arXiv research

New exotic 4D spaces found using knot slicing techniques.

The paper compares isotopic and diffeomorphic links in lens spaces.

This paper classifies links in 3D dynamical systems.

In this paper we investigate the relationship between isotopy classes of knots and links in S^3 and the diffeomorphism types of homeomorphic smooth 4-manifolds. As a corollary of this initial investigation, we begin to uncover the surprisingly rich structure of diffeomorphism types of manifolds homeomorphic to the K3 s…

The abstract link L_d of the complex isolated singularity x^2 + y^2 + z^2 + v^{2d} = 0 is diffeomorphic to S^3 \times S^2. We classify the embedded links of these singularities up to regular homotopies precomposed with diffeomorphisms of S^3 \times S^2. Let us denote by i_d the inclusion of L_d in S^7. We show that for…

Let $n \geq 2$. We prove a homological stability theorem for the diffeomorphism groups of $(4n+1)$-dimensional manifolds, with respect to forming the connected sum with $(2n-1)$-connected, $(4n+1)$-dimensional manifolds that are stably parallelizable. Our techniques involve the study of the action of the diffeomorphism…

New invariants defined for volume-preserving flows on 3-manifolds.

The paper proves infinitely many strong symplectic fillings for cusp singularity links.

The paper constructs many knotted and linked objects in higher dimensions.

Using techniques from the theory of Kirby calculus we give an explicit construction of a four dimensional hyperbolic link complement in a 4-manifold that is diffeomorphic to the standard 4-sphere.

Invariants from Seiberg-Witten equations link scalar curvature to diffeomorphisms.

Khovanov homology invariant proved for links in $\mathbb{RP}^3$.

Solves Cauchy problem for a KP hierarchy on non-formal operators and relates to diffeomorphisms.

Using techniques from the theory of Kirby calculus we give an explicit construction of a four dimensional hyperbolic link complement in a 4-manifold that is diffeomorphic to a standard $S^2 \times S^2$.

New modules derived from Khovanov homology for links.

Let $(X^{m+1}, g)$ be a globally hyperbolic spacetime with Cauchy surface diffeomorphic to an open subset of $\mathbb R^m$. The Legendrian Low conjecture formulated by Natário and Tod says that two events $x,y\inß$ are causally related if and only if the Legendrian link of spheres $\mathfrak S_x, \mathfrak S_y$ whose p…

We analyze the effect of adding, removing, and moving basepoints on link Floer homology. We prove that adding or removing basepoints via a procedure called quasi-stabilization is a natural operation on a certain version of link Floer homology, which we call $CFL_{UV}^\infty$. We consider the effect on the full link Flo…

Fintushel-Stern's knot surgery gave many pairs of exotic manifolds, which are homeomorphic but non-diffeomorphic. We show that if an elliptic fibration has two parallel, oppositely oriented vanishing circles (for example $S^2\times S^2$ or Matsumoto's $S^4$), then the knot surgery gives rise to standard manifolds. The …

The flag curvature of the Numata Finsler structures is shown to admit a nontrivial prolongation to the one-dimensional case, revealing an unexpected link with the Schwarzian derivative of the diffeomorphisms associated with these Finsler structures.

Starting from a divide, i.e. a generic immersion of finitely many copies of the interval [0,1] in the disk, we construct a classical link in the 3-sphere. We prove that the link's complement fibers over the circle, if the divide is connected. Moreover, we compute the monodromy diffeomorphism from the combinatorics of t…

Study pochette surgery on 4-manifolds, focusing on 4-spheres.

Study reveals a link between Ruelle-Pollicott resonances and cohomology eigenvalues for Anosov diffeomorphisms.

The set N of all null geodesics of a globally hyperbolic (d+1)-dimensional spacetime (M,g) is naturally a smooth (2d-1)-dimensional contact manifold. The sky of an event is the subset of N defined by all null geodesics through that event, and is an embedded Legendrian submanifold of N diffeomorphic to a (d-1)-dimension…

We slightly extend the notion of a natural fibre bundle by requiring diffeomorphisms of the base to lift to automorphisms of the bundle only infinitesimally, i.e. at the level of the Lie algebra of vector fields. Spin structures are natural only in this extended sense. We classify fibre bundles with this property, assu…

Divides help construct fibered links from singularities.

The abstract discusses detecting knotted spheres through their traces in high dimensions.

Theory is developed for linear-quadratic at infinity generating families for Legendrian knots in R^3. It is shown that the unknot with maximal Thurston--Bennequin invariant of -1 has a unique linear-quadratic at infinity generating family, up to fiber-preserving diffeomorphism and stabilization. From this, invariant ge…

Study shows boundary of Milnor fibre is invariant for certain singularities.

The paper provides a link between ergodic theory and symplectic topology. A classical notion of ergodic theory is a skew product map associated with a loop in a group of transformations. We study skew products which come from loops in the group of Hamiltonian diffeomorphisms of a symplectic manifold. Our main question …

Study confirms conjecture linking 4-spheres, finds new spheres.

A theorem of Kirby gives a necessary and sufficient condition for two framed links in S^3 to yield orientation-preserving diffeomorphic results of surgery. Kirby's theorem is an important method for constructing invariants of 3-manifolds. In this paper, we prove a variant of Kirby's theorem for null-homologous framed l…

Study equivariant isotopy in higher dimensions, finding exceptions.

In this note we present a new definition of the 4-manifold admitting inequivalent symplectic structures constructed by McMullen-Taubes which leads to the identification of a new symplectic structure. We prove moreover that it is diffeomorphic to one of the link surgery manifolds introduced by Fintushel-Stern.

The paper develops a bordered $\mathit{HF}^-$ theory using link surgery formula.

In this work we define and study the relations between Lorentzian Manifolds given by the diffeomorphisms which map causal future directed vectors onto causal future directed vectors. This class of diffeomorphisms, called proper causal relations, contains as a subset the well-known group of conformal relations and are d…

Proves new inequality linking spectral numbers of Lagrangians and their reductions.

Study on four-dimensional Dehn twists and Milnor fibrations, revealing new phenomena.

New classes defined for manifold pseudogroups, linking to cohomology and bundle structures.

We show that all versions of Heegaard Floer homology, link Floer homology, and sutured Floer homology are natural. That is, they assign concrete groups to each based 3-manifold, based link, and balanced sutured manifold, respectively. Furthermore, we functorially assign isomorphisms to (based) diffeomorphisms, and show…

The paper connects diffeomorphism groups and sphere embeddings, proving a group structure.

Study reveals failure of uniqueness in dynamical invariants for 3D volume-preserving diffeomorphisms.

In a recent paper of Akhmedov, Etnyre, Mark and Smith, it was shown that there exist infinitely many contact Seifert fibered 3-manifolds each of which admits infinitely many exotic (homeomorphic but pairwise non-diffeomorphic) simply-connected Stein fillings. Here we extend this result to a larger set of contact Seifer…

New homotopy types defined for links in thickened surfaces with higher genus.

New topological object connects complex dynamics and topology.

The paper proves critical point results for Frechet manifolds.

A dual pair is constructed for contact groups, linking submanifolds and orbits.

Knot lattice homology invariant of smooth knot type in rational homology spheres.

Perelman's proof confirmed, new method uses 4D topology.