arXiv research

The study classifies isotopy types of 3-periodic nets and their embeddings.

This paper studies isotopies of periodic tangles in 3-manifolds using finite covers.

Paper defines and analyzes mathematical equivalence of periodic tangles.

This paper investigates dynamics that persist under isotopy in classes of orientation-preserving homeomorphisms of orientable surfaces. The persistence of periodic points with respect to periodic and strong Nielsen equivalence is studied. The existence of a dynamically minimal representative with respect to these relat…

Paper characterizes isotopies and hyperbolicity of weaves using geodesics.

We study periodic wind-tree models, billiards in the plane endowed with $\mathbb{Z}^2$-periodically located identical connected symmetric right-angled obstacles. We show asymptotic formulas for the number of (isotopy classes of) closed billiard trajectories (up to $\mathbb{Z}^2$-translations) on the wind-tree billiard.…

New diagrams classify triply periodic entanglements.

A mapping class group of an oriented manifold is a quotient of its diffeomorphism group by the isotopies. We compute a mapping class group of a hypekahler manifold $M$, showing that it is commensurable to an arithmetic subgroup in SO(3, b_2-3). A Teichmuller space of $M$ is a space of complex structures on $M$ up to is…

In this article, we study the knots realized by periodic orbits of R-covered Anosov flows in compact 3-manifolds. We show that if two orbits are freely homotopic then in fact they are isotopic. We show that lifts of periodic orbits to the universal cover are unknotted. When the manifold is atoroidal, we deduce some fin…

The mapping class group of a Heegaard splitting is the group of automorphisms of the ambient 3-manifold that take the surface onto itself, modulo isotopies that keep the surface on itself. We characterize the mapping classes that restrict to periodic and reducible automorphisms of the surface.

Two elements generate extended mapping class groups of certain surfaces.

The purpose of this paper is to construct and study equivariant Khovanov homology - a version of Khovanov homology theory for periodic links. Since our construction works regardless of the characteristic of the coefficient ring it generalizes a previous construction by Chbili. We establish invariance under equivariant …

Links with isotopic preimages in $S^3$ are isotopic in $RP^3$.

We consider a parallelizable $2n$-manifold $F$ which has the homotopy type of the wedge product of $n$-spheres and show that the group of pseudo-isotopy classes of orientation preserving diffeomorphisms that keep the boundary $\partial F$ pointwise fixed and induce the trivial variation operator is a central extension …

Extends tangle theory to include undetermined crossings in periodic structures.

We classify isotopy classes of automorphisms (self-homeomorphisms) of 3-manifolds satisfying the Thurston Geometrization Conjecture. The classification is similar to the classification of automorphisms of surfaces developed by Nielsen and Thurston, except an automorphism of a reducible manifold must first be written as…

We present a technique for the enumeration of all isotopically distinct ways of tiling a hyperbolic surface of finite genus, possibly nonorientable and with punctures and boundary. This provides a generalization of the enumeration of Delaney-Dress combinatorial tiling theory on the basis of isotopic tiling theory. To a…

We construct an isotopy of a planar compactum that is not the restriction of an isotopy of any planar continuum.

A braid-like isotopy for links in 3-space is an isotopy which uses only those Reidemeister moves which occur in isotopies of braids. We define a refined Jones polynomial and its corresponding Khovanov homology which are, in general, only invariant under braid-like isotopies.

New obstructions show some 4-manifold homeomorphisms are pseudo-isotopic but not isotopic.

Approaches 4D Schoenflies via pseudo-isotopy.

In this article we analyze totally periodic pseudo-Anosov flows in graph three manifolds. This means that in each Seifert fibered piece of the torus decomposition, the free homotopy class of regular fibers has a finite power which is also a finite power of the free homotopy class of a closed orbit of the flow. We show …

Classifies curves up to symplectic isotopy.

In this paper we solve the following problem in the affirmative: Let $Z$ be a continuum in the plane $\complex$ and suppose that $h:Z\times [0,1]\to\complex$ is an isotopy starting at the identity. Can $h$ be extended to an isotopy of the plane? We will provide a new characterization of an accessible point in a planar …

In the first part of this dissertation, we give a new definition of a Laplace operator for Finsler metric as an average, with regard to an angle measure, of the second directional derivatives. This operator is elliptic, symmetric with respect to the Holmes-Thompson volume, and coincides with the usual Laplace--Beltrami…

Smooth isotopy on cube saves energy with extra dimensions.

A star-like isotopy for oriented links in 3-space is an isotopy which uses only Reidemeister II moves with opposite orientations and Reidemeister III moves with alternating orientations when checking the strands clockwise (or anticlockwise). We define a link polynomial derived from the Jones polynomial which is, in gen…

A long-standing open problem in systolic geometry asks whether a Riemannian metric on the real projective space whose volume equals that of the canonical metric, but is not isometric to it, must necessarily carry a periodic geodesic of length smaller than π. A contact-geometric reformulation of systolic geometry and th…

Refined 1-cocycle for knots helps quantify isotopies.

Classifies isotopy classes of links from Thompson's group F and its subgroup.

Uniform convergence of isotopies implies ambient isotopy, aiding knot equivalence.

Proves a Thom Isotopy Theorem for nonproper semialgebraic maps.

Minimal surfaces and curves can have singularities removed by isotopy.

Improved proofs for topological and smooth pseudo-isotopies of simply connected 4-manifolds.

The study constructs examples of pseudo-isotopic but not isotopic 4-manifold diffeomorphisms.

In this paper we define and study flexible links and flexible isotopy in projective space. Flexible links are meant to capture the topological properties of real algebraic links. We classify all flexible links up to flexible isotopy using Ekholms interpretation of Viros encomplexed writhe.

Veering triangulations link Thurston norm and isotopy of surfaces.

Proof confirms Hamiltonian isotopy of submanifolds.

The symplectic isotopy conjecture states that every smooth symplectic surface in $CP^2$ is symplectically isotopic to a complex algebraic curve. Progress began with Gromov's pseudoholomorphic curves [Gro85], and progressed further culminating in Siebert and Tian's proof of the conjecture up to degree 17 [ST05], but fur…

This paper introduces two virtual knot theory ``analogues'' of a well-known family of invariants for knots in thickened surfaces: the Grishanov-Vassiliev finite-type invariants of order two. The first, called the three loop isotopy invariant, is an invariant of virtual knots while the second, called the three loop fram…

We construct counterexamples to lifting properties of Hamiltonian and contact isotopies.

Proof of isotopy invariance in Khovanov link homology.

In this paper we survey recent developments in the classical theory of minimal surfaces in Euclidean spaces which have been obtained as applications of both classical and modern complex analytic methods; in particular, Oka theory, period dominating holomorphic sprays, gluing methods for holomorphic maps, and the Rieman…

Classifies smooth isotopy of a specific Lagrangian embedding.

We define an equivalence relation called A-isotopy between finitely determined map-germs, which is a strengthened version of A-equivalence. We consider the number of A-isotopy classes of equidimensional Morin singularities, and some other well-known low-dimensional singularities. We also give an application to stable p…

Study isotopy and concordance for metrics on manifolds with boundary.

Floer theory connects dynamics on surfaces to their chain-level theory.

Constructs pseudo-isotopies with Cerf diagrams and Hatcher-Wagoner invariants for barbell maps.