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.
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.
New symplectic instanton homology constructed via traceless character varieties.
problem Constructing symplectic instanton homology for 3-manifolds.
method Using traceless character varieties to define a Lagrangian Floer invariant.
result Symplectic instanton homology conjecturally equivalent to a gauge theoretic variant.
The paper develops a new homology theory for 3-manifolds and proves its functoriality.
problem Developing a new homology theory for 3-manifolds.
method Using symplectic geometry and Floer theory, constructing a new homology theory and proving its functoriality.
result The symplectic instanton homology is functorial with respect to cobordisms.
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.
We define a twisted version of Manolescu and Woodward's Symplectic Instanton homology, prove that this invariant fits into the framework of Wehrheim and Woodward's Floer Field theory, and describe its behaviour for connected sum and Dehn surgery.
New approach solves problems in gauge theory and symplectic geometry.
problem Instanton Floer homology and Lagrangian Floer theory.
method Combination of cobordism type argument and homological algebra.
result Various problems solved and proofs simplified.
Proves irreducible SU(2) representations for 3-surgery knots.
problem Irreducible SU(2) representations for 3-surgery knots.
method Instanton Floer homology and symplectic Floer homology.
result Proves irreducible SU(2) representations for 3-surgery knots.
Defines a new field theory in 1+1+1 dimensions.
problem Developing a new field theory in 1+1+1 dimensions.
method Defines an extended field theory as a quasi 2-functor with values in a completed 2-category, H a m ^ \widehat{\mathcal{H}am} H am . result Extends existing theories and provides a real analog of a construction by Moore and Tachikawa.
Develops gluing theory for contact instantons and pseudoholomorphic curves.
problem Constructing contact instanton Floer cohomology and Fukaya-type category.
method Gluing theory of contact instantons and pseudoholomorphic curves in symplectization context.
result Construction of Legendrian contact instanton homology and moduli spaces of holomorphic buildings.
This paper explores adiabatic solutions of Haydys-Witten equations for knot homology.
problem Investigating instanton Floer homology and its relation to Khovanov homology.
method Analyzes decoupled Haydys-Witten equations and their equivalence to EBE solutions.
result Proposes an equivalence between adiabatic solutions of decoupled Haydys-Witten equations and non-vertical paths in EBE moduli space.
Formulates a strategy to prove Atiyah-Floer conjecture.
problem Relate instanton Floer homology and Lagrangian Floer homology.
method Develops a strategy to construct the desired isomorphism.
result Formulates a possible approach to address singular space of flat connections.
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…
Study of instantons and knot representations in 3-manifolds.
problem Understanding instantons and knot representations in 3-manifolds.
method Study of U ( 3 ) U(3) U ( 3 ) instanton Floer homology, knot group representations, and sutured manifolds. result Established a surface decomposition theorem for U ( 3 ) U(3) U ( 3 ) instanton Floer homology of sutured manifolds. Novel gauge-theoretic Floer homologies defined for 7, 6, and 5-manifolds.
problem Defining novel gauge-theoretic Floer homologies for different dimensions.
method Topologically-twisted 8d N = 1 \mathcal{N}=1 N = 1 gauge theory on Spin(7)-manifolds. result Derivation of novel Floer homologies and A ∞ A_\infty A ∞ -categories. 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.
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 u 2 − 64 u^2-64 u 2 − 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…
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.
A group action on a moduli space is shown to be faithful.
problem Injectivity of a homomorphism from mapping class group to symplectic mapping class group.
method Instanton Floer homology, Atiyah-Floer Conjecture, Heegaard Floer strategy.
result The homomorphism is injective for surfaces of genus at least 2.
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.
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 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 ) (1,1) ( 1 , 1 ) -knots. 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 F 5 F_5 F 5 homology. result A spectral sequence connects instanton and Khovanov homologies.
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 instanton Floer homology for links in RP^3 and use it to detect knots.
problem Detecting knots in R P 3 \mathbb{RP}^3 RP 3 using instanton Floer homology. method Compute instanton Floer homology for links in R P 3 \mathbb{RP}^3 RP 3 and use spectral sequences. result Khovanov homology detects the unknot and projective unknot in R P 3 \mathbb{RP}^3 RP 3 . 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 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.
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.
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 ∞ I^\infty 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 λ I ( Y ) λ_I(Y) λ I ( 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 {\varphi} φ for knots in the 3-sphere using Donaldson invariants and Floer's instanton homology. result The invariant φ {\varphi} 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.
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.
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 q 2 q_2 q 2 from 3D cobordism to integers.
problem Understanding 4-manifolds with boundary and their homology.
method Instanton homology with Z / 2 Z/2 Z /2 coefficients. result Found a non-linear homomorphism q 2 q_2 q 2 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 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.