Study ribbon homology concordances using link Floer homology.
problem Understanding ribbon homology concordances and their effects on link Floer homology.
method Combining results from Daemi, Lidman, Vela-Vick, Wong, and Zemke, using link Floer homology and torsion submodules.
result Ribbon homology concordances induce split injections on HFL−. 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.
Determines conditions for ribbon cobordisms between lens spaces.
problem Conditions for ribbon rational homology cobordisms between lens spaces.
method Analyzes ribbon cobordisms and uses properties of lens spaces and linear lattices.
result If a lens space admits a ribbon rational homology cobordism to a different lens space, it must be homeomorphic to L(n,1), up to orientation-reversal. Rational homology ribbon cobordism defines a partial order on 3-manifolds.
problem Classifying 3-manifolds based on their homology
method Proving a partial order on homeomorphism classes of 3-manifolds
result Closed, connected, oriented 3-manifolds are partially ordered by rational homology ribbon cobordism.
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.
Study uses twisted Alexander polynomials to link fibered classes in 3-manifolds.
problem Linking fibered classes in 3-manifolds via Alexander polynomials.
method Applies twisted Alexander polynomials to fibered classes in ribbon homology cobordisms.
result Fibered classes of Y+ map to those of Y−. Study of equivariant ribbon concordance using Khovanov homology.
problem Understanding equivariant ribbon concordance.
method Functoriality of equivariant Khovanov homology under equivariant cobordisms, and induced split injection.
result Equivariant ribbon concordances induce a split injection on equivariant Khovanov homology.
We study 4-dimensional homology cobordisms without 3-handles, showing that they interact nicely with Thurston geometries, character varieties, and instanton and Heegaard Floer homologies. Using these, we derive obstructions to such cobordisms. As one example of these obstructions, we generalize other recent results on …
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.
Proves module structure on odd Khovanov homology and applies to ribbon 2-knots.
problem Proving module structure on odd Khovanov homology.
method Geometric description of module structure and link cobordism.
result Combinatorial proof of odd invariant for ribbon 2-knots.
Constructs TQFTs for cobordisms with cohomology class decorations.
problem Creating TQFTs for cobordisms with cohomology class decorations.
method Starting from an abelian group G and a factorizable ribbon Hopf G-bialgebra H, constructs a TQFT JH for connected framed cobordisms between connected surfaces with connected boundary decorated with cohomology classes with coefficients in G. result Our functor recovers a special case of Kerler-Lyubashenko TQFTs when restricted to trivial decorations.
Alternating links bound rational homology balls if their chessboard lattice is cubiquitous.
problem When do alternating links bound rational homology balls?
method Heegaard Floer homology and flows on planar graphs.
result The normalized determinant of the link's chessboard lattice is a necessary and sufficient condition for the link to bound a rational homology ball.
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…
Constructs homologies for ribbon graphs to recover Penrose polynomials.
problem Recovering Penrose polynomials from ribbon graphs.
method TQFT-valued cube of resolutions from two-dimensional cobordisms.
result Euler characteristics of constructed homologies recover Penrose polynomials.
Study shows how Khovanov homology behaves for split links and cobordisms.
problem Understanding Khovanov homology's sensitivity to cobordisms between split links.
method Proved that Khovanov homology map is determined by individual components of cobordism, not linking.
result Khovanov homology detects non-split links from split links.
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 uses instanton Floer theory to obstruct knot unknotting operations.
problem Obstructing knot unknotting operations and ribbon concordance.
method Equivariant singular instanton Floer theory with Chern--Simons filtration.
result For a large class of slice knots, any unknotting sequence must contain both signs.
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.
We investigate rational homology cobordisms of 3-manifolds with non-zero first Betti number. This is motivated by the natural generalization of the slice-ribbon conjecture to multicomponent links. In particular we consider the problem of which rational homology S1×S2's bound rational homology S1×D3'…
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.
We consider ribbon R4's, that is, smooth open 4-manifolds, homeomorphic to R4 and associated to h-cobordisms between closed 4-manifolds. We show that any generalized ribbon R4 associated to a sequence of h-cobordisms between non-diffeomorphic 4-manifolds is exotic. Notion of a positive ribbon R4 is defi…
New 3-manifolds bound rational 4-balls through specific operations.
problem Finding rational homology 3-spheres that bound rational homology 4-balls.
method Two operations that preserve lattice embedding obstruction to bounding rational homology balls.
result Explicit examples of rational surgeries on torus knots that bound rational homology balls.
New skein categories for non-semisimple settings, extending existing theory.
problem Extending skein theory to non-semisimple settings.
method Introducing skein categories based on tensor ideals in linear ribbon categories.
result Skein categories coincide with factorization homology in non-semisimple settings.
Extends Heisenberg homology to ribbon graphs.
problem Configurations in bounded surfaces.
method Regular thickening of ribbon graphs.
result Heisenberg homology applied to ribbon graphs.
Paper studies spectral invariants and monopole Floer homology for rational homology three-spheres.
problem Tackles the existence of positive scalar curvature metrics on ribbon homology cobordisms.
method Defines an R-filtration on the equivariant complex of monopole Floer homology via Chern-Simons-Dirac functional, leading to a spectral invariant.
result Shows that the spectral invariant provides an obstruction to the existence of positive scalar curvature metrics on ribbon homology cobordisms.
Lower bounds on ribbon distance using Bar-Natan and α-Homology.
problem Calculating the minimum number of ribbon operations to unknot a knot.
method Bar-Natan Homology and α-Homology approaches.
result Lower bounds on ribbon distance via both Bar-Natan and α-Homology.
The paper classifies homotopy ribbon discs with specific groups.
problem Classifying homotopy ribbon discs with given fundamental groups.
method Using geometric and algebraic properties of groups, particularly Farrell-Jones conjecture.
result Classification of homotopy ribbon discs for specific knot groups and Baumslag-Solitar groups.
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.
We introduce Khovanov homology for ribbon graphs and show that the Khovanov homology of a certain ribbon graph embedded on the Turaev surface of a link is isomorphic to the Khovanov homology of the link (after a grading shift). We also present a spanning quasi-tree model for the Khovanov homology of a ribbon graph.
Defines cobordism maps connecting Khovanov and instanton homologies.
problem No direct correspondence between Khovanov and instanton homology cobordism maps.
method Defines a cobordism map on the instanton cube complex as a filtered chain map.
result Proves the cobordism map recovers both Khovanov and instanton homology cobordism maps.
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 evidence refutes old conjectures about knot homology ranks, suggesting new congruences.
problem Determining the rank of knot homology theories modulo 4 for ribbon knots.
method Proved homomorphism of knot concordance group, checked conjectures for 2.4 million knots.
result Revised conjectures about knot homology ranks modulo 4 for ribbon knots hold true.
Study shows Khovanov homology's relation to decomposable Lagrangian cobordisms.
problem Understanding the relationship between Khovanov homology and decomposable Lagrangian cobordisms.
method Utilized previously defined filtered invariants to give obstructions.
result Partial answer to Ekholm, Honda, and Kálmán's question about Khovanov homology and decomposable Lagrangian cobordisms.
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.
It was recently proved by several authors that ribbon concordances induce injective maps in knot Floer homology, Khovanov homology, and the Heegaard Floer homology of the branched double cover. We give a simple proof of a similar statement in a more general setting, which includes knot Floer homology, Khovanov-Rozansky…
We show that a ribbon concordance between two links induces an injective map on Khovanov homology.
3-manifolds with specific homology are cobordant if homeomorphic.
problem Characterizing integer homology cobordism for Sol manifolds.
method Analyzing first homology groups and using cobordism theory.
result Integer homology cobordant Sol manifolds are homeomorphic.
We prove that the map on knot Floer homology induced by a strongly homotopy-ribbon concordance is injective.
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. Paper extends cobordism maps in Khovanov and instanton homologies.
problem Define and prove compatibility of cobordism maps in Khovanov and instanton homologies.
method Extend embedded cobordism map to immersed cobordisms and prove compatibility.
result Induced map on Khovanov homology is injective for certain concordances.
A cobordism between links in thickened surfaces consists of a surface S and a 3-manifold M, with S properly embedded in M×I. We show that there exist links in thickened surfaces such that if (S,M) is a cobordism between them in which S is simple, then M must be complex. That is, there…
Kanenobu has given infinite families of knots with the same HOMFLY polynomials. We show that these knots also have the same sl(n) and HOMFLY homologies, thus giving the first example of an infinite family of knots undistinguishable by these invariants. This is a consequence of a structure theorem about the homologies o…
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.
We prove that the map on knot Floer homology induced by a ribbon concordance is injective. As a consequence, we prove that the Seifert genus is monotonic under ribbon concordance. We also generalize a theorem of Gabai about the super-additivity of the Seifert genus under band connected sum. Our result gives evidence fo…
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 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…
Prove strong ribbon concordance induces a partial order on links, certify minimality for a handful of knots, and find minimal ribbon minimal knots.
problem Prove strong ribbon concordance induces a partial order on links.
method Use results from knot Floer homology to certify minimality under the ribbon partial order.
result Certify minimality for a handful of knots and find minimal ribbon minimal knots.
Oriented ribbon graphs (dessins d'enfant) are graphs embedded in oriented surfaces. A quasi-tree of a ribbon graph is a spanning subgraph with one face, which is described by an ordered chord diagram. We show that for any link diagram L, there is an associated ribbon graph whose quasi-trees correspond bijectively to …