Grid homology invariant proved for lens space links.
problem Proving combinatorial invariance of grid homology for lens space links.
method Combining combinatorial methods with sign assignments to prove invariance.
result Grid homology is a link invariant for lens space links.
New formulas link Milnor invariants to Heegaard Floer homology.
problem Understanding Milnor invariants through Heegaard Floer homology.
method Established new relationships between Milnor invariants and Heegaard Floer homology.
result Formula for the Milnor triple linking number from the link Floer complex.
New invariants derived from link homology for 4-manifolds.
problem Creating invariants for surfaces in smooth 4-manifolds.
method Skein lasagna modules based on equivariant and deformed glN link homology. result Non-vanishing and decomposition results for deformed glN skein lasagna modules. Study s-invariants from sl(3)-link homology, extending to other characteristics.
problem Deepen understanding of s-invariants from Khovanov's homology. method Use Mackaay-Vaz universal sl(3)-link homology, extend to other characteristics, use scanning algorithm for calculations. result Extend s-invariants to other characteristics, particularly p=3. Khovanov homology invariant proved for links in RP3.
problem Proving Khovanov homology invariant for links in RP3. method Established functoriality of Khovanov homology for link cobordisms in IimesRP3. result Khovanov homology is an invariant of links in unparametrized RP3. New 4-manifold invariants from Khovanov-Rozansky link homology.
problem Defining invariants for 4-manifolds.
method Using Khovanov-Rozansky gl(N) link homology, constructing skein modules from 4-categories with duals.
result Proof of the sweep-around property for well-defined link homologies in the 3-sphere.
Link homology compared with geometric link invariants using Bott-Samelson varieties.
problem Comparing different link homology theories with geometric link invariants.
method Using Khovanov-Rozansky homology and equivariant cohomology applied to Bott-Samelson varieties.
result Equivariant integral sl(n) link homology with specialized or universal potential.
Defines an odd analog of Plamenevskaya's invariant for transverse links.
problem No specific problem stated; focuses on extending an invariant.
method Defines and analyzes an odd analog invariant in Khovanov homology.
result The odd analog invariant is an invariant of transverse links with similar properties.
New homology invariant for links in surfaces, a deformation of APS.
problem Defining a new homology invariant for links in surfaces.
method A 1-parameter family of homology invariants, motivated by instanton Floer homology.
result The new invariant recovers APS homology and has a stronger detection property.
Khovanov-Floer theories are shown to be invariant under mutation.
problem Invariance of Khovanov-Floer theories under Conway mutation.
method Spectral sequences from Khovanov homology and proofs of conjectures.
result Strong Khovanov-Floer theories are mutation-invariant.
New obstructions show links with vanishing Milnor invariants may not be concordant to homology boundary links.
problem Understanding links with vanishing Milnor invariants and their concordance properties.
method Developing new obstructions and examples within the solvable filtration framework.
result Existence of links with vanishing Milnor invariants that are not concordant to homology boundary links.
New link homology theories for RP3 yield distinct invariants.
problem Extending link homologies for links in RP3. method Introducing new Lee and Bar-Natan spectral sequences for links in RP3. result New Lee and Bar-Natan invariants are distinct from existing ones.
Knot lattice homology invariant of smooth knot type in rational homology spheres.
problem Invariance of knot lattice homology in rational homology spheres.
method Proving knot lattice homology invariant through doubly-filtered homotopy type.
result Knot lattice homology invariant of smooth knot type in rational homology spheres.
The theory of signature invariants of links in rational homology spheres is applied to covering links of homology boundary links. From patterns and Seifert matrices of homology boundary links, an explicit formula is derived to compute signature invariants of their covering links. Using the formula, we produce fused bou…
Unified framework for proving functoriality of various link homology theories.
problem Functoriality and Reidemeister invariance of link homology theories.
method Unified framework leveraging Khovanov homology and various link homology theories.
result Stronger functoriality results by avoiding spectral sequences.
Link Floer homology is an invariant for links defined using a suitable version of Lagrangian Floer homology. In an earlier paper, this invariant was given a combinatorial description with mod 2 coefficients. In the present paper, we give a self-contained presentation of the basic properties of link Floer homology, incl…
Link homology theories connect to 4-manifold invariants and TQFTs.
problem Connecting link homology theories to 4-manifold invariants and TQFTs.
method Functorial link homologies, skein modules, handle decompositions.
result Link homology theories become diagram-independent and furnish invariants of 4-manifolds.
Paper defines new invariants from Khovanov homology, linking them to existing ones.
problem Developing new transverse link invariants from Khovanov homology.
method Using Mackaay-Vaz approach to universal sl3-homology, defining β3-invariants. result Established relationship between new invariants and existing ones like Plamenevskaya's and Wu's.
The spectral sequence's Ek-page is a link invariant for k≥3.
problem Proving the invariance of the spectral sequence pages.
method Analyzing the spectral sequence construction and its properties.
result The Ek-page is a link invariant for k≥3. We construct cobordism maps on link Floer homology associated to decorated link cobordisms. The maps are defined on a curved chain homotopy type invariant. We describe the construction, and prove invariance. We also make a comparison with the graph TQFT for Heegaard Floer homology.
New invariants for knotted surfaces derived from link homology.
problem Distinguishing knotted surfaces from unknotted ones.
method Link homology and A∞-algebras. result Invariant distinguishes unknotted sphere from certain knotted spheres.
M. Khovanov and L. Rozansky gave a categorification of the HOMFLY-PT polynomial. This study is a generalization of the Khovanov-Rozansky homology. We define a homology associated to the quantum (sln,∧Vn) link invariant, where ∧Vn is the set of the fundamental representations of the quantum group of $sl…
New theory for unoriented virtual links, extending classical invariants.
problem Invariants for unoriented virtual links not previously defined.
method Developed a new Khovanov homology theory for unoriented virtual links.
result Unoriented Jones polynomial for virtual links is a new invariant.
New invariants generalize Khovanov homology to sln-like structures.
problem Generalizing Khovanov homology to sln-like structures. method Defining and studying a family of link invariants HFKn(L) using holomorphic disc counts. result Conjecture of a spectral sequence from sln homology to HFKn(L). Paper introduces Bar-Natan homology for special links in a modified space.
problem Developing homology theory for null homologous links in \mathbb{RP}^3.
method Deformation of Khovanov homology, defining s-invariant, using twisted orientation and cobordisms.
result Established genus bounds using the new s-invariant.
New invariant for virtual links defined using homology.
problem Invariants for virtual links and their properties.
method Defining homological arrow polynomial and using it to study virtual links.
result New invariant for virtual links and its properties.
Symplectic Khovanov homology is an invariant of oriented links defined by Seidel and Smith and conjectured to be isomorphic to Khovanov homology. I define morphisms (up to a global sign ambiguity) between symplectic Khovanov homology groups, corresponding to isotopy classes of smooth link cobordisms in 4D between a fix…
Extends knot concordance invariant to balanced spatial graphs using grid homology.
problem Defining a concordance invariant for balanced spatial graphs.
method Using grid homology to extend the invariant from knots to spatial graphs.
result The combinatorial Υ invariant is a concordance invariant for balanced spatial graphs. New shifting chain map enhances quandle invariants for links.
problem Enhancing quandle invariants for links and surfaces.
method Introducing a shifting chain map σ and its pull-back σ# to transform cocycles.
result Shifting chain map σ# transforms 2-cocycles to 3-cocycles, enhancing invariants.
Paper discusses biquandle cohomology and invariants for surface-links.
problem Computing invariants for surface-links using biquandle theory.
method Developed (co)homology theory of biquandles, introduced new invariants using broken surface diagrams and marked graph diagrams.
result Constructs new invariants for surface-links using biquandle theory.
Study transverse and Legendrian invariants of certain link cables using Floer homology.
problem Transverse and Legendrian invariants of specific link cables.
method Combining grid diagrams and Floer homology, using inclusion maps of grid complexes.
result Transverse and Legendrian invariants of certain link cables are zero for large q/p. Holomorphic polygons help calculate link complements' Floer homology.
problem Calculating the Floer homology of link complements.
method Counts of holomorphic polygons in Heegaard multi-diagrams.
result Quasi-isomorphism to bordered-sutured invariant of link complements.
New algorithm calculates S-invariants for links efficiently.
problem Cannot be done by standard spectral sequence methods.
method Developed an algorithm for S-invariants of links. result Demonstrated efficiency and applicability to sl(3)-link homology.
Paper introduces a new algebraic link invariant related to knot Floer and Khovanov homologies.
problem Developing a new algebraic link invariant related to knot homologies.
method Constructing a chain complex C1±1(D) from plat braid diagrams, showing it is isomorphic to Khovanov homology and conjecturing it is a link invariant. result The total homology of the constructed complex is a link invariant, and it is conjectured to be isomorphic to δ-graded knot Floer homology.
We introduce the notion of a quandle with a good involution and its homology groups. Carter et al. defined quandle cocycle invariants for oriented links and oriented surface-links. By use of good involutions, quandle cocyle invariants can be defined for links and surface-links which are not necessarily oriented or orie…
New link homology theory supercategorifies Jones polynomial.
problem Constructing link homology theories.
method Skew version of KLR algebras for type A quiver, cyclotomic quotients of supercategory.
result Invariant is distinct from Khovanov homology and conjectured to be isomorphic to odd Khovanov homology.
We introduce a generalization of the Ozsváth-Szabó τ-invariant to links by studying a filtered version of link grid homology. We prove that this invariant remains unchanged under strong concordance and we show that it produces a lower bound for the slice genus of a link. We show that this bound is sharp for torus lin…
In this article, we give an elementary construction of homological invariants of links presented by braid closures. The Euler characteristic of this complex is equal to quantum polynomial invariant of link.
Link invariants fail to detect most links with high probability.
problem Detecting specific link types using invariants.
method Mathematical proof and big-data analysis.
result Link invariants have a zero probability of detecting alternating links.
New invariant restores symmetry in link homology and matches Hilbert scheme ideals.
problem Restoring missing symmetry in Khovanov-Rozansky homology.
method Defining a deformation of Khovanov-Rozansky homology with parameters yc. result Matches ideals in Haiman's description of isospectral Hilbert scheme.
Proves inequality linking instanton and Khovanov homologies.
problem Rank inequality on instanton and Khovanov homologies.
method Constructs spectral sequence relating pointed Khovanov homology to instanton invariant.
result Establishes rank inequality between instanton and Khovanov homologies.
GRID invariants block certain Lagrangian cobordisms in 3D.
problem Obstructing decomposable Lagrangian cobordisms in 3D.
method Filtered GRID invariants and link Floer homology.
result GRID invariants obstruct decomposable Lagrangian cobordisms.
Floer homology connects to quiver Hecke algebras in Coulomb branches.
problem Understanding the representation theory of quiver Hecke algebras.
method Using Lagrangian Floer homology in multiplicative Coulomb branches.
result Cylindrical KLRW algebras are realized by Floer homology.
Paper categorifies Vassiliev skein relation for Khovanov homology.
problem Clarifying the relation between Vassiliev invariants and Khovanov homology.
method Developed a categorified version of Vassiliev skein relation on Khovanov homology.
result Khovanov homology's genus-one operation leads to a crossing change, enabling invariance under Reidemeister moves and extending to singular links.
New link homologies categorify Jones polynomial at odd prime powers.
problem Categorify Jones polynomial at odd prime powers.
method Specialize Cautis differential to positive integers.
result Non-isomorphic link homologies for odd primes.
A new skein relation for Khovanov homology categorifies the θ-invariant.
problem Categorify the θ-invariant using Khovanov homology.
method Developed a skein relation for Khovanov homology using spectral sequences.
result Categorified the θ-invariant using Khovanov homology.
The study explores Legendrian invariants and half Giroux torsion in contact structures.
problem Understanding Legendrian invariants and their behavior with half Giroux torsion.
method Analysis of Legendrian links with non-vanishing contact invariants and the study of half Giroux torsion.
result Null-homologous links with irreducible complements have non-loose Legendrian realizations with non-zero invariants.
New spin on Khovanov-Rozansky homology categorifies spin link polynomial.
problem Categorify spin link polynomial using colored Khovanov-Rozansky homology.
method Equip Λ^n-colored sl_{2n} Khovanov-Rozansky homology with an involution.
result Categorifies spin-colored so_{2n+1} quantum link polynomial for n=1,2,3.