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.
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 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.
The paper proves infinitely many non-isotopic splitting 3-spheres for split sphere links in 4D.
problem Proving the existence of infinitely many non-isotopic splitting 3-spheres for split sphere links in 4D.
method Establishing a general sufficient condition for infinitely many topologically non-isotopic splitting 3-spheres in connected sums of 4-manifolds.
result Proves infinitely many non-isotopic splitting 3-spheres for split sphere links in 4D.
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.
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…
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.
Links can be transformed into many others using a specific operation.
problem Understanding the relationship between strongly quasipositive links and their concordance.
method Used a satellite operation with a slice knot to transform links.
result Strongly quasipositive links can be transformed into infinitely many other links.
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.
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.
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 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. 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…
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.
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.
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 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.
I construct "fake algebraic curves" in Cp2. More precisely, for any k>2, I construct infinitely many pairwise smoothly non-isotopic (and moreover not ambient diffeomorphic) smooth surfaces F⊂Cp2 homeomorphic to a non-singular algebraic curve of degree 2k, realizing the same homology class as such a curve a…
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.
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…
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.
The Kinoshita graph is the most famous example of a Brunnian theta graph, a nontrivial spatial theta graph with the property that removing any edge yields an unknot. We produce a new family of diagrams of spatial theta graphs with the property that removing any edge results in the unknot. The family is parameterized by…
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.
We find a geometric invariant of isotopy classes of strongly irreducible Heegaard splittings of toroidal 3-manifolds. Combining this invariant with a theorem of R Weidmann, proved here in the appendix, we show that a closed, totally orientable Seifert fibered space M has infinitely many isotopy classes of Heegaard spli…
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.
We show that if an orientable Seifert fibered space M with an orientable genus g base space admits a strongly irreducible horizontal Heegaard splitting then there is a one-to-one correspondence between isotopy classes of strongly irreducible horizontal Heegaard splittings and elements of Z2g. The corr…
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.
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.
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.
Engel structures on bundles over 3-manifolds in complex 3-space.
problem Embedding bundles over 3-manifolds into complex 3-space with Engel structures.
method Sufficient condition for S1-bundles to admit immersions/embeddings with complex tangencies defining Engel structures. result Every oriented S1-bundle over a closed, oriented 3-manifold admits an immersion with complex tangencies defining Engel structures. 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 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.
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.
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.
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.
For spin manifolds with boundary we consider Riemannian metrics which are product near the boundary and are such that the corresponding Dirac operator is invertible when half-infinite cylinders are attached at the boundary. The main result of this paper is that these properties of a metric can be preserved when the met…
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 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. Polynomial bound on surfaces in hyperbolic 3-manifolds.
problem Bounding the number of surfaces in hyperbolic 3-manifolds.
method Using polynomial functions of the volume of the manifold and the Euler characteristic.
result An upper bound for the number of compact essential surfaces is a polynomial function of the volume of the manifold.
The paper constructs infinite rank summands in diffeomorphism groups via Seiberg-Witten theory.
problem Constructing infinite rank summands in diffeomorphism groups.
method Using Seiberg-Witten theory, the paper constructs spherical families and computes invariants.
result Infinite rank summands in homotopy and homology groups of diffeomorphism groups.
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.
Non-isotopic Heegaard splittings of non-minimal genus were known previously only for very special 3-manifolds. We show in this paper that they are in fact a wide spread phenomenon in 3-manifold theory: We exhibit a large class of knots and manifolds obtained by Dehn surgery on these knots which admit such splittings. M…
New findings show that some knotted surfaces remain distinct even after many stabilizations.
problem Understanding the behavior of knotted surfaces after internal stabilizations.
method Analyzing the properties of smoothly knotted surfaces and their behavior under internal stabilizations.
result There is no upper bound on the number of internal stabilizations required for some knotted surfaces to remain distinct.
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.