Study semi-fibered concordance of links using virtual covers.
problem Understanding semi-fibered concordance of links and its applications.
method Use virtual covers and semi-fibered concordance to study links.
result Concordance class of virtual knots is a semi-fibered concordance invariant.
The study limits the number of ribbon concordant fibered knots.
problem Understanding the relationship between ribbon concordance and fibered knots.
method Combining Floer homology results with fixed points of monodromy and volume inequalities.
result There are only finitely many hyperbolic fibered knots ribbon concordant to any given knot.
Study ribbon concordance and minimal compressions, proving new results about fibered knots.
problem Understanding ribbon concordance and minimal compressions of surface homeomorphisms.
method Proving monotonicity of simplicial volume and dilatation under ribbon concordance, algorithmic enumeration of minimal compressions.
result Every fibered knot has only finitely many predecessors in the ribbon-concordance partial order.
Adding a braid closure to a fibered knot makes a link ribbon concordance minimal.
problem Finding the minimal ribbon concordance for links in S3. method Link Floer homology combined with classical techniques.
result Any link L in S3 can be made ribbon concordance minimal by adding a single unknot. Either fibered knots supporting the tight contact structure are unique in their smooth concordance class or there exists a fibered counterexample to the Slice-Ribbon Conjecture.
The Upsilon invariant helps classify fibered knots and their open book decompositions.
problem Classifying fibered knots and their open book decompositions.
method Using the Ozsváth-Stipsicz-Szabó concordance invariant Upsilon.
result Fibered knots satisfying a specific condition are either unique in their smooth concordance classes or provide counterexamples to the Slice-Ribbon Conjecture.
Collects properties of Seifert homology spheres for concordance studies.
problem Understanding concordance properties of fibers in Seifert homology spheres.
method Using Dehn surgery along Seifert fibers.
result Includes various properties in one place for concordance studies.
The paper proves tight fibered knots are minimal in a specific knot order.
problem Understanding the minimality of tight fibered knots.
method Proved ribbon concordance forms a partial order and used it to show tight fibered knots are minimal.
result All tight fibered knots are minimal in the ribbon concordance order.
The study establishes a link between fibered links and their concordance invariants.
problem Determining the conditions for fibered strongly quasi-positive links.
method Proved that an n-component fibered link L is strongly quasi-positive if and only if τ(L)=g3(L)+n−1. result Explicitly determined fibered prime links with at most 9 crossings and their properties.
This paper shows that only finitely many knots can be ribbon concordant to any given knot.
problem Understanding the relationship between ribbon concordance and fibered knots.
method Using knot Floer homology and bordered Heegaard Floer homology, the paper proves an inequality relating the knot Floer homology of a satellite knot and its companion.
result Every knot in S3 has only finitely many fibered predecessors under ribbon concordance. Sharp knots and iterated cables lead to ribbon knots or failure of slice-ribbon conjecture.
problem Understanding the concordance properties of knots and their cables.
method Using Seifert genus and concordance invariant γ0 from bordered Heegaard Floer homology.
result Connected sums of γ0-sharp fibered knots are ribbon only if they are of a specific form, or the slice-ribbon conjecture fails.
The paper explores fibered and quasi-positive links, introducing new families and invariants.
problem Understanding the structure of L-space links and their properties. method Using the H-function as a concordance link invariant.
result Introduced a subfamily of fibered strongly quasi-positive L-space links and an infinite family of non-quasi-positive L-space links. Links can be transformed into many others using a specific operation.
problem Understanding the relationship between strongly quasipositive links and their concordance.
method Used a satellite operation with a slice knot to transform links.
result Strongly quasipositive links can be transformed into infinitely many other links.
Study of knots with generalized Mazur patterns and their invariants.
problem Understanding the invariants and properties of knots with generalized Mazur patterns.
method Computational analysis of τ and ε invariants for n-twisted satellites. result None of the n-twisted patterns from the family act surjectively on the smooth or rational concordance group. Akbulut and Kirby conjectured that two knots with the same 0-surgery are concordant. In this paper, we prove that if the slice-ribbon conjecture is true, then the modified Akbulut-Kirby's conjecture is false. We also give a fibered potential counterexample to the slice-ribbon conjecture.
Proves certain knots are rationally slice and vanishing of certain invariants.
problem Rational sliceness of specific knots.
method Uses Heegaard Floer homology to show vanishing of invariants.
result Certain knots are rationally slice and invariants vanish.
This paper provides a rational model for fiberwise THH transfer using A-infinity algebras.
problem Rational models for fiberwise THH transfer of fibrations over a base space.
method Explicit description of Hochschild homology transfer in terms of A-infinity algebras.
result Rational models for Becker-Gottlieb transfer and fiberwise THH-simple structures.
The study explores how surface diffeomorphisms of knots relate to their topological properties.
problem Understanding how properties of surface diffeomorphisms of knots relate to their topological properties.
method Examining both braid and fibered knot perspectives to explore the relationship between surface diffeomorphisms and knot properties.
result Properties of surface diffeomorphisms may relate to four-dimensional topological properties of knots, such as the slice genus.
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.
The study connects twist positivity to L-space knots and concordance.
problem Understanding the relationship between twist positivity and knot concordance.
method Analyzing Alexander polynomials and braid indices of knots.
result For twist positive L-space knots, the braid index equals the bridge index.
New examples of knots with infinitely many inequivalent slice disks.
problem Existence of knots with infinitely many pairwise nonisotopic slice disks.
method Classification of fibered, homotopy-ribbon disks and use of satellite operations.
result Examples of fibered, hyperbolic knots bounding infinitely many fibered, ribbon disks that are pairwise nonisotopic modulo local knotting.
In this paper we examine the relationship between various types of positivity for knots and the concodance invariant tau discovered by Ozsvath and Szabo and independently by Rasmussen. The main result shows that, for fibered knots, tau characterizes strong quasipositivity. This is quantified by the statement that for K…
The main goal of this article is to obtain a condition under which an infinite collection F of satellite knots (with companion a positive torus knot and pattern similar to the Whitehead link) freely generates a subgroup of infinite rank in the smooth concordance group. This goal is attained by examining bot…
Every negative amphichiral knot is rationally slice.
problem Proving every negative amphichiral knot is rationally slice.
method Systematic analysis of the action induced by the negative amphichiral map on the JSJ decomposition of the link exterior.
result Every negative amphichiral link is rationally slice.
Infinitely many links are top concordant to Hopf but not smoothly.
problem Finding links topologically concordant to Hopf but not smoothly.
method Provided infinitely many 2-component links.
result Links have nontrivial Alexander polynomial.
Study shows differences between smooth and topological almost concordance of knots.
problem Disparity between smooth and topological almost concordance of knots.
method Defined and analyzed almost concordance in 3-manifolds, proving results through infinite families and specific examples.
result Infinite families of knots exist that are topologically concordant but not smoothly almost concordant.
Study algebraic concordance groups for non-trivial links.
problem Invariance of signature, Fox-Milnor condition, and Blanchfield pairing under concordance.
method Defined algebraic concordance groups using generalized Seifert matrices.
result Recovery of invariance of signature, Fox-Milnor condition, and Blanchfield pairing for concordant links.
Stability theorem for concordance embeddings with applications.
problem Stability of concordance embeddings.
method Proving a stability theorem.
result Various applications to diffeomorphisms and homeomorphisms.
The study defines and explores almost-concordance classes of knots in 3-manifolds.
problem Defining and exploring almost-concordance classes of knots in 3-manifolds.
method Action of the concordance group of knots in the three-sphere on concordances of knots in arbitrary 3-manifolds; definition of almost-concordance; use of modified tau-invariant to obstruct almost-concordances.
result Existence of non-trivial almost-concordance classes in all non-abelian 3-manifolds; infinitely many nullhomologous non almost-concordant knots in L(p,1).
The concordance genus of a knot is the least genus of any knot in its concordance class. It is bounded above by the genus of the knot, and bounded below by the slice genus, two well-studied invariants. In this paper we consider the concordance genus of 11--crossing prime knots. This analysis resolves the concordance ge…
Classical knots embed into virtual knots' concordance group.
problem Understanding concordance of virtual knots.
method Proved equivalence between classical and virtual sliceness.
result Embedding of classical knots' concordance group into virtual knots.
New homomorphisms from knot Floer homology help classify knots.
problem Classifying knots based on their concordance properties.
method Defined an infinite family of concordance homomorphisms using knot Floer complexes.
result Explicitly computable homomorphisms that are linearly independent.
This work shows non-abelian quotient groups in string link concordance.
problem Understanding the structure of string link concordance groups.
method Analyzing the quotient groups formed by string link concordance and pure braid groups.
result The quotient of each string link concordance group by its pure braid subgroup is non-abelian.
New knot invariants from instanton homology.
problem Bounding knot genus and double points.
method Instanton homology groups with local coefficients.
result 1-parameter family of homomorphisms fr from knot concordance group to reals. Khovanov homology shows (4,5) torus knot is a summand in its concordance class.
problem Understanding the global ribbon minima in knot concordance classes.
method Application of Khovanov homology to ribbon concordance.
result Khovanov homology of (4,5) torus knot is a summand in any knot's homology.
Study shows knots with similar Blanchfield forms can be homotopy ribbon concordant.
problem Understanding homotopy ribbon concordance for knots.
method Using Blanchfield pairings and twisted Alexander polynomials.
result Existence of infinite families of knots with same Blanchfield form but not homotopy ribbon concordant.
Injective map on Khovanov homology induced by ribbon concordance.
problem Understanding ribbon concordance between links.
method Injective map on Khovanov homology.
result Injective map on Khovanov homology induced by ribbon concordance.
New homomorphism from Khovanov homology for knot concordance.
problem Understanding smooth concordance of knots.
method Modulo equivalence relation on Khovanov chain complex.
result Strictly stronger than Rasmussen invariants.
The concordance genus of a knot K is the minimum Seifert genus of all knots smoothly concordant to K. Concordance genus is bounded below by the 4-ball genus and above by the Seifert genus. We give a lower bound for the concordance genus of K coming from the knot Floer complex of K. As an application, we prove that ther…
The paper defines and studies new types of links in 4-manifolds.
problem Understanding and classifying links in 4-dimensional spaces.
method Defining and studying r-shake slice and r-shake concordant links, providing examples and characterizations.
result Characterizations of shake slice and shake concordant links for r=0, and proving invariants of shake concordance.
Kirby and Lickorish showed that every knot in the 3-sphere is concordant to a prime knot, equivalently, every concordance class contains a prime knot. We prove here that their result can be strengthened: Every knot in the 3-sphere is invertibly concordant to a prime knot. A consequence is that every double concordance …
Introduces slice knots and concordance, linking to exotic smooth structures.
problem Understanding slice knots and their equivalence through concordance.
method Explains connections and introduces filtrations, satellite operations.
result Connections between slice knots and exotic smooth structures on \(\mathbb{R}^4\).
The concordance genus of a knot is the least genus of any knot in its concordance class. Although difficult to compute, it is a useful invariant that highlights the distinction between the three-genus and four-genus. In this paper we define and discuss the stable concordance genus of a knot, which describes the behavio…
Abstract: Proves non-abelian group of equivariant concordance.
problem Equivariant concordance group non-abelian
method Infinite family of nontrivial commutators
result Equivariant concordance group is not abelian
Knot Floer homology detects ribbon concordance.
problem Detecting ribbon concordance between knots.
method Using knot Floer homology to study ribbon concordance.
result The map induced by ribbon concordance on knot Floer homology is injective.
New hyperbolic knots not concordant to algebraic ones found.
problem Identifying knots not concordant to algebraic knots.
method Constructing hyperbolic L-space knots.
result Found hyperbolic knots that are not concordant to algebraic knots.
Functorial properties of knot invariants under specific concordances.
problem Computing obstructions to regular Lagrangian concordances.
method Functoriality of EH class and LOSS invariant under Lagrangian concordances in Weinstein cobordisms.
result Computable obstructions to regular Lagrangian concordances.
Study shows how concordance surgery impacts a 4D knot invariant.
problem Understanding how concordance surgery affects a specific 4D knot invariant.
method Used sutured Floer TQFT and a perturbed version of sutured Floer homology.
result Formula involving the graded Lefschetz number of the concordance map on knot Floer homology.