arXiv research

Here we discuss an example of topologically isotopic but smoothly non-isotopic pair of 2-spheres in a simply connected 4-manifold, which become smoothly isotopic after stabilizing by connected summing with S^2 x S^2.

The paper finds infinitely many 4-manifolds with non-isotopic sections.

Study finds non-isotopic transverse tori in Engel manifolds.

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

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

New ribbon disks in 4D space, non-isotopic to each other.

We construct infinite families of topologically isotopic but smoothly distinct knotted spheres in many simply connected 4-manifolds that become smoothly isotopic after stabilizing by connected summing with $S^2 \times S^2$, and as a consequence, analogous families of diffeomorphisms and metrics of positive scalar curva…

We construct infinitely many smooth oriented 4-manifolds containing pairs of homotopic, smoothly embedded 2-spheres that are not topologically isotopic, but that are equivalent by an ambient diffeomorphism inducing the identity on homology. These examples show that Gabai's recent "Generalized" 4D Lightbulb Theorem does…

Low entropy hypersurfaces in 4D are isotopic to a sphere.

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

We construct a pair of isotopic link configurations that are not thick isotopic while preserving total length.

New spheres can split a 4D link in ways not possible in 3D.

New non-isotopic Seifert surfaces found in 4-ball.

We use the methods of Hedden, Juhasz, and Sarkar to exhibit a set of arborescent knots that bound large numbers of non-isotopic minimal genus spanning surfaces. In particular, we describe a sequence of prime knots K_{n} which will bound at least 2^{2n-1} non-isotopic minimal spanning surfaces of genus n.

In this paper, we study stable equivalence of exotically knotted surfaces in 4-manifolds, surfaces that are topologically isotopic but not smoothly isotopic. We prove that any pair of embedded surfaces in the same homology class become smoothly isotopic after stabilizing them by handle additions in the ambient 4-manifo…

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

Given an $m$-component link $L$ in $S^3$ ($m \ge 2$), we construct a family of links which are link homotopic, but not link isotopic, to $L$. Every proper sublink of such a link is link isotopic to the corresponding sublink of $L$. Moreover, if $L$ is an unlink then there exist links that in addition to the above prope…

We give examples of contactomorphisms in every dimension that are smoothly isotopic to the identity but that are not contact isotopic to the identity. In fact, we prove the stronger statement that they are not even symplectically pseudo-isotopic to the identity. We also give examples of pairs of contactomorphisms which…

Consider a smooth $4$-manifold $X$ and a diffeomorphism $f : X \to X$. We give an obstruction in the form of an adjunction inequality for an embedded surface in $X$ to be isotopic to its image under $f$. It follows that the minimal genus of a surface representing a given homology class and which is isotopic to its imag…

The study examines the realizability of a 4-manifold invariant for homeomorphisms.

We use microlocal sheaf theory to show that if two knots have Legendrian isotopic conormal tori, then the knots are isotopic or mirror images.

For any pair of integers $n\geq 1$ and $q\geq 2$, we construct an infinite family of mutually non-isotopic symplectic tori representing the homology class $q[F]$ of an elliptic surface E(n), where $[F]$ is the homology class of the fiber. We also show how such families can be non-isotopically and symplectically embedde…

We considered a surgery, called Lagrangian attaching disk surgery, that can be applied to a Lagrangian surface L at the presence of a Lagrangian attaching disk D, to obtain a new Lagrangian surface L' which is always smoothly isotopic to L. We showed that this type of surgery includes all even generalized Dehn twists a…

For each member of an infinite family of homology classes in the K3-surface E(2), we construct infinitely many non-isotopic symplectic tori representing this homology class. This family has an infinite subset of primitive classes. We also explain how these tori can be non-isotopically embedded as homologous symplectic …

Every knot can be transformed into an unknot without cutting.

New gauge theory invariant detects non-smooth isotopy of $P^2$-knots.

Manufacturing infinite sets of knotted and unknotted surfaces in 4-manifolds.

We show that after generic filling along a torus boundary component of a 3-manifold, no two closed, 2-sided, essential surfaces become isotopic, and no closed, 2-sided, essential surface becomes inessential. That is, the set of essential surfaces (considered up to isotopy) survives unchanged in all suitably generic Deh…

Paper presents a method for identifying isotope envelopes in MALDI-ToF data.

Infinitely many hyperbolic links in lens space have isotopic lifts in 3-sphere.

Study weak conjugacy in surface homeomorphisms.

New surfaces in a 4-manifold are found that are not isotopic but homotopic.

We construct (infinitely many) examples in all dimensions of contactomorphisms of closed overtwisted contact manifolds that are smoothly isotopic but not contact-isotopic to the identity.

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

We prove a neighbourhood theorem for arbitrary knots in contact 3-manifolds. As an application we show that two topologically isotopic Legendrian knots in a contact 3-manifold become Legendrian isotopic after suitable stabilisations.

New knots bound multiple non-isotopic ribbon disks.

New proof associates partitions to isotopic pseudo-Anosov homeomorphisms.

In this paper, we construct the first families of distinct Lagrangian ribbon disks in the standard symplectic 4-ball which have the same boundary Legendrian knots, and are not smoothly isotopic or have non-homeomorphic exteriors.

Study equivariant isotopy in higher dimensions, finding exceptions.

In this paper we give examples of closed smooth submanifolds of RP^n which are isotopic to nonsingular projective subvarieties of RP^n but they can not be isotopic to the real parts of nonsingular complex projective subvarieties of CP^n.

Study on topological pseudo-isotopies of 4-manifolds, proving some cases and finding counterexamples.

The purpose of this short note is to prove the uniqueness of Hamiltonian volume minimizing Lagrangian submanifolds which are Hamiltonian isotopic to RP^n in CP^n modulo isometric group actions.

Let E(1)_K denote the closed 4-manifold that is homotopy equivalent (hence homeomorphic) to the rational elliptic surface E(1) and is obtained by performing Fintushel-Stern knot surgery on E(1) using a knot K in S^3. We construct an infinite family of homologous non-isotopic symplectic tori representing a primitive hom…

New 3D handlebodies in 4-sphere and 5-ball are not isotopic even with same boundary.

Single stabilization is not always enough to make exotic surfaces isotopic.

Basket links are shown to be isotopic to $T(2,n+1)$.

We construct an infinite family of homologous, non-isotopic, symplectic surfaces of any genus greater than one in a certain class of closed, simply connected, symplectic four-manifolds. Our construction is the first example of this phenomenon for surfaces of genus greater than one.

We prove that two Legendrian knots in a contact structure which is trivializable as a plane bundle are Legendrian isotopic provided that (1) they are isotopic as framed knots, (2) they have the same rotation number with respect to some parallelization of the contact structure, and (3) there is an overtwisted disk disjo…