The symplectic isotopy conjecture states that every smooth symplectic surface in 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…
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48 results for “isotopy problem”
Improved proofs for topological and smooth pseudo-isotopies of simply connected 4-manifolds.
problem Proving topological and smooth pseudo-isotopies of simply connected 4-manifolds.
method Provided different arguments that bypass the replacement criterion, thus completing Quinn's proofs.
result Corrected and completed Quinn's proofs of both topological and stable smooth pseudo-isotopy theorems.
In this paper we solve the following problem in the affirmative: Let 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 be extended to an isotopy of the plane? We will provide a new characterization of an accessible point in a planar …
New examples show limits of physical link isotopies.
problem Limits of physical link isotopies in thick and thickly embedded links.
method Construction of specific examples of thick and thickly embedded links.
result Explicit examples of links that cannot be split through thick homotopies.
Finite type invariants separate PL links in 3D space.
problem If two PL links are isotopic, are they PL isotopic?
method Proved PL isotopic links are indistinguishable by finite type invariants.
result Finite type invariants separate PL links in 3D space if and only if certain conjectures hold.
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…
The paper solves isotopy problems on 4-manifolds and classifies symplectic structures.
problem Isotopy problems on 4-manifolds and uniqueness of symplectic structures.
method Generalization of Dax invariant to embedded closed surfaces, symplectic topology techniques.
result Infinitely many non-isotopic symplectic forms on irrational ruled surfaces.
We give an overview of various recent results concerning the topology of symplectic 4-manifolds and singular plane curves, using branched covers and isotopy problems as a unifying theme. While this paper does not contain any new results, we hope that it can serve as an introduction to the subject, and will stimulate in…
We construct an isotopy of a planar compactum that is not the restriction of an isotopy of any planar continuum.
The deformation problem for pseudoholomorphic curves and related geometrical properties of the total moduli space of pseudoholomorphic curves are studied. A sufficient condition for the saddle point property of the total moduli space is established. The local symplectic isotopy problem is formulated and solved for the …
The Markov Theorem Without Stabilization (MTWS) established the existence of a calculus of braid isotopies that can be used to move between closed braid representatives of a given oriented link type without having to increase the braid index by stabilization. Although the calculus is extensive there are three key isoto…
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.
problem Tackling the difference between pseudo-isotopy and isotopy for 4-manifolds.
method Defining and showing realizable obstructions in Whitehead groups.
result Found homeomorphisms that are pseudo-isotopic but not isotopic.
In this note, we discuss the mean curvature flow of graphs of maps between Riemannian manifolds. Special emphasis will be placed on estimates of the flow as a non-linear parabolic system of differential equations. Several global existence theorems and applications to isotopy problems in geometry and topology will be pr…
In this article, the author provides full details of the proof of the concordance/isotopy problem. The first published proof, [5], accomplished this task only partially since there was an error, see the erratum [6], which damaged the main argument of [5, Theorem 2.9], and, consequently, the proof of [5, Theorem A].
Approaches 4D Schoenflies via pseudo-isotopy.
problem Smooth 4D Schoenflies conjecture.
method Pseudo-isotopy theory.
result Offers new approach to 4D Schoenflies.
The Markov Theorem Without Stabilization (MTWS) (see math.GT/0310279) established the existence of a calculus of braid isotopies that can be used to move between closed braid representatives of a given oriented link type without having to increase the braid index by stabilization. Although the calculus is extensive the…
Classifies curves up to symplectic isotopy.
problem Classifying rational cuspidal curves up to symplectic isotopy.
method Topological tools, pseudoholomorphic techniques, and birational transformations.
result Classifies rational cuspidal curves of degrees 6 and 7 up to symplectic isotopy.
Solves a problem about deforming symplectic forms on a Klein bottle.
problem Deforming symplectic forms on a nonorientable Lagrangian surface.
method Analyzes a specific Lagrangian Klein bottle in a symplectic 4-manifold.
result Shows how far symplectic forms can be deformed before isotopy classes contain no Lagrangians.
This paper solves the equivalence problem for projectivizations of knots in 3D.
problem Determining if different projectivizations of the same knot are equivalent in .
method Adapting Hatcher's embedding space idea, the paper provides an algorithm to produce explicit isotopies between projectivizations of knots.
result The paper offers a constructive solution to the equivalence problem for knots in .
Smooth isotopy on cube saves energy with extra dimensions.
problem Constructing an isotopy with infinite kinetic energy.
method Explicit construction of isotopy on .
result Existence of isotopy with infinite kinetic energy.
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…
New relation found in 4D symplectic mapping class group.
problem Relation between Dehn twists in symplectic 4-manifolds.
method Holomorphic curve techniques, symplectic isotopy problem solution.
result Relation between two products of Dehn twists.
Refined 1-cocycle for knots helps quantify isotopies.
problem Quantify knot isotopies using refined tangle equations.
method Refined combinatorial 1-cocycle for regular isotopies of knots.
result Refined tangle equations provide quantitative knot information.
Classifies isotopy classes of links from Thompson's group F and its subgroup.
problem Classify isotopy classes of links from Thompson's group F and its subgroup.
method Introduced a method to produce links from elements of Thompson's group F and its subgroup.
result Classified isotopy classes of links from Thompson's group F and its subgroup.
Uniform convergence of isotopies implies ambient isotopy, aiding knot equivalence.
problem Determining when uniform convergence of isotopies leads to ambient isotopies.
method Using a diagrammatic condition to offload uniform convergence, constructing examples of tame knots.
result Constructing tame knots with countably-many crossings, distinguishing them from wild curves.
Proves a Thom Isotopy Theorem for nonproper semialgebraic maps.
problem Nonproper semialgebraic maps without properness.
method Proves a Thom Isotopy Theorem for nonproper semialgebraic maps.
result Validates the Thom Isotopy Theorem for nonproper semialgebraic maps.
Minimal surfaces and curves can have singularities removed by isotopy.
problem Removing singularities of minimal surfaces and curves.
method Isotopy through conformal minimal surfaces and null holomorphic curves.
result Branch points and complete ends of finite total curvature can be removed.
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…
If a continuous map f: X->Q is approximable arbitrary closely by embeddings X->Q, can some embedding be taken onto f by a pseudo-isotopy? This question, called Isotopic Realization Problem, was raised by Shchepin and Akhmet'ev. We consider the case where X is a compact n-polyhedron, Q a PL m-manifold and show that the …
The study constructs examples of pseudo-isotopic but not isotopic 4-manifold diffeomorphisms.
problem Understanding pseudo-isotopy and its relation to isotopy for 4-manifolds.
method Construction of examples and analysis of pseudo-isotopy obstructions.
result Not all pseudo-isotopic diffeomorphisms are isotopic, providing new insights into 4-manifold topology.
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.
problem Understanding the relationship between Thurston norm and isotopy of surfaces.
method Analyzing veering triangulations and their relation to Thurston norm and isotopy.
result Veering triangulations specify faces of Thurston norm balls and link isotopy of surfaces.
Proof confirms Hamiltonian isotopy of submanifolds.
problem Hamiltonian isotopy of submanifolds in symplectic geometry.
method Complete proof of isotopy lemma for symplectic submanifolds.
result All submanifolds are Hamiltonian isotopic.
This paper has been withdrawn by the authors, due to a mistake pointed out by Lenny Ng and Josh Sabloff.
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.
On Khovanov complexesmath.GT
Proof of isotopy invariance in Khovanov link homology.
problem Isotopy invariance of Khovanov link homology.
method Explicit description of retractions and chain homotopies.
result Isotopy invariance of Khovanov link homology proved.
Unknotting Lagrangian in math.SG
Classifies smooth isotopy of a specific Lagrangian embedding.
problem Classifying Lagrangian embeddings of a specific form.
method Smooth isotopy classification for all dimensions .
result Lagrangian embeddings are fully classified up to isotopy.
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.
problem Understanding isotopy and concordance for metrics on manifolds with boundary.
method Developed tools for studying isotopy and concordance, introduced variants of minimal concordance.
result Obtained a complete picture for surfaces, introduced new variants of concordance.
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…
Constructs pseudo-isotopies with Cerf diagrams and Hatcher-Wagoner invariants for barbell maps.
problem Understanding and constructing pseudo-isotopies for barbell maps.
method Constructs specific pseudo-isotopies with Cerf diagrams and Hatcher-Wagoner invariants.
result Every pseudo-isotopy with vanishing first Hatcher-Wagoner invariant can be isotoped to a composition of standard barbell pseudo-isotopies.
We establish the uniqueness up to Hamiltonian isotopy of the Lagrangian spheres in some four dimensional Stein manifolds.
Grid Moviesmath.GT
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…
Paper characterizes isotopies and hyperbolicity of weaves using geodesics.
problem Characterizing isotopies and hyperbolicity of weaves.
method Using diagrams of closed geodesics and normal positions of essential surfaces.
result Weaves are hyperbolic and cannot have essential Conway spheres.
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 -dimensional Lipschitz submanifolds in . It turns out that due to a smoothing effect any seq…
We call two Engel structures isotopic if they are homotopic through Engel structures by a homotopy that fixes the characteristic line field. In the present paper we define an isotopy invariant of Engel structures on oriented circle bundles over closed oriented three-manifolds and apply it to give an isotopy classificat…