Infinitely many splitting spheres found for unlinked 2-spheres in 4-space.
problem Existence of pairwise non-isotopic splitting spheres for unlinked 2-spheres in 4-space.
method Analytical proof showing non-isotopic spheres.
result Infinitely many non-isotopic splitting spheres found.
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. 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.
Study contact structures on specific manifolds, proving infinite non-isotopic structures and tight/overtwisted classifications.
problem Classifying contact structures with specific contact surgery numbers on Brieskorn spheres and lens spaces.
method Algorithm for computing Euler class from rational contact surgery, Legendrian knot classification, and analysis of contact structures.
result Infinitely many non-isotopic contact structures on Brieskorn spheres and lens spaces cannot be obtained by a single rational contact surgery.
A modular tensor category C gives rise to a Reshetikhin-Turaev type topological quantum field theory which is defined on 3-dimensional bordisms with embedded C-coloured ribbon graphs. We extend this construction to include bordisms with surface defects which in turn can meet along line defects. …
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…
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 contact structures on projective spaces, proving infinite non-isotopic structures.
problem Classify contact structures on projective spaces with specific surgery numbers.
method Detailed analysis of Gompf's Γ-invariant and d_3-invariant of tangential 2-plane fields.
result Infinitely many non-isotopic contact structures on real projective 3-space.
New 3D handlebodies in 4-sphere and 5-ball are not isotopic even with same boundary.
problem Existence of non-isotopic handlebodies with identical boundary.
method Construction of specific genus-g handlebodies in S4 and B5. result Proves Budney-Gabai conjecture for genus at least 2.
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.
The abstract discusses detecting knotted spheres through their traces in high dimensions.
problem Detecting knotted spheres in high-dimensional spaces.
method Generalizing the RBG link construction to all dimensions and using surgery.
result Existence of non-isotopic smooth (n−2)-knots with diffeomorphic traces. 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 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.
We construct a family of pairs of non-isotopic symplectic surfaces in the standard symplectic 4-disk such that they are bounded by the same transverse knot in the standard contact 3-sphere and fundamental groups of their complements are isomorphic. In the appendix, we prove explicitly that one can obtain a symplect…
Classifies essential annuli in a genus two handlebody exterior.
problem Classifying essential annuli in a genus two handlebody exterior.
method Building on JSJ-graph classification and essential annuli classification.
result Characterizes the numbers of different types of essential annuli in an infinite family.
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…
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.
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 …
The paper constructs many knotted and linked objects in higher dimensions.
problem Understanding knotted and linked objects in higher dimensions.
method Using barbell diffeomorphisms to construct examples.
result Infinitely many knotted and linked objects in 4 and 5 dimensions.
We review Giroux's contact handles and contact handle attachments in dimension three and show that a bypass attachment consists of a pair of contact 1 and 2-handles. As an application we describe explicit contact handle decompositions of infinitely many pairwise non-isotopic overtwisted 3-spheres. We also give an alter…
Given a transverse link in the standard contact 3-sphere, we study the contact manifold that arises as a branched double cover of the sphere. We give a contact surgery description of such manifolds, which allows to determine the Heegaard Floer contact invariants for some of them. By example of the knots of Birman--Mena…
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.
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…
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.
Let E(1)_p denote the rational elliptic surface with a single multiple fiber f_p of multiplicity p. We construct an infinite family of homologous non-isotopic symplectic tori representing the primitive class [f_p] in E(1)_p when p>1. As a consequence, we get infinitely many non-isotopic symplectic tori in the fiber cla…
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.
New Legendrian knots found with equivalent Stein traces.
problem Characterizing slopes and Legendrian isotopy types.
method Contact annulus twist, Weinstein handlebody equivalences, dualizable patterns.
result First example of Legendrian knots with equivalent Stein traces.
Study infinite symplectic forms on ruled surfaces.
problem Uniqueness of symplectic structures on four-manifolds.
method Provided an infinite family of non-isotopic symplectic forms.
result Symplectic forms on ruled surfaces are pairwise non-isotopic.
The study limits the number of 2-holed tori in knot exteriors.
problem Bounding the number of 2-holed tori in knot exteriors.
method Continuing Motegi's program, the paper applies universal bounds to hyperbolic knots.
result There are at most six non-isotopic, nested, essential 2-holed tori in the complement of every hyperbolic knot.
The paper classifies all tight contact structures on a solid torus.
problem Classifying tight contact structures on a solid torus with specified dividing sets.
method Writing down a closed formula for the number of non-isotopic tight contact structures with any given dividing set.
result The complete classification of tight contact structures on a solid torus.
Contact homology for Legendrian submanifolds in standard contact (2n+1)-space is rigorously defined using moduli spaces of holomorphic disks with Lagrangian boundary conditions in complex n-space. It provides new invariants of Legendrian isotopy. Using these invariants the theory of Legendrian isotopy is shown to b…
New examples show transverse knots are determined by their branched covers.
problem Transverse knots and their isotopy classes.
method Constructing and analyzing non-isotopic transverse knots with contactomorphic cyclic branched covers.
result Transverse isotopy classes of many transverse knots are determined by the contactomorphism type of their cyclic branched covers.
The paper finds non-isotopic exact Lagrangians in symplectic manifolds with C∗-actions.
problem Finding non-isotopic exact Lagrangians in symplectic manifolds.
method Using contracting C∗-actions, the paper constructs families of non-isotopic closed exact Lagrangian submanifolds. result The Floer cohomologies of these Lagrangians are topological, recovering ordinary cohomologies of intersections.
The paper defines and studies contact surgery numbers for contact 3-manifolds.
problem Understanding the minimal number of components of a surgery link describing a contact 3-manifold.
method Defined and studied various versions of contact surgery numbers, relating them to other invariants and computing specific cases.
result There exist infinitely many non-isotopic contact structures on certain manifolds that cannot be obtained by a single rational contact surgery from the standard tight contact 3-sphere.
Bridge trisections and knotted surfaces connected via tri-plane diagrams.
problem Computing and understanding knotted surfaces using bridge trisections.
method Using tri-plane diagrams to compute normal Euler number, fundamental group, and analyze bridge trisections of ribbon surfaces.
result Produced an infinite family of knotted spheres with non-isotopic bridge trisections of minimal complexity.
The paper finds non-isotopic Legendrian unit conormal bundles in high dimensions.
problem Identifying non-isotopic Legendrian unit conormal bundles in high dimensions.
method Defined strip Legendrian contact homology and coproduct for Legendrian submanifolds.
result Found non-isotopic Legendrian unit conormal bundles using topological and string topology methods.
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. The paper creates non-isotopic but homotopic diffeomorphisms in 4-manifolds.
problem Creating non-isotopic but homotopic diffeomorphisms in 4-manifolds.
method Surgery along Θ-graphs and use of a twisted characteristic class.
result Constructs countably many non-isotopic but homotopic diffeomorphisms.
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.
Unique 3-balls in S^4 can be non-isotopic.
problem Existence and uniqueness of spanning 3-balls in S^4.
method Introducing barbell diffeomorphisms, implantations, and twistings; 2-parameter calculus of embeddings; framed cobordism method.
result Non-uniqueness of spanning 3-balls in S^4.
A regular circle-valued Morse function on the knot complement C(K) = S^3\K is a function f from C(K) to S^1 which separates critical points and which behaves nicely in a neighborhood of the knot. Such a function induces a handle decomposition on the knot exterior E(K) = S^3\N (K), with the property that every regular l…
New model for links uses meander diagrams and combinatorics.
problem Modeling and analyzing random links.
method Random meander model based on meander diagrams and graphs, proving properties using combinatorics.
result Trivial links are unlikely, and there's a lower bound on non-isotopic knots.
New invariant detects more elements in 4D diffeomorphism group.
problem Detecting more elements in the mapping class group of 4D handlebodies.
method Defined and computed a new invariant (W3)m for π0Diff(aturalmS1imesD3,∂). result Infinitely many non-isotopic separating 3-balls found.
New findings on hyperbolicity of augmented links in thickened surfaces.
problem Proving hyperbolicity of links in thickened surfaces.
method Extending hyperbolicity results to generalized augmented cellular alternating links.
result Generalized augmented cellular alternating links in thickened surfaces are hyperbolic.
We describe for each postive integer k a 3-manifold with Heegaard surfaces of genus 2k and 2k−1 such that any common stabilization of these two surfaces has genus at least 3k−1. We also show that for every positive n, there is a 3-manifold that has n pairwise non-isotopic Heegaard splittings of the same gen…
The purpose of this paper is to investigate the following problem: For a fixed 2-dimensional homology class K in a simply connected symplectic 4-manifold, up to smooth isotopy, how many connected smoothly embedded symplectic submanifolds represent K? We show that when K can be represented by a symplectic torus, there a…
New diagrams classify triply periodic entanglements.
problem Classifying triply periodic entanglements.
method Introducing diagrams for links in 3-torus and defining additional moves.
result New diagrams allow classification of non-isotopic links with same preimage.