arXiv research

The study sets constraints on 4-manifold forms linked to specific invariants.

The paper studies hyperbolic three-manifolds and their geometric constraints.

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…

This note explores two questions: (1) Which bigraded groups arise as the knot Floer homology of a knot in the three-sphere? (2) Given a knot, how many distinct knots share its Floer homology? Regarding the first, we show there exist bigraded groups satisfying all previously known constraints of knot Floer homology whic…

Classifies links with small Khovanov homology ranks.

We investigate constraints on embeddings of a non-orientable surface in a $4$-manifold with the homology of $M \times I$, where $M$ is a rational homology $3$-sphere. The constraints take the form of inequalities involving the genus and normal Euler class of the surface, and either the Ozsváth--Sazbó $d$-invariants or …

The ``classical BRST construction'' as developed by Batalin-Fradkin-Vilkovisky is a homological construction for the reduction of the Poisson algebra $P = C^\infty (W)$ of smooth functions on a Poisson manifold $W$ by the ideal $I$ of functions which vanish on a constraint locus. This ideal is called first class if $I$…

Manolescu correction terms are numerical invariants of homology three-spheres arising from $\mathrm{Pin}(2)$-equivariant Seiberg-Witten theory that contain information about homology cobordism. We discuss several constraints on these invariants for homology spheres obtained by Dehn surgery on a knot in the three-sphere…

Study cosmetic surgeries on knots in homology spheres using Casson-Walker invariant.

Solved a conjecture about rational homology projective planes with quotient singularities.

Graph manifolds with small homology have non-trivial SU(2) representations.

The paper calculates involutive Heegaard Floer homology for specific 3-manifolds.

We consider the question of which Dehn surgeries along a given knot bound rational homology balls. We use Ozsváth and Szabó's correction terms in Heegaard Floer homology to obtain general constraints on the surgery coefficients. We then turn our attention to the case of integral surgeries, with particular emphasis on p…

Study shows most knots up to 10 crossings can't be chirally cosmetic.

The study shows a 3D manifold's macroscopic dimension is 1 under specific curvature constraints.

Study Markov staircases in symplectic embeddings of rational homology ellipsoids.

Using the conjugation symmetry on Heegaard Floer complexes, we define a three-manifold invariant called involutive Heegaard Floer homology, which is meant to correspond to $\mathbb{Z}_4$-equivariant Seiberg-Witten Floer homology. Further, we obtain two new invariants of homology cobordism, $\underline{d}$ and $\bar{d}$…

We give multiplicity results for the solutions of a nonlinear elliptic equation, with an asymmetric double well potential of Van der Waals-Allen--Cahn--Hilliard type, satisfying a linear volume constraint, on a bounded Lipschitz domain $Ω\subset\mathds R^N$. The number of solutions is estimated in terms of topological …

Study extends contact cosmetic surgeries to non-trivial Legendrian knots in L-spaces.

We compute Khovanov homology for tangles using TQFT.

The study limits the number of cosmetic surgeries for certain knots in specific 3-manifolds.

We prove a Smith-type inequality for regular covering spaces in monopole Floer homology. Using the monopole Floer / Heegaard Floer correspondence, we deduce that if a 3-manifold Y admits a p^n-sheeted regular cover that is a Z/pZ-L-space (for p prime), then Y is a Z/pZ-L-space. Further, we obtain constraints on surgeri…

Developed Gompf connected sum for orbifolds, constructing symplectic and K-contact manifolds.

A functorial semi-norm on singular homology is a collection of semi-norms on the singular homology groups of spaces such that continuous maps between spaces induce norm-decreasing maps in homology. Functorial semi-norms can be used to give constraints on the possible mapping degrees of maps between oriented manifolds. …

Many methods for reducing and simplifying differential equations are known. They provide various generalizations of the original symmetry approach of Sophus Lie. Plenty of relations between them have been noticed and in this note a unifying approach will be discussed. It is rather close to the classical differential co…

We focus on L-spaces for which the boundary maps of the Heegaard Floer chain complexes vanish. In previous paper \cite{Usui}, we collect such manifolds systematically by using the smoothing order on links. In this paper, we classify such L-spaces under appropreate constraint.

We show that, if a rational homology 3-sphere $Y$ bounds a positive definite smooth 4-manifold, then there are finitely many negative definite lattices, up to the stable-equivalence, which can be realized as the intersection form of a smooth 4-manifold bounded by $Y$. To this end, we make use of constraints on definite…

We use invariants of Hendricks and Manolescu coming from involutive Heegaard Floer theory to find constraints on possible configurations of singular points of a rational cuspidal curve of odd degree in the projective plane. We show that the results do not carry over to rational cuspidal curves of even degree.

This paper shows that only finitely many knots can be ribbon concordant to any given knot.

Defines a Rasmussen invariant for links in RP^3.

Study on contact type hypersurfaces in 4-space, proving no Brieskorn spheres can embed.

Suppose C is a singular curve in CP^2 and it is topologically an embedded surface of genus g; such curves are called cuspidal. The singularities of C are cones on knots K_i. We apply Heegaard Floer theory to find new constraints on the sets of knots {K_i} that can arise as the links of singularities of cuspidal curves.…

Embeds 3-manifolds in symplectic 4-manifolds with constraints.

In an earlier paper, we used the absolute grading on Heegaard Floer homology to give restrictions on knots in $S^3$ which admit lens space surgeries. The aim of the present article is to exhibit stronger restrictions on such knots, arising from knot Floer homology. One consequence is that all the non-zero coefficients …

Let Y be a closed oriented 3-manifold with a contact form such that all Reeb orbits are nondegenerate. The embedded contact homology (ECH) index associates an integer to each relative 2-dimensional homology class of surfaces whose boundary is the difference between two unions of Reeb orbits. This integer determines the…

Study slopes on knot manifolds to understand their fundamental groups.

Defines a new Upsilon torsion function for knot Floer homology.

In relation to the 4-dimensional smooth Poincaré conjecture we construct a tentative invariant of homotopy 4-spheres using embedded contact homology (ECH) and Seiberg-Witten theory (SWF). But for good reason it is a constant value independent of the sphere, so this null-result demonstrates that one should not try to us…

Fold maps are higher dimensional versions of Morse functions and fundamental and important tools in studying algebraic and differential topological properties of manifolds: as the theory established by Morse and the higher dimensional version, started by Thom and Whitney, later actively studied by Eliashberg, Levine et…

The study connects ECH capacities to Anosov flows, proving infinite capacities and obstructions.

In two seminal papers Kontsevich used a construction called_graph homology_ as a bridge between certain infinite dimensional Lie algebras and various topological objects, including moduli spaces of curves, the group of outer automorphisms of a free group, and invariants of odd dimensional manifolds. In this paper, we s…

TopInG improves graph interpretability using persistent homology.

Study PL bordism theories with quantitative bounds on filling simplices.

Arithmetic study of knots connects homology and SL2 representations.

Topology-GS improves 3D GS for better structural and feature integrity.

Lengths of simple closed geodesics on hyperbolic surfaces in prescribed homology classes

High throughput sequencing techniques have highly impactedon modern biology, widening the gap between sequenced andannotated data. Automatic annotation tools are thereforeof the foremost importance to guide biologists' experiments. However, most of the state-of-the-art methods rely on annotation transfer, offering reli…

The paper explores geometric relationships in manifolds with curvature constraints, proving new inequalities and rigidity results.