SU(3) instanton homology counts Tait colorings for webs and foams.
problem Counting Tait colorings for webs and foams.
method SU(3) gauge theory with structure group, eigenspace decomposition, skein exact triangles.
result SU(3) homology counts Tait colorings, Euler characteristic interpretable as polynomial invariant value.
Proves SU(2) representations for certain 3-spheres with embedded tori.
problem Characterizing SU(2) representations for specific 3-manifolds.
method Instanton Floer homology, surgery exact triangle, holonomy perturbations, non-vanishing results, and cable surgery results.
result Fundamental groups of certain 3-manifolds admit irreducible SU(2) representations.
The study connects knot representations to Seifert hypersurfaces and instanton Floer homology.
problem Existence of Seifert hypersurfaces and irreducible representations for 2-knots.
method Quantitative instanton Floer homology and related techniques.
result Existence of at least four irreducible SU(2) representations for certain knots.
Surgery exact triangles in various 3-manifold Floer homology theories provide an important tool in studying and computing the relevant Floer homology groups. These exact triangles relate the invariants of 3-manifolds, obtained by three different Dehn surgeries on a fixed knot. In this paper, the behavior of SU(N)-ins…
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.
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.
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. 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.
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…
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.
We recently defined invariants of contact 3-manifolds using a version of instanton Floer homology for sutured manifolds. In this paper, we prove that if several contact structures on a 3-manifold are induced by Stein structures on a single 4-manifold with distinct Chern classes modulo torsion then their contact invaria…
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.
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 irreducible SU(2) representations for knots in 3D.
problem Existence of irreducible SU(2) representations for cyclic branched covers of knots.
method Combining covering space techniques, SICUP matrices, and instanton Floer homology.
result Irreducible SU(2) representations exist for certain knots and branched covers.
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.
New proof shows certain 3D shapes can't be instanton L-spaces.
problem Proving certain 3D shapes cannot be instanton L-spaces.
method Proving by gluing complements of knots and analyzing their fundamental groups.
result Proven that the resulting 3-manifold cannot be an instanton L-space.
We study knots in S3 with infinitely many SU(2)-cyclic surgeries, which are Dehn surgeries such that every representation of the resulting fundamental group into SU(2) has cyclic image. We show that for every such nontrivial knot K, its set of SU(2)-cyclic slopes is bounded and has a unique limit point, whic…
Formula established for instanton homology of dual knots.
problem Calculating the dimension of instanton homology for dual knots.
method Dimension formula for unreduced singular instanton homology of dual knots.
result Dimension formula for instanton homology of dual 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.
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.
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.
In arXiv:1611.09927, we constructed a well-defined Lagrangian Floer invariant for any closed, oriented 3-manifold Y via the symplectic geometry of so-called traceless SU(2)-character varieties. This invariant, SI(Y), which we refer to as the symplectic instanton homology of Y, was also shown…
Constructs instantons on a specific G_2-manifold limit.
problem Finding instantons on a particular G_2-manifold limit.
method Dynamical systems approach involving perturbations of abelian and cone solutions.
result Obtains a 1-parameter family of SU(2) instantons with bounded curvature.
New method uses Floer homology to create exotic disk pairs.
problem Creating exotic disk pairs in 4-manifolds.
method Singular instanton Floer homology with Chern--Simons filtration.
result Constructs exotic pairs of slice disks and knots.
Study of instantons and knot representations in 3-manifolds.
problem Understanding instantons and knot representations in 3-manifolds.
method Study of U(3) instanton Floer homology, knot group representations, and sutured manifolds. result Established a surface decomposition theorem for U(3) instanton Floer homology of sutured manifolds. New one-parameter families of SU(2)2-invariant instantons found on Calabi-Yau 3-folds.
problem Behavior of Calabi-Yau instantons and monopoles with SU(2)2-symmetry. method Gauge theory on asymptotically conical Calabi-Yau 3-folds with SU(2)2 co-homogeneity one action. result New one-parameter families of invariant instantons found.
Study conically singular instantons over SU(3)-manifolds, proving existence and dimension formulas.
problem Existence and dimension of moduli spaces of conically singular instantons.
method Develop Fredholm deformation theory, investigate cokernel of instanton operator.
result Formula for virtual dimension of moduli space of conically singular instantons with structure group P(U(n)).
In 1985 lectures at MSRI, A. Casson introduced an interesting integer valued invariant for any oriented integral homology 3-sphere Y via beautiful constructions on representation spaces (see [1] for an exposition). The Casson invariant λ(Y) is roughly defined by measuring the oriented number of irreducible representati…
We call a knot in the 3-sphere SU(2)-simple if all representations of the fundamental group of its complement which map a meridian to a trace-free element in SU(2) are binary dihedral. This is a generalisation of being a 2-bridge knot. Pretzel knots with bridge number ≥3 are not SU(2)-simple. We provide an …
Study G2-instantons on R4imesS3 with invariant properties.
problem Existence of G2-instantons on R4imesS3. method Explicit construction and study of invariant properties.
result Existence of 1-parameter families of G2-instantons. 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…
In this paper we explicitly calculate the analogue of the 't Hooft SU(2) Yang--Mills instantons on Gibbons--Hawking multi-centered gravitational instantons which come in two parallel families: the multi-Eguchi--Hanson, or A_k ALE gravitational instantons and the multi-Taub--NUT, or A_k ALF gravitational instantons. We …
New method to compute tangle sums in character varieties.
problem Computing character varieties for tangle sums.
method Cut-and-paste methods to construct perturbed character varieties.
result Bounding cochains can be used to recover differential of Iatural. We initiate the systematic study of G2-instantons with SU(2)2-symmetry. As well as developing foundational theory, we give existence, non-existence and classification results for these instantons. We particularly focus on R4×S3 with its two explicitly known distinct holonomy G2 metrics, whic…
The first irreducible solution of the SU(2) self-duality equations on the Euclidean Schwarzschild (ES) manifold was found by Charap and Duff in 1977, only 2 years later than the famous BPST instantons on R4 were discovered. While soon after, in 1978, the ADHM construction gave a complete descrip…
New G2-instantons found on 3-sphere's spinor bundle.
problem Classifying and constructing G2-instantons. method Used SU(2)3-symmetries and asymptotically conical, co-homogeneity one G2-metric. result Found new examples of G2-instantons with obstructed deformations. We present a general procedure to construct 6-dimensional manifolds with SU(3)-structure from SU(2)-structure 5-manifolds. We thereby obtain half-flat cylinders and sine-cones over 5-manifolds with Sasaki-Einstein SU(2)-structure. They are nearly Kahler in the special case of sine-cones over Sasaki-Einstein 5-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 on Yang-Mills equation near instanton-anti-instanton configurations with energy constraints.
problem Understanding Yang-Mills connections near instanton-anti-instanton configurations.
method Analyzing the Uhlenbeck limit and bubble configurations, determining obstructions and proving solutions.
result Instantons are the only solutions with energy less than $4π^2 \left( |κ| + 2
ight) + \varepsilon_κ$.
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.
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.
We present a construction of self-dual Yang-Mills connections on the Taub-NUT space. We illustrate it by finding explicit expressions for all SU(2) instantons of instanton number one and generic monodromy at infinity.
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.
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. We define an elementary relatively Z/4 graded Lagrangian-Floer chain complex for restricted immersions of compact 1-manifolds into the pillowcase, and apply it to the intersection diagram obtained by taking traceless SU(2) character varieties of 2-tangle decompositions of knots. Calculations for torus 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.