Grid homology confirms the Upsilon invariant in knot theory.
problem Verifying the equivalence of Upsilon invariants in knot theory.
method Reconstructed Upsilon invariant using grid homology and proved equivalence.
result Upsilon invariants in knot Floer and grid homology are equivalent.
Knot Floer homology is a knot invariant derived from Heegaard Floer homology.
problem Understanding and computing knot invariants.
method Derived from Heegaard Floer homology, with contributions from many researchers.
result Knot Floer homology is a useful tool for studying knots.
Homology from ternary algebras gives knot and surface invariants.
problem Creating knot and surface invariants using algebraic structures.
method Defining ternary algebra homology based on Reidemeister moves and quasigroups.
result Normalized homology yields invariants of knots and knotted surfaces.
Defines knot concordance invariant using instanton homology and Donaldson invariants.
problem Knot concordance and its classification.
method Defines an invariant φ for knots in the 3-sphere using Donaldson invariants and Floer's instanton homology. result The invariant φ coincides with a special case of an invariant defined by Froyshov. New spectral sequences define knot invariants.
problem Understanding strongly invertible knots.
method Two spectral sequences in knot Floer homology.
result Numerical invariant defined for strongly invertible knots.
New knot invariants discovered using mirror symmetry.
problem Categorify knot homology groups and explain their meaning.
method Homological mirror symmetry for new families of manifolds.
result Explicitly computable invariants for any Lie algebra.
New method computes transverse knot invariant using braids.
problem Compute transverse invariant of knots and links.
method Define new combinatorial complex to compute knot Floer homology.
result Identify BRAID invariant of transverse knots and links.
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.
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. Study finite type invariants for knots in rational homology 3-spheres.
problem Understanding invariants of knots in rational homology 3-spheres.
method Combinatorial description and analysis of null Lagrangian-preserving surgeries.
result Universal finite type invariants for null-homologous knots with trivial Alexander polynomial.
Using the combinatorial approach to knot Floer homology, we define an invariant for Legendrian knots in the three-sphere, which takes values in link Floer homology. This invariant can be used to also construct an invariant of transverse knots.
New operations on Khovanov homology refine knot invariants.
problem Understanding finer knot invariants through homological operations.
method Developed an algebra of homological operations on Khovanov homology and lifted it to integral Khovanov homology.
result Conjectured infinite algebras of homological operations and provided evidence.
We give a combinatorial treatment of transverse homology, a new invariant of transverse knots that is an extension of knot contact homology. The theory comes in several flavors, including one that is an invariant of topological knots and produces a three-variable knot polynomial related to the A-polynomial. We provide …
Combinatorial proof shows knot invariant in Lipshitz's grid homology.
problem Proving knot invariance in Lipshitz's grid homology.
method Purely combinatorial proof.
result Proves 'minus' version of Lipshitz's double-point enhanced grid homology is a knot invariant.
It is known that knot Floer homology detects the genus and Alexander polynomial of a knot. We investigate whether knot Floer homology of K detects more structure of minimal genus Seifert surfaces for K. We define an invariant of algebraically slice, genus one knots and provide examples to show that knot Floer homol…
New invariant fully describes finite type invariants of knots in homology 3-spheres.
problem Constructing a universal finite type invariant for knots in homology 3-spheres.
method Refined construction of a new invariant that is strictly stronger and universal.
result New invariant fully describes the graded space of finite type invariants of knots in homology 3-spheres.
New invariant connects knot homology and BPS series for plumbed knot complements.
problem Understanding invariants of plumbed knot complements.
method Introducing an invariant unifying knot lattice homology and BPS series, proving a surgery formula.
result Proved a surgery formula relating the new invariant to the weighted graded root of the surgered 3-manifold.
New spectral sequences link knot homology to instantons, revealing concordance invariants.
problem Understanding the relationship between knot homology and instantons.
method Established spectral sequences connecting Bar-Natan homology to instanton homology.
result Derived concordance invariants from instanton homology groups.
Researchers link tangle invariants for Khovanov and knot Floer homologies.
problem Relating tangle invariants for Khovanov and knot Floer homologies.
method Constructing algebraic DA bimodules for tangles and open braids, showing homotopy equivalence to Ozsvath-Szabo bimodules.
result Homotopy equivalence of DA bimodules for tangles and knot Floer homology.
Enhanced knot contact homology uniquely identifies knots.
problem Identifying knots uniquely up to smooth isotopy.
method Contact homology with Legendrian contact homology enhancements.
result Knots are uniquely determined by their contact homology.
We show that there exists a Legendrian knot with maximal Thurston-Bennequin invariant whose contact homology is trivial. We also provide another Legendrian knot which has the same knot type and classical invariants but nonvanishing contact homology.
Study cosmetic surgeries on knots in homology spheres using Casson-Walker invariant.
problem Cosmetic surgeries on knots in homology spheres and their constraints.
method Rational surgery formula of the Casson-Walker invariant for 2-component links.
result Constraints on knots and surgery slopes for cosmetic surgeries.
New knot invariant Υ defined without holomorphic theory.
problem Defining knot invariants without holomorphic theory.
method Developed grid homology theory to define Υ. result Υ is a well-defined knot invariant and provides a lower bound on unknotting number. Homologically fibered knots are knots whose exteriors satisfy the same homological conditions as fibered knots. In our previous paper, we observed that for such a knot, higher-order Alexander invariants defined by Cochran, Harvey and Friedl are generally factorized into the part of the Magnus matrix and that of a certa…
Modified bikei homology enhances knot and surface invariants.
problem Enhancing knot and surface invariants using bikei.
method Using bikei 2-cocycles to modify homology and cohomology.
result Improved counting invariant for knots and surfaces.
New invariants from Heegaard Floer homology for knot concordance.
problem Understanding knot concordance.
method Truncated Heegaard Floer homology.
result Construction of new concordance invariants νn(K). Generalizes knot τ-invariant to rational homology spheres.
problem Defining knot invariants for knots in rational homology spheres.
method Generalizing knot filtration approach to rational homology spheres.
result Some τ-invariants provide lower bounds for surface genus. Study on weaving knots W(3,n) confirms they are fibered and provides normal distribution evidence.
problem Investigate polynomial knot invariants and homologies of weaving knots.
method Compute polynomial knot invariants and homologies for W(3,n), analyze Khovanov homology ranks.
result W(3,n) are fibered knots and Khovanov homology ranks asymptotically follow a normal distribution.
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.
New knot homology invariant grows exponentially with color.
problem Constructing and understanding colored torus knot homology.
method Invariant construction and recursive formula for reduced HOMFLY homology.
result Doubly-graded invariant of positive torus knots grows exponentially in color.
New homology categorifies knotoid polynomial.
problem Generalizing knot theory to open-ended diagrams.
method Defined winding homology as a triply-graded invariant.
result Categorifies Turaev's polynomial invariant of knotoids.
Formula for Heegaard Floer multicurves of double tangles from knot complements.
problem Computing Heegaard Floer multicurve invariants of double tangles.
method Using a simple formula derived from knot complements, comparing with Khovanov homology.
result New characterisation of L-space knots and first example of thin knot Floer homology.
We use monopole Floer homology for sutured manifolds to construct invariants of Legendrian knots in a contact 3-manifold. These invariants assign to a knot K in Y elements of the monopole knot homology KHM(-Y,K), and they strongly resemble the knot Floer homology invariants of Lisca, Ozsváth, Stipsicz, and Szabó. We pr…
New invariant for surface-knots in 4D from marked graphs.
problem Invariants for surface-knots in 4D.
method Marked graph diagrams for surface-knots.
result Invariant constructed for surface-knots.
Study contact invariants using Floer homology to understand knots.
problem Understanding contact invariants of knots through surgeries.
method Used Heegaard Floer based contact invariants, sutured Floer homology, and limit constructions.
result Established relations between contact invariants of knots through surgeries.
New knot invariant from Kauffman states and algebraic modules.
problem Developing a new knot invariant related to Alexander polynomial.
method Defining a bigraded knot invariant using Kauffman states and differential graded algebras.
result The invariant's homology corresponds to the Alexander polynomial.
Study knot Floer homology to create concordance invariants and slice genus bounds.
problem Developing concordance invariants using knot Floer homology.
method Using knot Floer homology, define and analyze equivariant concordance invariants.
result Showed a family of strongly invertible slice knots with arbitrarily large equivariant slice genus.
New invariant for surface-knots in 4D space.
problem Invariants for surface-knots in 4D space.
method Diagrammatic approach in 3D space.
result Invariant constructed for surface-knots.
Automates machine learning of correlations between knot invariants.
problem Discovering and validating new relationships between knot invariants.
method Trained a neural network on 200,000 sets of knot invariants to predict an output invariant.
result Found novel correlations not explained by known results in knot theory.
New algebraic method for knot Floer homology computation.
problem Computing knot Floer homology efficiently.
method Extending bordered Floer homology to partial knot projections and establishing a pairing result.
result Identification of knot Floer homology with its algebraic definition.
Loose Legendrian knots in rational homology spheres are Legendrian isotopic if they have the same classical invariants.
problem Classifying loose Legendrian knots in rational homology spheres.
method Combining Dymara's result on loose Legendrian knots and Eliashberg's classification of overtwisted contact structures.
result Loose Legendrian knots in rational homology spheres are Legendrian isotopic if and only if they have the same classical invariants.
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.
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.
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.
The paper calculates the slicing degree of knots using advanced homology theories.
problem Determining the minimum slicing degree of knots.
method Rasmussen's s-invariant, knot Floer homology, and singular instanton homology.
result Computed slicing degrees for many small knots and some families of torus knots.
New invariants from framed instanton homology for knot concordance.
problem Concordance of knots and their properties.
method Framed instanton homology to define invariants.
result Computations and bounds on knot concordance invariants.
Researchers compute knot invariants in Seifert manifolds using Wilson loops.
problem Calculating BPS invariants for knots in Seifert manifolds.
method Analytically continued Chern-Simons level and representation from Wilson loop operator.
result Homological blocks for knots in Seifert manifolds and Seifert integer homology spheres.
We summarize recent work on a combinatorial knot invariant called knot contact homology. We also discuss the origins of this invariant in symplectic topology, via holomorphic curves and a conormal bundle naturally associated to the knot.