arXiv research

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

The inclusion of the space of all knots of a prescribed writhe in a particular isotopy class into the space of all knots in that isotopy class is a weak homotopy equivalence.

Legendrian Lavrentiev links are shown to be equivalent to smooth links.

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…

We present a grid diagram analogue of Carter, Rieger and Saito's smooth movie theorem. Specifically, we give definitions for grid movies, grid movie isotopies and present a definition of grid planar isotopy as a particular subset of the grid diagram moves: stabilization, destabilization and commutation. We show that gr…

New distances defined on Legendrian spaces without positive loops.

Proves optimal regularity for sphere minimizers in 3-sphere.

Knots in 3-manifolds are equivalent if isotopic, except in special cases.

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…

This is the less official, English version of the proof of the fact that every closed atoroidal 3-manifold carries finitely many isotopy classes of tight contact structures.

For M_r = #_r(S^p \times S^p), p=3, 7, we calculate the group of isotopy classes of orientation preserving diffeomorphisms of $M_r$ modulo isotopy classes with representatives which are the identity outside a 2p-disc and also the group of homotopy classes of orientation preserving homotopy equivalences of M_r.

Proves any three or more knots can form a genus-zero link in a 3-manifold.

The paper defines a condition for the finiteness of mapping class groups of Heegaard splittings.

Veering triangulations link Thurston norm and isotopy of surfaces.

New algebraic theory classifies symplectic curves in complex projective space.

There is a natural way to associate with a transformation of an isotopy class of rational tangles to another, an element of the modular group. The correspondence between the isotopy classes of rational tangles and rational numbers follows, as well as the relation with the braid group $B_3$.

Defines Lorentzian distance on contactomorphisms, proving continuity and finite conditions.

An elementary stabilization of a Legendrian link $L$ in the spherical cotangent bundle $ST^*M$ of a surface $M$ is a surgery that results in attaching a handle to $M$ along two discs away from the image in $M$ of the projection of the link $L$. A virtual Legendrian isotopy is a composition of stabilizations, destabiliz…

In this paper, we establish compactness for various geometric curvature energies including integral Menger curvature, and tangent-point repulsive potentials, defined a priori on the class of compact, embedded $m$-dimensional Lipschitz submanifolds in ${\mathbb{R}}^n$. It turns out that due to a smoothing effect any seq…

Refined 1-cocycle for knots helps quantify isotopies.

New method for classifying disk embeddings in 4-manifolds.

Entangled embedded periodic nets and crystal frameworks are defined, along with their dimension type, homogeneity type, adjacency depth and periodic isotopy type. We obtain periodic isotopy classifications for various families of embedded nets with small quotient graphs. We enumerate the 25 periodic isotopy classes of …

New examples show limits of physical link isotopies.

The study finds non-simple isotopy classes of links in 3-manifolds, including Legendrian and pseudo-Legendrian examples.

Links in lens spaces may be defined to be equivalent by ambient isotopy or by diffeomorphism of pairs. In the first case, for all the combinatorial representations of links, there is a set of Reidemeister-type moves on diagrams connecting isotopy equivalent links. In this paper we provide a set of moves on disk, band a…

New unknots with geometric constraints exist, proving a long-standing conjecture.

Embeddings of surfaces in 5-manifolds are classified by an invariant.

New algebraic structure for 2-string links and long knots.

We define a suitably tame class of singular symplectic curves in 4-manifolds, namely those whose singularities are modeled on complex curve singularities. We study the corresponding symplectic isotopy problem, with a focus on rational curves with irreducible singularities (rational cuspidal curves) in the complex proje…

Two elements generate extended mapping class groups of certain surfaces.

New transformations preserve link isotopy, showing complexity differences.

Proves a theorem for comparing surfaces in 3D space.

Scharlemann constructed a connected simplicial 2-complex $Γ$ with an action by the group ${\mathcal H_{2}}$ of isotopy classes of orientation preserving homeomorphisms of $S^3$ that preserve the isotopy class of an unknotted genus 2 handlebody $V$. In this paper we prove that the 2-complex $Γ$ is contractible. Therefor…

We show that given a partially flat angled ideal triangulation for a 3-manifold $M$ with boundary (as defined by Lackenby), there is an algorithm to produce a list of Heegaard splittings for $M$ such that below a given genus $g$, each isotopy class appears exactly once. In particular, this algorithm determines precisel…

Tait's flyping conjecture, stating that two reduced, alternating, prime link diagrams can be connected by a finite sequence of flypes, is extended to reduced, alternating, prime diagrams of 4-regular graphs in S^3. The proof of this version of the flyping conjecture is based on the fact that the equivalence classes wit…

We consider two pairs: the standard unknotted $n$-sphere in $S^{n+2}$, and the product of two $p$-spheres trivially embedded in $S^{2p+2}$, and study orientation preserving diffeomorphisms of these pairs. Pseudo-isotopy classes of such diffeomorphisms form subgroups of the mapping class groups of $S^n$ and $S^p\times S…

New relation found in 4D symplectic mapping class group.

In this paper we classify, up to rigid isotopy, non-singular real rational curves of degrees less than or equal to 6 in a quadric homeomorphic to the 3-sphere. We also study their connections with rigid isotopy classes of real rational knots in $\mathbb{RP}^3$.

Fractional Dehn twists give a measure of the difference between the relative isotopy class of a homeomorphism of a bordered surface and the Thurston representative of its free isotopy class. We show how to estimate and compute these invariants. We discuss the the relationship of our work to stabilization problems in cl…

We give a combinatorial description of the Legendrian differential graded algebra associated to a Legendrian knot in PxR, where P is a punctured Riemann surface. As an application we show that for any integer k and any homology class h in H_1(PxR) there are k Legendrian knots all representing h which are pairwise smoot…

Smoothly isotopic surfaces in a 4-manifold are stable isotopic if trivial in the 2-homology.

Proves the bending map is proper for hyperbolic 3-manifolds.

Projection maps virtual Legendrian knots to classical ones.

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

New findings on hyperbolicity of fine curve graphs and their subgraphs.

David Gabai recently proved a smooth 4-dimensional "Light Bulb Theorem" in the absence of 2-torsion in the fundamental group. We extend his result to 4-manifolds with arbitrary fundamental group by showing that an invariant of Mike Freedman and Frank Quinn gives the complete obstruction to "homotopy implies isotopy" fo…

The paper studies how knots and links behave under connected sum operations.

New link found that can't be smoothly deformed.