Study calculates nilpotency in instanton homology and related framed instanton homology.
problem Understanding instanton homology and framed instanton homology for surfaces times a circle.
method Computes nilpotency degree and uses moduli space of stable rank two holomorphic bundles.
result Framed instanton homology closely related to kernel of u2−64. We construct a spectral sequence from the reduced odd Khovanov homology of a link converging to the framed instanton homology of the double cover branched over the link, with orientation reversed. Framed instanton homology counts certain instantons on the cylinder of a 3-manifold connect-summed with a 3-torus. En route…
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.
We introduce a class of links strictly containing quasi-alternating links for which mod 2 reduced Khovanov homology is always thin. We compute the framed instanton homology for double branched covers of such links. Aligning certain dotted markings on a link with bundle data over the branched cover, we also provide many…
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.
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. 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.
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. Instanton Floer homology matches Heegaard Floer for almost-rational plumbings.
problem Matching instanton Floer homology with Heegaard Floer for specific 3-manifolds.
method Utilizes lattice homology and a recent cobordism map decomposition theorem.
result Isomorphism between framed instanton Floer homology and Heegaard Floer for almost-rational plumbings.
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.
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.
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.
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.
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.
Instantons prove knots are fibered and strongly quasipositive.
problem Proving properties of knots using instantons and L-space surgeries.
method New cobordism maps in framed instanton Floer homology.
result Proves fibered and strongly quasipositive knots have specific homomorphisms.
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.
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.
Study calculates ring structure in instanton homology for a surface with points.
problem Calculating the ring structure in instanton homology for a surface with points.
method Used singular instanton Floer homology and excision formula.
result Proved an excision formula for instanton Floer homology when n=1.
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 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. We study the representation spaces R(K;i) as appearing in Kronheimer and Mrowka's framed instanton knot Floer homology, for a class of pretzel knots. In particular, for pretzel knots P(p,q,r) with p,q,r pairwise coprime, these appear to be non-degenerate and comprise representations in SU(2) that are not b…
The paper introduces a new family of instanton-invariants for four-manifolds and connects them to Khovanov homology.
problem Understanding the relationship between gauge-theoretic instanton invariants and Khovanov homology.
method Develops a one-parameter family of Haydys-Witten instanton Floer homology groups and investigates dimensional reductions of the Haydys-Witten equations.
result Establishes a connection between the new instanton invariants and Khovanov homology, particularly for geometric blow-ups of three-manifolds.
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 instanton homology for webs counts Tait colorings.
problem Counting Tait colorings of complex webs.
method Introducing a local system of coefficients to deform instanton homology.
result Rank of deformed instanton homology equals number of Tait colorings.
Sutured instanton homology uses Heegaard diagrams to bound homology dimensions.
problem Bounding the dimension of sutured instanton homology.
method Using admissible Heegaard diagrams, proving a bound on the number of generators of the associated sutured Heegaard Floer complex.
result Strong L-spaces are instanton L-spaces.
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.
Establishes a spectral sequence linking instanton and Khovanov homologies.
problem Connecting instanton and Khovanov homologies.
method Develops a spectral sequence specializing invariants from characteristic-2 F5 homology. result A spectral sequence connects instanton and Khovanov homologies.
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. New instanton Floer homology for sutured manifolds with tangles defined.
problem Defining instanton Floer homology for complex geometric structures.
method Extending excision theorems to handle singularities and tangles.
result Detection of unlink and trivial braid closures in annular Khovanov homology.
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 instanton homology theories for 3-manifolds and bundles.
problem Defining instanton homology for a class of 3-manifolds and bundles.
method Functorial construction for 4-manifold cobordisms, algebraic construction of homology theories for dg-modules.
result Isomorphic to Floer's instanton homology for admissible bundles, calculates I∞ in all cases it is defined. New instanton invariants for rational homology spheres defined and shown to be functorial.
problem Defining and proving invariance of instanton homology groups for rational homology spheres.
method Novel suspended flow category technique to handle obstructed cobordisms and prove wall-crossing formula.
result Instanton invariant λI(Y) conjecturally equals Casson-Walker invariant for rational homology spheres. 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 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.
Khovanov homology can identify trefoil knots.
problem Detecting trefoil knots using Khovanov homology.
method Floer homology, contact geometry, open books, instanton Floer setting, bypass exact triangle, spectral sequence.
result Khovanov homology detects trefoils.
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.
Proves naturalness of Symplectic Instanton homology and defines cobordism maps.
problem Naturalness and cobordism maps of Symplectic Instanton homology.
method Proves naturalness and defines maps associated with four-dimensional cobordisms.
result Defines representations of mapping class groups and fundamental groups.
Kronheimer-Mrowka's instanton homology dimension equals Tait colorings.
problem Calculating the dimension of a specific homology group for plane trivalent graphs.
method Using SO(3) instanton Floer homology, the dimension is shown to be equal to the number of Tait colorings.
result The dimension of J#(G) is equal to the number of Tait colorings of G.
Formula calculates instanton homology dimensions for knot surgeries over arbitrary fields.
problem Calculating instanton homology dimensions for knot surgeries over arbitrary fields.
method Established a dimension formula for framed instanton homology of knot surgeries over arbitrary fields.
result Formula generalizes instanton homology dimensions to arbitrary fields, including new results for p/q and knots. New formula splits instanton Floer homology invariants.
problem Expressing Seiberg-Witten invariants in terms of other invariants.
method Observation of a splitting formula in instanton Floer homology.
result A new formula for instanton Floer homology invariants.
Study calculates instanton homology for simple braids, linking to Fano variety quantum cohomology.
problem Calculating instanton homology for specific braids.
method Using local coefficients and algebraic curves, calculates homology groups and their module structures.
result Equivalent to computing quantum cohomology of a moduli space of parabolic bundles.
New homology q2 from 3D cobordism to integers.
problem Understanding 4-manifolds with boundary and their homology.
method Instanton homology with Z/2 coefficients. result Found a non-linear homomorphism q2 with bounds on 4-manifolds. Paper generalizes sutured Floer homologies with new algorithms and polytopes.
problem Computing and understanding sutured Floer homologies for complex manifolds.
method Grading shifting property, algorithms, polytopes, canonical decompositions.
result Established new bounds and detection results for sutured Floer homologies.
Paper introduces new homology for links in an annulus.
problem Detecting links in an annulus and distinguishing braids from other tangles.
method Defined annular instanton Floer homology and constructed a spectral sequence.
result Annular Khovanov homology detects the unlink in the thickened annulus.
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.
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 symplectic instanton homology properties and applications.
problem Understanding symplectic instanton homology under various three-manifold operations.
method Proved Künneth-type formula, defined twisted versions, and used surgery exact sequences.
result Computed ranks of groups associated to branched covers and surgeries.