Computes upper bounds for instanton knot homology.
problem Computing the exact dimension of instanton knot homology.
method Developed an algorithm using knot diagrams and Heegaard diagrams.
result Algorithm provides sharp bounds for knots up to seven crossings.
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.
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.
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 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.
Study calculates instanton Floer homology for surgeries on L-space knots.
problem Computing the framed instanton Floer homology of surgeries on L-space knots.
method Used Baldwin-Sivek contact invariant and proved homogeneity of the invariant.
result Framed instanton Floer homology agrees with Heegaard Floer homology for L-space knots.
New homological action on sutured instanton homology defined.
problem Detecting link splitting in knot homology.
method Defining a homological action on sutured instanton Floer homology.
result Instanton knot homology detects link splitting for two-component links.
New proofs of knot detection using instanton Floer homology.
problem Detecting specific knots using homology theories.
method Using instanton Floer homology instead of knot Floer homology.
result Proves detection of specific knots using instanton homology.
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. 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 instanton Floer homology for links in RP^3 and use it to detect knots.
problem Detecting knots in RP3 using instanton Floer homology. method Compute instanton Floer homology for links in RP3 and use spectral sequences. result Khovanov homology detects the unknot and projective unknot in RP3. We show that instanton knot homology is mutation-invariant, as a consequence of earlier work of the third author.
This paper studies knots in sutured manifolds achieving minimum instanton homology rank.
problem Properties of knots achieving minimum instanton homology rank in sutured manifolds.
method Analyzes sutured manifolds and instanton homology to find knots achieving minimum rank.
result Knots achieving minimum instanton homology rank in sutured manifolds are the unknot.
Study knot invariants and surgeries, proving properties of instanton homologies.
problem Characterize 3-manifolds as Dehn surgery on knots.
method Establish conjugation symmetry, use cobordism maps, analyze homology cobordism group.
result Prove ν♯(K) is always zero or odd. Proves a formula for instanton Floer homology of knots.
problem Calculating changes in instanton Floer homology for knots.
method Uses sutured instanton homology and the octahedral lemma.
result Derives an integral surgery formula for framed instanton homology.
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…
New invariants help study satellite knots and their concordance.
problem Understanding the concordance of satellite knots.
method Filtered instanton homology and classical knot theory techniques.
result Criteria for satellite knots to be linearly independent.
We prove that the instanton knot homology KHI(K) as defined by Kronheimer and Mrowka (Knots, sutures and excision, preprint), recovers the Alexander polynomial for knots K in the 3-sphere.
Enhances knot surgery formulae for instanton Floer homology.
problem Calculating instanton Floer homology for various 3-manifolds.
method Integral surgery formulae and rational surgery formulae for null-homologous knots.
result Characterizes instanton Floer homology of specific 3-manifolds and knots.
Instanton homology detects 2-torsion in fibered knots.
problem Detecting 2-torsion in instanton homology for fibered knots.
method Using sutured instanton theory to derive a formula for I♯(Y,K;C) and comparing dimensions. result Proves the presence of 2-torsion in instanton homology for null-homologous fibered knots.
The paper introduces new knot invariants using singular instanton gauge theory.
problem Developing new knot invariants using singular instanton gauge theory.
method Using SU(2) singular instanton gauge theory, the paper constructs invariants and Morse chain complexes. result The constructions lead to a triad of groups and several concordance invariants.
For each partial flag manifold of SU(N), we define a Floer homology theory for knots in 3-manifolds, using instantons with codimension-2 singularities.
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.
Detects knots in thickened surfaces using instanton homology.
problem Detecting knots in thickened surfaces using homology.
method Uses Asaeda-Przytycki-Sikora (APS) homology and sutured instanton homology.
result Detects the unknot in (−1,1)imesΣ and characterizes minimal sutured instanton homology. 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.
Study instanton homology and its connection to Froyshov's invariant.
problem Determining framed instanton homology for Dehn surgeries on knots.
method Calculating the homology groups and relating them to Froyshov's invariant.
result Shows a relationship between framed instanton homology and Froyshov's invariant.
The instanton Floer homology of a knot in the three-sphere is a vector space with a canonical mod 2 grading. It carries a distinguished endomorphism of even degree,arising from the 2-dimensional homology class represented by a Seifert surface. The Floer homology decomposes as a direct sum of the generalized eigenspaces…
A spectral sequence is established, from Bar-Natan's variant of Khovanov homology to a deformation of instanton homology for knots and links. This spectral sequence arises as a specialization of a spectral sequence from a characteristic-2 version of F5 homology, in Khovanov'sclassification. Concordance invariants of…
New method proves cosmetic surgery conjecture for certain knots.
problem Proving the cosmetic surgery conjecture for specific knots.
method Using filtered instanton homology and Chern-Simons filtration.
result Reduced the conjecture to surgeries of ±2 on genus 2 knots. We prove that Khovanov homology detects the trefoils. Our proof incorporates an array of ideas in Floer homology and contact geometry. It uses open books; the contact invariants we defined in the instanton Floer setting; a bypass exact triangle in sutured instanton homology, proven here; and Kronheimer and Mrowka's spe…
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.
Paper proves depth bounds for taut foliations using instanton Floer homology.
problem Finding depth bounds for taut foliations in sutured manifolds.
method Sutured instanton Floer homology, adapted to monopole and Heegaard Floer settings.
result Dimension of sutured instanton Floer homology bounds minimal depth of taut foliations.
The paper examines 2-torsion in instanton Floer homology for knots and 3-manifolds.
problem Existence of 2-torsion in instanton Floer homology for knots and 3-manifolds.
method Analyzes framed and singular instanton Floer homology, uses lemmas from Heegaard Floer theory.
result Shows existence of 2-torsion in instanton Floer homology for various knots and 3-manifolds.
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…
New proof shows L-space knots are fibered and have specific properties.
problem Proving L-space knots are fibered and have strong quasipositive properties.
method Conceptually simpler proof using Heegaard Floer homology, translated to instanton Floer homology.
result Generalization of L-space knot properties to other Floer homologies.
New knot homology theory from symplectic geometry.
problem Developing a symplectic counterpart to instanton knot homology.
method Using symplectic character varieties and Floer homology, a new invariant for knots in 3-manifolds is constructed.
result Symplectic instanton knot homology (SIK) is a new invariant of knots and links in 3-manifolds.
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.
New insights into knot surgeries via instanton 2-torsion.
problem Understanding rational surgeries on knots and their implications.
method Extending earlier results on integral surgeries to rational surgeries using framed instanton homology.
result If framed instanton homology is 2-torsion-free, the knot is an instanton L-space knot and the surgery parameter is greater than 2g(K)-1.
Khovanov homology ranks 2 for certain knots in a specific bundle.
problem Generalizing Khovanov homology to IimesT2. method Proving rank 2 for specific knots in (−1,1)imesT2. result Khovanov homology ranks 2 for isotopic knots in {0}imesT2. We prove that a knot is the unknot if and only if its reduced Khovanov cohomology has rank 1. The proof has two steps. We show first that there is a spectral sequence beginning with the reduced Khovanov cohomology and abutting to a knot homology defined using singular instantons. We then show that the latter homology i…
We introduce explicit holonomy perturbations of the Chern-Simons functional on a 3-ball containing a pair of unknotted arcs. These perturbations give us a concrete local method for making the moduli spaces of flat singular SO(3) connections relevant to Kronheimer and Mrowka's singular instanton knot homology non-degene…
For any link of two components in an integral homology sphere, we define an instanton Floer homology whose Euler characteristic is the linking number between the components of the link. We relate this Floer homology to the Kronheimer-Mrowka instanton Floer homology of knots. We also show that, for two-component links i…
Paper classifies Heegaard Floer minimal knots in sutured manifolds.
problem Classifying minimal knots in sutured manifolds.
method Uses Heegaard Floer homology.
result Agrees with instanton Floer homology classification when applicable.
Enhanced Euler characteristic improves knot homology detection.
problem Detecting knots in instanton homology.
method Decomposing sutured instanton homology with torsion.
result Enhanced Euler characteristic equals SFH Euler characteristic.
Null-homotopic knots in certain 3-manifolds are uniquely identified by their complements.
problem Identifying null-homotopic knots in specific 3-manifolds.
method Instanton Floer homology and SU(2)-representation varieties.
result Null-homotopic knots are determined by their complements in certain 3-manifolds.
New knot classification based on SU(2) representations and instanton homology.
problem Classifying knots based on SU(2) representations and Alexander polynomials.
method Large surgery formula connecting instanton knot homology and framed instanton homology.
result Non-SU(2)-abundant knots are prime with restricted Alexander polynomial coefficients. In earlier work of the authors, the Khovanov complex of a knot or link appeared as the first page in a spectral sequence abutting to the instanton homology. The quantum and (co)homological gradings on Khovanov homology do not survive as gradings, but we show that they survive as filtrations.
For a given smooth 2-knot in S4, we relate the existence of a smooth Seifert hypersurface of a certain class to the existence of irreducible SU(2)-representations of its knot group. For example, we see that any smooth 2-knot having the Poincaré homology 3-sphere as a Seifert hypersurface has at least four i…