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.
Culler and Shalen, and later Yoshida, give ways to construct incompressible surfaces in 3-manifolds from ideal points of the character and deformation varieties, respectively. We work in the case of hyperbolic punctured torus bundles, for which the incompressible surfaces were classified by Floyd and Hatcher. We conver…
For every closed hyperbolic Haken 3-manifold and, more generally, for any hyperbolic 3-manifold M which is homeomorphic to the interior of a Haken manifold, the number 0.286 is a Margulis number. If M has non-zero first Betti number, or if M is closed and contains a semi-fiber, then 0.292 is a Margulis number for M.
Let M be a closed, orientable hyperbolic 3-manifold and φ a homomorphism of its fundamental group onto Z that is not induced by a fibration over the circle. For each natural number n we give an explicit lower bound, linear in n, on rank of the fundamental group of the cover of M corresponding to $φ…
Let M be a compact, orientable, irreducible, atoroidal 3-manifold with boundary an incompressible torus. Techniques based on the characteristic submanifold theory are used to bound the intersection number of two slopes αand βon the boundary of M. The method applies when βis the boundary slope of an essential surface F …
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.
Study links' concordance using signatures and nullity.
problem Link concordance invariance of Levine-Tristram signatures.
method Determine invariance for complex numbers on the unit circle.
result Signature and nullity give link concordance invariants.
Satellite operations have infinite rank in smooth concordance group.
problem Understanding satellite operations in the smooth concordance group.
method Reduction to winding number zero satellites and use of SO(3) gauge theory. result Provides a criterion for satellite operations to generate infinite rank subgroups.
Survey on knot concordance invariants from Heegaard Floer homology.
problem Understanding knot concordance using Heegaard Floer homology.
method Analysis of knot concordance invariants derived from Heegaard Floer homology.
result Stable equivalence of knot Floer complexes for concordant knots.
Injective map proven in knot Floer homology for strong homotopy-ribbon concordances.
problem Injectivity of knot Floer homology maps under strong homotopy-ribbon concordances.
method Knot Floer homology and properties of strongly homotopy-ribbon concordances.
result Injective map proven in knot Floer homology for strong homotopy-ribbon concordances.
Knot concordance linked to involutive knot Floer homology equivalence.
problem Understanding knot concordance through algebraic topology.
method Involutive knot Floer homology and stable equivalence.
result Concordant knots have equivalent involutive knot Floer complexes.
Study shows knots in homology spheres can be equivalent to knots in 3-sphere after any filtration step.
problem Detecting knots in homology spheres using the solvable filtration.
method Proved that for any knot in a homology sphere, there exists a knot in the 3-sphere equivalent modulo any term of the solvable filtration.
result Knots in homology spheres can be equivalent to knots in 3-sphere after any filtration step.
Proves non-solvability of concordance groups using Milnor invariants.
problem Non-solvability of concordance groups of 2-string links and strongly invertible knots.
method Using Milnor invariants to prove non-solvability.
result Proves non-solvability of C(2) and equivariant concordance groups of strongly invertible knots. Knots can be ordered by ribbon concordance, solving a long-standing question.
problem Ordering knots by ribbon concordance.
method Representation varieties of knot groups to SO(N) and relations induced by ribbon concordance. result Ribbon concordance forms a partial ordering on the set of knots.
It was shown by Jim Davis that a 2-component link with Alexander polynomial one is topologically concordant to the Hopf link. In this paper, we show that there is a 2-component link with Alexander polynomial one that has unknotted components and is not smoothly concordant to the Hopf link, answering a question of Jim D…
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. The Mazur pattern acts by the identity up to topological concordance.
problem Understanding the action of the Mazur pattern up to topological concordance.
method Comparing satellite operators and using topological concordance properties.
result Evidence that the Mazur pattern acts by the identity up to topological concordance.
We define a new smooth concordance homomorphism based on the knot Floer complex and an associated concordance invariant, epsilon. As an application, we show that an infinite family of topologically slice knots are independent in the smooth concordance group.
Study on knots in S1imesS2 with unique smooth concordance class for winding number 1.
problem Understanding concordance classes of knots in S1imesS2. method Defined winding number and used it to classify knots, demonstrated distinctions between smooth and topological categories.
result Unique smooth concordance class for winding number 1, infinitely many topological concordance classes for other winding numbers.
New knot concordance invariants from sl(n) cohomology.
problem Developing new invariants to distinguish knots.
method Construct piecewise linear maps from [0,1] to R, derived from sl(n) knot cohomology.
result Verify properties analogous to Upsilon invariant and define new concordance invariants.
New findings on knot concordance show limitations to primary decompositions.
problem Understanding the structure of knot concordance groups.
method Analyzing homomorphisms and polynomial factorizations of Alexander polynomials.
result Primary decompositions of topologically slice knots cannot exist.
New knot concordance invariants derived from regions in the plane.
problem Knot concordance and distinguishing knots from thin or algebraic ones.
method Associate invariants to regions in the plane, compute for specific knots, and use to obstruct concordances.
result Compute and use new invariants to obstruct concordances to specific types of knots.
Obstructs 2-torsion in rational knot concordance group.
problem Identifying 2-torsion elements in rational knot concordance group.
method Localized von Neumann ρ-invariant.
result Provides an obstruction for knots of order 2 in algebraic rational concordance group from being of finite order in rational knot concordance group.
Study of concordances between links in 4-space, defining new invariants.
problem Investigate the set of all embedded concordances between two fixed links in 4-space.
method Define Milnor-type invariants of the set of concordances, modulo indeterminacy.
result For slice links, these invariants classify the set of concordances up to link-homotopy.
Study on 2-bridge knots, proving equivariant concordance order is infinite.
problem Equivariant concordance of 2-bridge knots.
method Formula for butterfly polynomial, two proofs of non-equivariant sliceness, new invariant for strongly invertible knots.
result Equivariant concordance order of 2-bridge knots is infinite.