Computational topology shows knots can converge to stick knots.
problem Understanding convergence of smooth knots to stick knots.
method Computational experiments and mathematical visualization.
result Sufficient conditions for isotopic equivalence of knots.
We prove that the control polygon of a Bezier curve B becomes homeomorphic and ambient isotopic to B via subdivision, and we provide closed-form formulas to compute the number of iterations to ensure these topological characteristics. We first show that the exterior angles of control polygons converge exponentially to …
Generalizing Milnor's result that an FTC (finite total curvature) knot has an isotopic inscribed polygon, we show that any two nearby knotted FTC graphs are isotopic by a small isotopy. We also show how to obtain sharper constants when the starting curve is smooth. We apply our main theorem to prove a limiting result f…
When approximating a space curve, it is natural to consider whether the knot type of the original curve is preserved in the approximant. This preservation is of strong contemporary interest in computer graphics and visualization. We establish a criterion to preserve knot type under approximation that relies upon pointw…
A set of control points can determine a Bezier surface and a triangulated surface simultaneously. We prove that the triangulated surface becomes homeomorphic and ambient isotopic to the Bezier surface via subdivision. We also show that the total Gaussian curvature of the triangulated surface converges to the total Gaus…
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.
problem Finding non-isotopic sections in 4-manifolds.
method Using Nielsen equivalence, the paper constructs infinitely many 4-manifolds with non-isotopic sections.
result Infinitely many 4-manifolds have non-isotopic sections.
Study finds non-isotopic transverse tori in Engel manifolds.
problem Identifying distinct transverse tori in Engel manifolds.
method Constructing an infinite family of non-isotopic transverse tori, introducing a homological invariant.
result Found an infinite family of non-isotopic transverse tori that are smoothly isotopic.
New examples show contact structures can be isotopic but not contact-isotopic.
problem Identifying contact structures that are isotopic but not contact-isotopic.
method Constructing specific contactomorphisms in all dimensions.
result Infinitely many examples of contactomorphisms that are isotopic but not contact-isotopic.
Links with isotopic preimages in S3 are isotopic in RP3.
problem Characterizing isotopy in RP3 from S3 preimages. method Analysis of isotopy invariants under double covering map.
result Isotopic preimages in S3 imply isotopic links in RP3. 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.
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 S2×S2, and as a consequence, analogous families of diffeomorphisms and metrics of positive scalar curva…
New ribbon disks in 4D space, non-isotopic to each other.
problem Non-isotopic ribbon disks in 4D.
method Using corks to construct diffeomorphic ribbon disks.
result Non-isotopic ribbon disks constructed in 4D.
Knots with isotopic conormal tori are either identical or mirror images.
problem Determining if two knots are identical or mirror images.
method Microlocal sheaf theory applied to Legendrian isotopy of conormal tori.
result Two knots with isotopic conormal tori are either identical or mirror images.
New links homotopic but not isotopic to a given link.
problem Constructing links that are homotopic but not isotopic.
method Constructing a family of links with specific properties.
result Existence of links homotopic but not isotopic to a given link.
Stabilizing surfaces in 4-manifolds makes exotically knotted pairs smoothly isotopic.
problem Stable equivalence of exotically knotted surfaces in 4-manifolds.
method Stabilizing surfaces by handle additions in the ambient 4-manifold, with standard attachments.
result Exotically knotted pairs of surfaces become smoothly isotopic after a single stabilization.
Smoothly isotopic surfaces in a 4-manifold are stable isotopic if trivial in the 2-homology.
problem Investigating whether topologically isotopic surfaces are smoothly isotopic in stabilised 4-manifolds.
method Analyzing the triviality of surfaces in the 2-homology of the ambient 4-manifold and constructing stabilisations.
result The result holds for surfaces trivial in the 2-homology of the 4-manifold and for certain fundamental groups.
The paper explores isotopic triples of triangles in 3D space.
problem Determining if triples of triangles are combinatorially isotopic.
method Continuous motion of triangles with disjoint outlines, algorithmic checks, and elementary proofs.
result Different types of triples of disjoint triangles are not isotopic.
Low entropy hypersurfaces in 4D are isotopic to a sphere.
problem Characterizing hypersurfaces with low entropy in 4D.
method Proving isotopy to the standard 3-sphere for hypersurfaces with entropy ≤ cylinder entropy.
result Closed hypersurfaces with low entropy are isotopic to the standard 3-sphere.
The paper introduces an inequality to detect when surfaces in 4-manifolds are not smoothly isotopic.
problem Detecting when surfaces in 4-manifolds are not smoothly isotopic.
method An adjunction inequality to distinguish non-isotopic surfaces.
result The minimal genus of a surface isotopic to its image under a diffeomorphism is generally larger.
New 4D examples show sphere equivalence doesn't imply isotopy.
problem Understanding when sphere equivalence implies topological isotopy in 4-manifolds.
method Constructing specific 4-manifolds with non-isotopic but equivalent 2-spheres.
result Gabai's 4D Lightbulb Theorem does not generalize to arbitrary 4-manifolds.
New spheres can split a 4D link in ways not possible in 3D.
problem Exploring how splitting spheres behave in 4D space.
method Constructing specific 2-component surface-links in S4. result Found non-isotopic splitting spheres in S4∖Lm,n. New disks fill Legendrian knots without being smoothly isotopic.
problem Finding non-isotopic Lagrangian fillings of Legendrian knots.
method Constructing distinct Lagrangian ribbon disks with same boundary.
result Found non-isotopic Lagrangian disks with same boundary.
New non-isotopic Seifert surfaces found in 4-ball.
problem Finding non-isotopic Seifert surfaces for knots and links.
method Using double branched covers and Seifert forms, with an algorithm based on quadratic forms.
result Families of knots and links with non-isotopic Seifert surfaces in D4. 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.
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.
The paper studies Legendrian mean curvature flow in η-Einstein Sasakian manifolds.
problem Existence and asymptotic behavior of Legendrian curves in η-Einstein Sasakian manifolds.
method Legendrian mean curvature flow, stability condition, Thomas-Yau conjecture.
result Existence and asymptotic convergence of long-time solutions.
The study examines the realizability of a 4-manifold invariant for homeomorphisms.
problem Realizing the Casson-Sullivan invariant for homeomorphisms of 4-manifolds.
method Investigation of the invariant's realizability and application to surface isotopy.
result The invariant can be realized fully after stabilizing with a single S2imesS2 for all orientable pairs of homeomorphic 4-manifolds. 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 any pair of integers n≥1 and q≥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…
Every knot can be transformed into an unknot without cutting.
problem Understanding the transformation of knots into simpler forms.
method Using a special annulus and homeomorphism to transform knots.
result Every knot is semi-isotopic to an unknot.
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 …
New gauge theory invariant detects non-smooth isotopy of P2-knots.
problem Detecting non-smooth isotopy of P2-knots. method Real Seiberg-Witten theory to construct a gauge theoretic invariant.
result Found a family of P2-knots that are topologically isotopic but not smoothly isotopic. Manufacturing infinite sets of knotted and unknotted surfaces in 4-manifolds.
problem Creating infinite sets of knotted and unknotted surfaces in 4-manifolds.
method Recent constructions of inequivalent smooth structures.
result Infinite sets of pairwise smoothly non-isotopic nullhomologous 2-tori and spheres.
New isotropic tori found in complex space, not Hamiltonian isotopic.
problem Finding non-Hamiltonian isotopic isotropic tori in complex space.
method Analyzing isotropic tori in Cm for m>n≥2. result At least two exact isotropic n-tori in Cm are not Hamiltonian isotopic. Paper presents a method for identifying isotope envelopes in MALDI-ToF data.
problem Deisotoping of isotopic peaks in MALDI-ToF molecular imaging data.
method Uses Mamdani-Assilan fuzzy system and spatial maps of molecular distribution to identify isotope envelopes.
result Proposed method detects overlapping envelopes and analyzes large data sets.
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…
Infinitely many hyperbolic links in lens space have isotopic lifts in 3-sphere.
problem Finding isotopic links in 3-sphere with specific properties.
method Using double covers and Reidemeister moves to construct and analyze links.
result Infinitely many non-isotopic hyperbolic links in lens space have isotopic lifts in 3-sphere.
In this paper we prove genus bounds for closed embedded minimal surfaces in a closed 3-dimensional manifold constructed via min-max arguments. A stronger estimate was announced by Pitts and Rubistein but to our knowledge its proof has never been published. Our proof follows ideas of Simon and uses an extension of a fam…
New surfaces in a 4-manifold are found that are not isotopic but homotopic.
problem Finding non-isotopic surfaces that are homotopic in a specific 4-manifold.
method Applying the Norman trick to a fixed immersed surface, using non-homotopic tubing arcs.
result Infinitely many embedded tori in T4#(S2imesS2) are non-isotopic but homotopic. Study weak conjugacy in surface homeomorphisms.
problem Understanding weak conjugacy in homeomorphisms of surfaces.
method Exploring the group of homeomorphisms isotopic to the identity.
result New insights into weak conjugacy relations.
The study finds non-simple isotopy classes of links in 3-manifolds, including Legendrian and pseudo-Legendrian examples.
problem Characterizing isotopy classes of links in 3-manifolds, especially in contact structures.
method Developed theory of links transverse to a nowhere-zero vector field, constructing examples in both Legendrian and pseudo-Legendrian settings.
result Non-simple isotopy classes of links exist, including Legendrian and pseudo-Legendrian examples.
New link pairs exist that can't be deformed without increasing length.
problem Link isotopy and thick isotopy constraints.
method Constructed isotopic link pairs with preserved total length.
result Link pairs exist that are not thick isotopic.
Examples of non-trivial contactomorphisms in various dimensions.
problem Examples of contactomorphisms not isotopic to the identity.
method Smooth isotopy, symplectic pseudo-isotopy, contact isotopy.
result Contactomorphisms not contact isotopic to the identity.
New knots bound multiple non-isotopic ribbon disks.
problem Finding knots that bound multiple non-isotopic ribbon disks.
method Classification of fibered, homotopy-ribbon disks for generalized square knots.
result Infinitely many knots bound infinitely many pairwise non-isotopic ribbon disks.
New proof associates partitions to isotopic pseudo-Anosov homeomorphisms.
problem Stable and unstable foliations for pseudo-Anosov homeomorphisms.
method Geometric realization of Fathi's result for isotopic homeomorphisms.
result Associated stable and unstable partitions for isotopic pseudo-Anosov homeomorphisms.
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.
Study equivariant isotopy in higher dimensions, finding exceptions.
problem Under what conditions are equivariant isotopic diffeomorphisms also equivariantly isotopic?
method Construct an invariant valued in the homology of an infinite cover to distinguish isotopic from equivariantly isotopic diffeomorphisms.
result Many equivariant diffeomorphisms are isotopic but not equivariantly isotopic in higher dimensions.