arXiv research

New exotic 4D spaces found using knot slicing techniques.

We construct maps on hat Heegaard Floer homology for cobordisms decorated with graphs. The graph TQFT allows for cobordisms with disconnected ends. Our construction uses Juhász's sutured Floer TQFT. We compute the maps for several elementary graph cobordisms. As an application, we compute the action of the fundamental …

Study corrects previous work on knot Floer homology of certain pretzel knots.

Using Taubes' periodic ends theorem, Auckly gave examples of toroidal and hyperbolic irreducible integer homology spheres which are not surgery on a knot in the three-sphere. We give an obstruction to a homology sphere being surgery on a knot coming from Heegaard Floer homology. This is used to construct infinitely man…

We construct a graph TQFT for the minus flavor of Heegaard Floer homology. Our graph TQFT extends Ozsváth and Szabó's TQFT for closed and connected 3-manifolds, and allows for cobordisms with disconnected ends. As an application, we give an explicit formula for the chain homotopy type of the $π_1$-action on Heegaard Fl…

The study distinguishes exotic R^4's using knot Floer homology and Casson handles.

Proves irreducible SU(2) representations for 3-surgery knots.

We give a complete classification of the spherical 3-manifolds that bound smooth rational homology 4-balls. Furthermore, we determine the order of spherical 3-manifolds in the rational homology cobordism group of rational homology 3-spheres. To this end, we use constraints for 3-manifolds to bound rational homology bal…

We investigate symmetry properties of peculiar modules, a Heegaard Floer invariant of 4-ended tangles which the author introduced in [arXiv:1712.05050]. In particular, we give an almost complete answer to the geography problem for components of peculiar modules of tangles. As a main application, we show that Conway mut…

Constructs exotic proper 2-knots from open 2-handles.

With a 4-ended tangle $T$, we associate a Heegaard Floer invariant $\operatorname{CFT^\partial}(T)$, the peculiar module of $T$. Based on Zarev's bordered sutured Heegaard Floer theory, we prove a glueing formula for this invariant which recovers link Floer homology $\operatorname{\widehat{HFL}}$. Moreover, we classify…

Extends Khovanov homology spectral sequence using Heegaard Floer homology.

Study links with annuli using sutured Floer homology.

Knot Floer homology matches fixed point Floer for fibred knots.

New link detection results using knot and link Floer homology.

Paper classifies Heegaard Floer minimal knots in sutured manifolds.

New colored knot Floer homology defined using infinite full twists.

New algebraic method for knot Floer homology computation.

Link Floer homology detects split links.

Lecture notes on Heegaard Floer homology for 3-manifolds and knots.

These lecture notes, which were designed for the Summer School "Heegaard-Floer Homology and Khovanov Homology" in Marseilles, 29th May - 2nd June, 2006, provide an elementary introduction to Khovanov homology. The intended audience is graduate students with some minimal background in low-dimensional and algebraic topol…

Maps from rational homology solid tori yield rank inequalities in Heegaard Floer homology.

Study volume growth in Milnor fibers using real Lagrangians.

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

Study on knot concordance and homology cobordism using Heegaard Floer homology.

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

Lecture notes on Heegaard Floer homology for beginners.

Study shows rank of knot Floer homology detects Hopf links and classifies second smallest links.

Introduces Floer lasagna modules using link Floer homology.

The purpose of this thesis is to define a "local" version of Ozsváth and Szabó's Heegaard Floer homology $\operatorname{\widehat{HFL}}$ for links in the 3-dimensional sphere, i.e. a Heegaard Floer homology $\operatorname{\widehat{HFT}}$ for tangles in the closed 3-ball. After studying basic properties of $\operatorname…

Real Heegaard Floer Homology extends Li's real monopole Floer homology.

Study shows inequality in Floer homologies for 3-manifold covers.

New method for Lagrangian Floer homology groups using flow trees.

Study Brieskorn spheres using Floer homology, generating infinite rank summands in homology cobordism.

Contact gluing maps are shown to be equivalent in sutured Floer homology.

Formula for Heegaard Floer multicurves of double tangles from knot complements.

Survey of bordered Floer homology for 3-manifolds with boundary.

Paper extends link Floer homology detection to almost braided links.

Heegaard Floer homology study confirms composition maps match up to homotopy.

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

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…

Identifies Heegaard Floer homology solid tori via Dehn fillings.

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 of knot Floer homology and its relation to Heegaard Floer homology via equivariant surgery.

In this paper, we construct the Rabinowitz-Floer homology for the coupled Dirac system \begin{equation*} \left\{ \begin{aligned} Du=\frac{\partial H}{\partial v}(x,u,v)\hspace{4mm} {\rm on} \hspace{2mm}M,\\ Dv=\frac{\partial H}{\partial u}(x,u,v)\hspace{4mm} {\rm on} \hspace{2mm}M, \end{aligned} \right. \end{equation*}…

We derive symmetries and adjunction inequalities of the knot Floer homology groups which appear to be especially interesting for homologically essential knots. Furthermore, we obtain an adjunction inequality for cobordism maps in knot Floer homologies. We demonstrate the adjunction inequalities and symmetries in explic…

For any three-manifold presented as surgery on a framed link (L,Λ) in an integral homology sphere, Manolescu and Ozsváth construct a hypercube of chain complexes whose homology calculates the Heegaard Floer homology of Λ-framed surgery on Y. This carries a natural filtration that exists on any hypercube of chain comple…

Geometric interpretation of tangle invariants using immersed curves.