared to other concordance types.
result Proved structural results on torsion and satellite operators in the Lagrangian zigzag concordance classes.
The study connects knot Floer homology and Lagrangian cobordisms.
problem Obstructing decomposable Lagrangian cobordisms in Weinstein cobordisms.
method Using knot Floer homology and transverse invariants, the study constructs maps between knot Floer homology groups.
result The invariant obstructs decomposable Lagrangian cobordisms of arbitrary genus.
We explain the notion of a grope cobordism between two knots in a 3-manifold. Each grope cobordism has a type that can be described by a rooted unitrivalent tree. By filtering these trees in different ways, we show how the Goussarov-Habiro approach to finite type invariants of knots is closely related to our notion of …
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…
Diagrammatic method calculates knot invariant from tangle decompositions.
problem Computing Rasmussen's invariant for knots.
method Diagrammatic approach using tangles and cobordisms.
result Computed s-invariants of all 3-strand pretzel knots. Defines a new Upsilon torsion function for knot Floer homology.
problem Obtaining constraints on knot cobordisms.
method Defines a one-parameter family of Heegaard Floer torsion invariants.
result Provides new obstructions related to the Gordian distance between knots.
Lower bounds for a knot invariant are derived using computations and cobordism inequality.
problem Calculating the concordance invariant s# for knots. method Computation for torus knots, cobordism inequality of s#, and arguments for slice-torus invariants. result Lower bounds for s# are derived for knots. Study symplectic cobordisms to empty sets, called hats, in 3-sphere and blow-ups.
problem Constraints on fillings of 3-sphere double covers.
method Relative symplectic cobordisms, focusing on 3-sphere and blow-ups of CP^2.
result Constraints on algebraic topology of fillings.
In the symplectization of standard contact 3-space, R×R3, it is known that an orientable Lagrangian cobordism between a Legendrian knot and itself, also known as an orientable Lagrangian endocobordism for the Legendrian knot, must have genus 0. We show that any Legendrian knot has a non-or…
Study on the cobordism distance between knots and their reverses.
problem Understanding the relationship between a knot and its reverse in terms of cobordism distance.
method Examined the cobordism distance d(K,K^r) for various knots and their properties.
result For knots with g_4(K) = g_3(K), d(K,K^r) is strictly less than twice g_4(K).
We consider the question of when a rational homology 3-sphere is rational homology cobordant to a connected sum of lens spaces. We prove that every rational homology cobordism class in the subgroup generated by lens spaces is represented by a unique connected sum of lens spaces whose first homology embeds in any other …
Ribbon cobordism forms a partial order in 3-manifolds.
problem Understanding partial orders in 3-manifolds.
method Utilizing recent methods from Ian Agol's work on knot concordance.
result Ribbon rational homology cobordism forms a partial order.
The study counts critical points in knot cobordisms using abelian and metacyclic invariants.
problem Counting critical points in knot cobordisms.
method Using homological invariants from cyclic and metacyclic branched covering spaces.
result For each pair of integers g and n, there exists a ribbon knot K with at least n critical points of each index in any genus g cobordism from K to its reverse.
Efficient cobordisms show minimal signature values on certain links.
problem Finding minimal signature values for specific link types.
method Constructing topological cobordisms between torus links and connected sums of trefoil knots.
result The signature invariant σω at ω=ζ6 takes minimal values on torus links. Study 4D homology cobordisms without 3-handles, deriving obstructions.
problem Understanding 4D homology cobordisms without 3-handles.
method Utilizing Thurston geometries, character varieties, and homologies, derive obstructions.
result Generalize knot Floer homology obstructions for ribbon concordances.
Floer homology bounds knot properties like bridge index and ribbon distance.
problem Bounding knot properties using Floer homology.
method Floer homology applied to knot cobordisms.
result Lower bounds on bridge index and ribbon distance.
We investigate the ramifications of the Legendrian satellite construction on the relation of Lagrangian cobordism between Legendrian knots. Under a simple hypothesis, we construct a Lagrangian concordance between two Legendrian satellites by stacking up a sequence of elementary cobordisms. This construction narrows the…
New techniques in Khovanov homology help distinguish exotic surfaces in 4-ball.
problem Distinguishing exotic surfaces in the 4-ball that are not diffeomorphic.
method Developed new techniques for distinguishing cobordism maps on Khovanov homology using knot symmetries and braid factorizations.
result Distinguishes smooth surfaces in the 4-ball that are exotically knotted.
This paper studies cobordism and concordance for virtual knots. We define the affine index polynomial, prove that it is a concordance invariant for knots and links (explaining when it is defined for links), show that it is also invariant under certain forms of labeled cobordism and study a number of examples in relatio…
The paper introduces new inequalities for knots in 4D cobordisms.
problem Understanding constraints on smooth cobordisms between knots.
method Establishes relative adjunction inequalities using Heegaard Floer homology.
result Produces concordance invariants for knots and links.
In this paper we define and investigate Z/2-homology cobordism invariants of Z/2-homology 3-spheres which turn out to be related to classical invariants of knots. As an application we show that many lens spaces have infinite order in the Z/2-homology cobordism group and we prove a lower bound for the slice genus of a k…
Study cobordism distances between 3-braid links and trefoil knots.
problem Understanding the geometric relationship between 3-braid links and trefoil knots.
method Determined cobordism distances between 3-braid links and trefoil knots, and explored limits of Coxeter's braid group result.
result Found cobordism distances between 3-braid links and trefoil knots, up to a constant error.
Ribbon cobordisms form a partial order on 3-manifolds.
problem Understanding the partial order structure of 3-manifolds.
method Building on Agol's proof for knots, we extend the partial order to 3-manifolds.
result Ribbon rational homology cobordism forms a partial order.
New homology cobordism invariants from instanton Floer theory.
problem Understanding the structure of homology cobordism groups.
method Introduced homology cobordism invariants rs(Y) using instanton Floer theory. result Found new homology cobordism group properties and computed specific values.
The paper contains an essentially self-contained treatment of Khovanov homology, Khovanov-Lee homology as well as the Rasmussen invariant for virtual knots and virtual knot cobordisms which directly applies to classical knot and classical knot cobordisms. To do so, we give an alternate formulation for the Manturov defi…
In 1987 S Cappell and J Shaneson constructed an s-cobordism H from the quaternionic 3-manifold Q to itself, and asked whether H or any of its covers are trivial product cobordism? In this paper we study H, and in particular show that its 8-fold cover is the product cobordism from S^3 to itself. We reduce the triviality…
In the present paper, we construct a simple invariant which provides a sliceness obstruction for {\em free knots}. This obstruction provides a new point of view to the problem of studying cobordisms of curves immersed in 2-surfaces, a problem previously studied by Carter, Turaev, Orr, and others. The obstruction to sli…