Proves properties of instanton knot Floer homology and connected sum formula.
problem Properties of instanton knot Floer homology.
method General properties of sutured Floer theories, Heegaard Floer and monopole Floer settings.
result Connected sum formula for instanton knot Floer homology.
Study satellite knots using bordered Floer theory, proving non-thinness and calculating genus.
problem Properties of twisted Mazur pattern satellite knots.
method Use bordered Floer theory to analyze knots and calculate genus.
result Prove non-thinness of Qn(K) and calculate 3-genus in terms of n and K. Notes on Khovanov and knot Floer theories' stable homotopy types.
problem Understanding stable homotopy types in Khovanov and knot Floer theories.
method Introduction to Khovanov and knot Floer theories' stable homotopy types.
result Introduction of stable homotopy types in Khovanov and knot Floer theories.
Study shows how knot Floer homology and bordered Floer theory are linked.
problem Understanding the relationship between knot Floer homology and bordered Floer theory.
method Proves local equivalence and establishes algebraic formulas.
result Involutive knot Floer homology and bordered Floer theory determine each other under certain conditions.
The paper develops a new theory for knots and 3-manifolds with involutions.
problem Developing a new theory for knots and 3-manifolds with involutions.
method Establishing a version of Seiberg-Witten Floer K-theory for knots and 3-manifolds with involutions.
result 10/8-type inequalities for knots and involutions, yielding lower bounds on stabilizing numbers and relative genera.
L-space knots lack essential Conway spheres, proven with Floer theory.
problem Essential Conway spheres in L-space knots.
method Floer theoretic invariant for tangles.
result L-space knots have no essential Conway spheres.
We define Floer homology theories for oriented, singular knots in S^3 and show that one of these theories can be defined combinatorially for planar singular knots.
Algebraic methods prove knot primality using Floer homology.
problem Proving the primality of knots.
method Knot Floer homology, metacyclic representations, and twisted homology.
result Primality tests have proven primality for over 99.6% of knots.
Enhances knot Floer homology with algebraic representation theory.
problem Compatibility between summands in bordered knot Floer homology.
method Categorifies intertwining property of higher representations.
result New algebraic reformulation of compatibility property.
Knot Floer homology is a knot invariant defined using holomorphic curves. In more recent work, taking cues from bordered Floer homology,the authors described another knot invariant, called "bordered knot Floer homology", which has an explicit algebraic and combinatorial construction. In the present paper, we extend the…
Satellite formula connects knot concordance invariants to surgery.
problem Understanding knot concordance invariants.
method Excision theorem for real Floer homotopy types.
result Concordance invariants depend only on zero-framed surgery.
New link detection results using knot and link Floer homology.
problem Detecting specific links and knots using Floer homology.
method Inspired by Khovanov homology, uses knot and link Floer homology.
result Detects specific links and knots with high precision.
This work extends knot homology theory to links, proving exact triangles and categorifying link signatures.
problem Extending knot homology theory to links and proving exact triangles.
method Equivariant singular instanton Floer theory, circle-equivariant Morse-Floer theory, cobordism constructions.
result Established unoriented skein exact triangles and categorified link signatures.
Knot Floer homology is an invariant for knots discovered by the authors and, independently, Jacob Rasmussen. The discovery of this invariant grew naturally out of studying how a certain three-manifold invariant, Heegaard Floer homology, changes as the three-manifold undergoes Dehn surgery along a knot. Since its origin…
Defines real link Floer homology for specific types of links.
problem Developing a new homology theory for certain types of links.
method Combining real Heegaard Floer homology and real sutured Heegaard Floer homology, using real grid diagrams in S3. result Observes structural and property properties of strongly invertible knots.
We use the Ozsvath-Szabo theory of Floer homology to define an invariant of knot complements in three-manifolds. This invariant takes the form of a filtered chain complex, which we call CF_r. It carries information about the Floer homology of large integral surgeries on the knot. Using the exact triangle, we derive inf…
Tests using knot Floer homology detect prime knots with high accuracy.
problem Detecting prime knots efficiently.
method Knot Floer homology and polynomial irreducibility tests.
result Over 96% of non-hyperbolic prime knots up to 20 crossings are identified as prime.
Proves exact triangle linking knot instanton Floer homology to surgeries.
problem Relating knot instanton Floer homology to surgeries.
method Proves an exact triangle using integer coefficients.
result Poincaré Homology Sphere is not an instanton L-space with integer coefficients.
The singular instanton Floer homology was defined by Kronheimer and Mrowka in connection with their proof that the Khovanov homology is an unknot detector. We study this theory for knots and two-component links using equivariant gauge theory on their double branched covers. We show that the special generator in the sin…
Study corrects previous work on knot Floer homology of certain pretzel knots.
problem Computing the Knot Floer Homology of specific pretzel knots.
method Applied peculiar modules theory for Floer homology of 4-ended tangles, using immersed curve interpretation.
result Corrected the rank of Knot Floer Homology for some pretzel knots.
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.
A well-known conjecture of Rasmussen states that for any knot K in S3, the rank of the reduced Khovanov homology of K is greater than or equal to the rank of the reduced knot Floer homology of K. This rank inequality is supposed to arise as the result of a spectral sequence from Khovanov homology to knot Flo…
New cobordism maps for instanton knot homology help compute surgeries on knots.
problem Computing surgeries on knots using instanton knot homology.
method Construct cobordism maps for instanton knot homology.
result Compute the framed instanton Floer homology of certain knots.
Combinatorial method computes Legendrian knot invariant.
problem Computing the Heegaard Floer contact invariant for Legendrian knots.
method Combining Plamenevskaya's combinatorial description with Heegaard Floer theory.
result Hat version of LOSS invariant can be computed combinatorially.
Floer homology linked to Milnor fibers for certain singularities.
problem Understanding Floer homology of singularities using Milnor fibers.
method Combining Floer theory with Milnor fibers, using combinatorial formulas.
result Equality of Casson invariants in Donaldson and Seiberg-Witten theories.
Single twist can unknot certain knots, study shows.
problem Can a knot be unknotted with a single twist?
method Classical knot invariants, Casson-Gordon invariants, and Heegaard Floer theory.
result Obstructions to unknotting with a single twist exist.
The paper defines grid homologies for singular links in lens spaces and constructs a resolution cube for knot Floer homology.
problem Defining and constructing a resolution cube for knot Floer homology of singular links in lens spaces.
method Defining grid homologies for singular links in lens spaces and using them to construct a resolution cube.
result A complete description of singular knot theory in lens spaces and a signed combinatorial resolution cube for knot Floer homology.
We unify two existing approaches to the tau invariants in instanton and monopole Floer theories, by identifying τG, defined by the second author via the minus flavors KHI− and KHM− of the knot homologies, with τG♯, defined b…
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.
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…
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.
Characterizes (1,1) knots with large Dehn surgeries.
problem Understanding (1,1) knots and their surgeries. method Diagrammatic characterization and Heegaard Floer homology analysis.
result Characterizes (1,1) knots admitting large Dehn surgeries. Using a knot concordance invariant from the Heegaard Floer theory of Ozsvath and Szabo, we obtain new bounds for the Thurston-Bennequin and rotation numbers of Legendrian knots in S^3. We also apply these bounds to calculate the knot concordance invariant for certain knots.
In this paper, we generalize the work of the second author and prove a grading shifting property, in sutured monopole and instanton Floer theories, for general balanced sutured manifolds. This result has a few consequences. First, we offer an algorithm that computes the Floer homologies of a family of sutured handle-bo…
The study examines how twisting a knot affects its homology and stability properties.
problem Investigating the impact of twisting a knot on its homology and stability.
method Use bordered Floer homology and immersed curve invariants.
result Total dimension, τ(K_m), and thickness of K_m are linear functions of m for large m.
New models found for guts of nearly fibered knots.
problem Characterizing nearly fibered knots topologically.
method Provided three models for the guts of nearly fibered knots.
result Nearly fibered condition can be purely topologically characterized.
We review the construction of Heegaard Floer homology for closed three-manifolds and also for knots and links in the three-sphere. We also discuss three applications of this invariant to knot theory: studying the Thurston norm of a link complement, the slice genus of a knot, and the unknotting number of a knot. We emph…
New technique connects instanton Floer homology to Heegaard diagrams for knots and 3-manifolds.
problem Accessing structural properties of instanton Floer homology.
method Developed a new technique using Heegaard diagrams and sutures.
result Established a relation between instanton knot homology and Heegaard diagrams for (1,1)-knots. New symplectic annular Khovanov homology connects knot theory to Floer homology.
problem Understanding the relationship between knot theory and Floer homology.
method Introducing a new version of symplectic annular Khovanov homology and establishing spectral sequences.
result Established spectral sequences linking different knot homologies.
Classifies knots in the Poincaré sphere, using fixed points and folding automata.
problem Classifying knots in the Poincaré sphere and understanding their properties.
method Theory of train tracks, folding automata, and knot Floer homology.
result Almost completely classified genus-two, hyperbolic, fibered knots.
Survey on knot theory's impact on four-dimensional topology.
problem Understanding invariants of smooth four-manifolds.
method Using Kirby diagrams and Heegaard Floer theory.
result Progress in detecting exotic structures.
Study proves naturality and functoriality in a type of Heegaard Floer homology.
problem Proving naturality and functoriality in a specific type of Heegaard Floer homology.
method Used the doubling model for the involution and variations to prove results.
result First-order naturality of involutive Heegaard Floer homology proved.
New knot homologies detect non-fibered knots, expanding on previous results.
problem Detecting non-fibered knots using knot homologies.
method Developed new knot homologies (knot Floer, Khovanov, HOMFLY) and proved a technical theorem.
result New homologies detect infinitely many non-fibered knots, expanding on previous results.
This paper shows how the Formal Knot Theory state model for the Alexander-Conway polynomial is related to Knot Floer Homology. In particular we prove a parity result about the states in this model that clarifies certain relationships of the model with Knot Floer Homology.
Floer homology vanishes on certain 3-manifolds formed by knots and their mirrors.
problem Understanding the homology cobordism group of 3-manifolds formed by knots and their mirrors.
method Using involutive Floer homology and gauge theory.
result Locally trivial involutive Floer homology on certain splices of knots and their mirrors.
Combinatorial approach to compute satellite knot invariants using graph theory.
problem Computing knot invariants for satellite knots using bordered Heegaard Floer homology.
method Construct weighted A∞-modules using decorated planar graphs and prove their isomorphism. result Combinatorial proof of A∞ structure relations for the constructed modules. Ozsvath and Szabo conjectured that knot Floer homology detects fibred knots. We propose a strategy to approach this conjecture based on Gabai's theory of sutured manifold decomposition and contact topology. We implement this strategy for genus-one knots, obtaining as a corollary that, if rational surgery on a knot K …
New hyperbolic knots with convex Upsilon invariants constructed.
problem Constructing knots with convex Upsilon invariants.
method Combinatorial method for (1,1)-knots and connected sum operation. result Infinitely many mutually non-concordant hyperbolic knots with convex Upsilon invariants.