arXiv research

New method proves cosmetic surgery conjecture for certain knots.

Defines cobordism maps connecting Khovanov and instanton homologies.

New conditions prevent non-trivial relations in local equivalence group.

We construct an obstruction for the existence of embeddings of homology $3$-sphere into homology $S^3\times S^1$ under some cohomological condition. The obstruction is defined as an element in the filtered version of the instanton Floer cohomology due to R.Fintushel-R.Stern. We make use of the $\mathbb{Z}$-fold coverin…

Study calculates instanton Floer homology for surgeries on L-space knots.

Proves properties of instanton knot Floer homology and connected sum formula.

Instanton Floer homology matches Heegaard Floer for almost-rational plumbings.

Proves exact triangle linking knot instanton Floer homology to surgeries.

New technique connects instanton Floer homology to Heegaard diagrams for knots and 3-manifolds.

Study calculates ring structure in instanton homology for a surface with points.

Paper generalizes sutured Floer homologies with new algorithms and polytopes.

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…

Study calculates Floer homology for binary polyhedral spaces.

New homological action on sutured instanton homology defined.

Study instanton Floer homology for links in RP^3 and use it to detect knots.

Paper proves depth bounds for taut foliations using instanton Floer homology.

New proofs of knot detection using instanton Floer homology.

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 cobordism maps for instanton knot homology help compute surgeries on knots.

Unified tau invariants in instanton and monopole Floer theories.

Sutured instanton homology uses Heegaard diagrams to bound homology dimensions.

We prove an excision theorem for the singular instanton Floer homology that allows the excision surfaces to intersect the singular locus. This is an extension of the non-singular excision theorem by Kronheimer and Mrowka and the genus-zero singular excision theorem by Street. We use the singular excision theorem to def…

Defines knot concordance invariant using instanton homology and Donaldson invariants.

Paper classifies Heegaard Floer minimal knots in sutured manifolds.

Proves a formula for instanton Floer homology of knots.

This paper studies knots in sutured manifolds achieving minimum instanton homology rank.

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…

Khovanov-Floer theories are a class of homological link invariants which admit spectral sequences from Khovanov homology. They include Khovanov homology, Szab{ó}'s geometric link homology, singular instanton homology, and various Floer theories applied to branched double covers. In this short note we show that certain …

The study connects knot representations to Seifert hypersurfaces and instanton Floer homology.

New knot homology theory from symplectic geometry.

Study uses instanton Floer theory to obstruct knot unknotting operations.

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…

Proves complex homology three-spheres can be bounded by many handles.

This work extends knot homology theory to links, proving exact triangles and categorifying link signatures.

New proof shows L-space knots are fibered and have specific properties.

In a recent paper, Lin, Ruberman and Saveliev proved a splitting formula expressing the Seiberg-Witten invariant $λ_{SW}(X)$ of a smooth $4$-manifold with rational homology of $S^1\times S^3$ in terms of the Frøyshov invariant $h(X)$ and a Lefschetz number in reduced monopole Floer homology. In this note we observe tha…

Computes upper bounds for instanton knot homology.

The paper examines 2-torsion in instanton Floer homology for knots and 3-manifolds.

Since its inception, Floer homology has been an important tool in low-dimensional topology. Floer theoretic invariants of $3$-manifolds tend to be either gauge theoretic or symplecto-geometric in nature, and there is a general philosophy that each gauge theoretic Floer homology should have a corresponding symplectic Fl…

We define four versions of equivariant instanton Floer homology ($I^+, I^-, I^\infty$ and $\widetilde I$) for a class of 3-manifolds and $SO(3)$-bundles over them including all rational homology spheres. These versions are analogous to the four flavors of monopole and Heegaard Floer homology theories. This construction…

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…

We consider a family of corks, denoted $W_n$, constructed by Akbulut and Yasui. Each cork gives rise to an exotic structure on a smooth 4-manifold via a twist $τ$ on its boundary $Σ_n = \partial W_n$. We compute the instanton Floer homology of $Σ_n$ and show that the map induced on the instanton Floer homology by $τ: Σ…

In this paper, we introduce the annular instanton Floer homology which is defined for links in a thickened annulus. It is an analogue of the annular Khovanov homology. A spectral sequence whose second page is the annular Khovanov homology and which converges to the annular instanton Floer homology is constructed. As an…

We show that there is a well-defined cap-product structure on the Fintushel-Stern spectral sequence. Hence we obtain the induced cap-product structure on the ${\BZ}_8$-graded instanton Floer homology. The cap-product structure provides an essentially new property of the instanton Floer homology, from a topological poin…

The paper introduces new knot invariants using singular instanton gauge theory.

Taubes proved that the Casson invariant of an integral homology 3-sphere equals half the Euler characteristic of its instanton Floer homology. We extend this result to all closed oriented 3-manifolds with positive first Betti number by establishing a similar relationship between the Lescop invariant of the manifold and…

Around 1988, Floer introduced two important theories: instanton Floer homology as invariants of 3-manifolds and Lagrangian Floer homology as invariants of pairs of Lagrangians in symplectic manifolds. Soon after that, Atiyah conjectured that the two theories should be related to each other and Lagrangian Floer homology…

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.