arXiv research

New homological action on sutured instanton homology defined.

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

Proves exact triangle linking knot instanton Floer homology to surgeries.

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.

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

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

Sutured instanton homology uses Heegaard diagrams to bound homology dimensions.

Establishes a spectral sequence linking instanton and Khovanov homologies.

New invariants from framed instanton homology for knot concordance.

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

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

Computes upper bounds for instanton knot homology.

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

Defines knot concordance invariant using instanton homology and Donaldson invariants.

New instanton invariants for rational homology spheres defined and shown to be functorial.

New proofs of knot detection using instanton Floer homology.

Kronheimer-Mrowka's instanton homology dimension equals Tait colorings.

Study calculates instanton homology for simple braids, linking to Fano variety quantum cohomology.

New homology $q_2$ from 3D cobordism to integers.

Paper generalizes sutured Floer homologies with new algorithms and polytopes.

Paper extends cobordism maps in Khovanov and instanton homologies.

Proves a formula for instanton Floer homology of knots.

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

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.

A deformation of the authors' instanton homology for webs is constructed by introducing a local system of coefficients. In the case that the web is planar, the rank of the deformed instanton homology is equal to the number of Tait colorings of the web.

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…

The SO(3) instanton homology recently introduced by the authors associates a finite-dimensional vector space over the field of two elements to every embedded trivalent graph (or "web"). The present paper establishes a skein exact triangle for this instanton homology, as well as a realization of the octahedral axiom. Fr…

Study knot invariants and surgeries, proving properties of instanton homologies.

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…

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

In the description of the instanton Floer homology of a surface times a circle due to Muñoz, we compute the nilpotency degree of the endomorphism $u^2-64$. We then compute the framed instanton homology of a surface times a circle with non-trivial bundle, which is closely related to the kernel of $u^2-64$. We discuss th…

Detects knots in thickened surfaces using instanton homology.

Study instanton homology and its connection to Froyshov's invariant.

We show that instanton knot homology is mutation-invariant, as a consequence of earlier work of the third author.

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 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…

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…

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

New method proves cosmetic surgery conjecture for certain knots.

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

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…

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.

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

Instanton homology detects 2-torsion in fibered knots.

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.

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…