We construct an isotopy of a planar compactum that is not the restriction of an isotopy of any planar continuum.
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48 results for “Isotopy”
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.
Approaches 4D Schoenflies via pseudo-isotopy.
problem Smooth 4D Schoenflies conjecture.
method Pseudo-isotopy theory.
result Offers new approach to 4D Schoenflies.
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.
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 …
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…
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.
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.
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.
We construct counterexamples to lifting properties of Hamiltonian and contact isotopies.
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…
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…
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…
We establish the uniqueness up to Hamiltonian isotopy of the Lagrangian spheres in some four dimensional Stein manifolds.
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.
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.
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…
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…
New distances defined on Legendrian spaces without positive loops.
problem Defining distances on Legendrian spaces without positive loops.
method Constructing unbounded invariant distances on Legendrian isotopy classes.
result Invariant distances on Legendrian isotopy classes are discrete.
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.
Geometric aspects of the filtration on classical links by k-quasi-isotopy are discussed, including the effect of Whitehead doubling, relations with Smythe's n-splitting and Kobayashi's k-contractibility. One observation is: ω-quasi-isotopy is equivalent to PL isotopy for links in a homotopy 3-sphere (resp. contractible…
The paper studies pseudo-isotopies of spherical 3-manifolds and computes ranks of certain groups.
problem Computing ranks of abelian groups related to spherical 3-manifolds.
method Surgery on theta-graphs embedded in spherical 3-manifolds, study of pseudo-isotopy behavior under suspension.
result Lower bounds of ranks of abelian groups are computed.
Subspaces of Knot Spacesmath.GT
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.
New smoothing techniques for topological surfaces in 4-manifolds.
problem Topological isotopy of surfaces in smooth 4-manifolds.
method Combining Quinn's smoothing theory with Gabai's light bulb theorem and other developments.
result Proves topological = smooth results for certain disks and spheres.
We construct real analytic flat Moebius strips of arbitrary isotopy types, whose centerlines are geodesics or lines of curvature.
New transformations preserve link isotopy, showing complexity differences.
problem Link isotopy preservation with complexity differences.
method Introducing multiflypes of rectangular diagrams of links.
result Two diagrams of same complexity not related by simpler moves.
New unknots with geometric constraints exist, proving a long-standing conjecture.
problem Existence of distinct isotopy classes of physical unknots with geometric constraints.
method Parametrised thickness and geometric thresholds to fragment isotopy classes.
result Existence of gordian unknots with prescribed geometric constraints.
Proves optimal regularity for sphere minimizers in 3-sphere.
problem Finding optimal regularity for sphere minimizers.
method Proves regularity for minimizers of prescribed mean curvature over isotopy classes.
result Proves optimal regularity for minimizers.
Virtual Crossing Realizationmath.GT
We study virtual isotopy sequences with classical initial and final diagrams, asking when such a sequence can be changed into a classical isotopy sequence by replacing virtual crossings with classical crossings. An example of a sequence for which no such virtual crossing realization exists is given. A conjecture on con…
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.
Unified Lie structures in homotopy and isotopy calculus.
problem Compatibility of Lie structures in homotopy and isotopy calculus.
method New technical tool: bracket on total homotopy fibres of collapsing cubes of wedge sums.
result Unified understanding of Lie structures in homotopy and isotopy calculus.
Legendrian Lavrentiev linksmath.SG
Legendrian Lavrentiev links are shown to be equivalent to smooth links.
problem Understanding equivalence of Legendrian and smooth links.
method Definition and analysis of Legendrian isotopies.
result Equivalence classes of Legendrian Lavrentiev links coincide with smooth links.
In part I it was shown that for each k>0 the generalized Sato-Levine invariant detects a gap between k-quasi-isotopy of link and peripheral structure preserving isomorphism of the finest quotient G_k of its fundamental group, `functorially' invariant under k-quasi-isotopy. Here we show that Cochran's derived invariant …
All knots are fused isotopic to the unknot via a process known as virtualization. We extend and adapt this process to show that, up to fused isotopy, classical links are classified by their linking numbers.