arXiv research

We introduce and study combinatorial equivariant analogues of the Kronheimer--Mrowka homology theory of planar trivalent graphs.

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

New invariant connects symplectic fillings and contact structures.

This note is an exposition of the proof of Thom's conjecture by Kronheimer and Mrowka, using the new Seiberg-Witten invariants.

Proves surgery exact triangle for monopole Floer homology over integers.

We present a large family of knots for which the Rasmussen s-invariants of arbitrary satellites do not detect sliceness. This answers a question of Hedden. The proof hinges on work of Kronheimer-Mrowka and Cochran-Harvey-Horn.

Kronheimer and Mrowka constructed a variant of Seiberg-Witten invariants for a 4-manifold $X$ with contact boundary in 1997. Using Furuta's finite dimensional approximation, we refine this invariant in the case $H^1(X, \partial X; \mathbb{R})=0$.

We observe inequalities involving the Herzlich volume of a 4-dimensional asymptotically complex hyperbolic Einstein manifold and its Euler characteristic provided the metrics is either Kaehler or selfdual. In the selfdual case we have to assume furthermore that the Kronheimer-Mrowka invariant is non vanishing.

We use grid diagrams to investigate the Ozsvath-Szabo concordance invariant tau, and to prove that |tau(K_1)-tau(K_2)|<=g, whenever there is a genus g knot cobordism joining K_1 to K_2. This leads to an entirely grid diagram-based proof of Kronheimer-Mrowka's theorem, formerly known as the Milnor conjecture.

These are notes of a talk given at the Mathematische Arbeitstagung 2005 in Bonn. Following ideas of Ozbagci-Stipsicz, a proof based on contact Dehn surgery is given of Eliashberg's concave filling theorem for contact 3-manifolds. The role of that theorem in the Kronheimer-Mrowka proof of property P for nontrivial knots…

This is an expansion on my talk at the Geometry and Topology conference at McMaster University, May 2004. We outline a program to relate the Heegaard Floer homologies of Ozsvath-Szabo, and Seiberg-Witten-Floer homologies as defined by Kronheimer-Mrowka. The center-piece of this program is the construction of an interme…

We prove that an infinite family of virtually overtwisted tight contact structures discovered by Honda on certain circle bundles over surfaces admit no symplectic semi-fillings. The argument uses results of Mrowka, Ozsvath and Yu on the translation-invariant solutions to the Seiberg-Witten equations on cylinders and th…

There are a number of homological knot invariants, each satisfying an unoriented skein exact sequence, which can be realized as the limit page of a spectral sequence starting at a version of the Khovanov chain complex. Compositions of elementary 1-handle movie moves induce a morphism of spectral sequences. These morphi…

New invariant recovers known contact element and considers finite coverings.

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

Classifies links with small Khovanov homology ranks.

New inequality linking geodesic length and volume in complex projective plane.

Lower bounds for a knot invariant are derived using computations and cobordism inequality.

We introduce two invariants called sl(3) Khovanov module and pointed sl(3) Khovanov homology for spatial webs (bipartite trivalent graphs). Those invariants are related to Kronheimer-Mrowka's instanton invariants $J^\sharp$ and $I^\sharp$ for spatial webs by two spectral sequences. As an application of the spectral seq…

New real invariants for 3-manifolds and links.

Proves SU(2) representations for certain 3-spheres with embedded tori.

We construct a new spectral sequence beginning at the Khovanov homology of a link and converging to the Khovanov homology of the disjoint union of its components. The page at which the sequence collapses gives a lower bound on the splitting number of the link, the minimum number of times its components must be passed t…

We prove that every Einstein metric on the unit ball B^4 of C^2, asymptotic to the Bergman metric, is equal to it up to a diffeomorphism. We need a solution of Seiberg--Witten equations in this infinite volume setting. Therefore, and more generally, if M^4 is a manifold with a CR-boundary at infinity, an adapted spinc-…

Boundary Dehn twist on $K3$ surfaces becomes trivial after abelianization.

We show that any compact symplectic manifold (W,ω) with boundary embeds as a domain into a closed symplectic manifold, provided that there exists a contact plane ξon dW which is weakly compatible with omega, i.e. the restriction ω|ξdoes not vanish and the contact orientation of dW and its orientation as the boundary of…

Link groups can only have certain SU(2) representations.

Paper classifies Heegaard Floer minimal knots in sutured manifolds.

Let K be a knot in the 3-sphere. A slope p/q is said to be characterising for K if whenever p/q surgery on K is homeomorphic, via an orientation-preserving homeomorphism, to p/q surgery on another knot K' in the 3-sphere, then K and K' are isotopic. It was an old conjecture of Gordon, proved by Kronheimer, Mrowka, Ozsv…

Unknot recognition is one of the fundamental questions in low dimensional topology. In this work, we show that this problem can be encoded as a validity problem in the existential fragment of the first-order theory of real closed fields. This encoding is derived using a well-known result on SU(2) representations of kno…

Defines monopole Floer homology for 3-manifolds with toroidal boundaries.

Study of unoriented SL(4) foams in 3-manifolds.

For a homotopically energy-minimizing map $u: N^3\to S^1$ on a compact, oriented $3$-manifold $N$ with boundary, we establish an identity relating the average Euler characteristic of the level sets $u^{-1}\{θ\}$ to the scalar curvature of $N$ and the mean curvature of the boundary $\partial N$. As an application, we ob…

New findings contradict the Thom conjecture for high degree hypersurfaces in $CP^3$.

Boundary Dehn twists become trivial after abelianization.

New invariant shows Dehn twist on connected sum of homology tori is not isotopic to identity.

New contactomorphisms found via Dehn twists on 3-manifold sums.

New example of non-smooth isotopy after stabilization in 4-manifolds.

Integrase proteins acting on circular double-stranded DNA often change its topology by transforming unknotted circles into torus knots and links. Two systems of tangle equations--corresponding to the two initial DNA sequences--arise when modelling this transformation: direct and inverted. With no a priori assumptions o…

For a harmonic map $u:M^3\to S^1$ on a closed, oriented $3$--manifold, we establish the identity $2π\int_{θ\in S^1}χ(Σ_θ)\geq \frac{1}{2}\int_{θ\in S^1}\int_{Σ_θ}(|du|^{-2}|Hess(u)|^2+R_M)$ relating the scalar curvature $R_M$ of $M$ to the average Euler characteristic of the level sets $Σ_θ=u^{-1}\{θ\}$. As our prima…

Monopole invariant studies contact structures on 3-manifolds.

New insights into knot surgeries via instanton 2-torsion.

Detects knots in thickened surfaces using instanton homology.

Detects torus knots using SL(2,C) representations and instanton Floer homology.

Study irreducible SU(2) representations for knots in 3D.

Third paper in series defines monopole Floer homology and gluing theorem for 3-manifolds.

This paper upgrades instanton TQFT to infinity-categories for better simplification.

Kronheimer and Mrowka recently suggested a possible approach towards a new proof of the four color theorem that does not rely on computer calculations. Their approach is based on a functor $J^\sharp$, which they define using gauge theory, from the category of webs and foams to the category of vector spaces over the fie…