This paper describes a method to construct standard 4-balls from homotopy 4-balls in C2.
problem The problem is whether every homotopy 4-ball in S4 is standard. method The approach is to use Stein surfaces and pseudoconvex domains to construct a diffeomorphic domain that is the union of three pseudoconvex domains, ensuring it is a standard 4-ball.
result The construction method ensures that the domain is a standard 4-ball, providing a compelling reimbedding construction for homotopy 4-balls in C2. Proves condition for 4-manifolds with sphere boundary to be standard.
problem Determining when acyclic 4-manifolds with sphere boundary are standard.
method Uses Turaev's shadows to provide a sufficient condition for diffeomorphism to the standard 4-ball.
result If a compact, smooth, acyclic 4-manifold with sphere boundary has shadow-complexity at most 2, it is diffeomorphic to the standard 4-ball.
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.
The study finds infinitely many Lagrangian fillings for most Legendrian torus links.
problem Infinitely many Lagrangian fillings for Legendrian torus links except for a few.
method Constructing infinite order Lagrangian concordances and using actions of modular and mapping class groups.
result There exist infinitely many Lagrangian fillings for most Legendrian torus links.
If a Legendrian knot Λ in the standard contact 3-sphere bounds an orientable exact Lagrangian surface Σ in the standard symplectic 4-ball, then the genus of Σ is equal to the slice genus of (the smooth knot underlying) Λ, the sum of the Thurston-Bennequin number of L and the Euler characteristic of Σ is zero …
The paper constructs new rational homology 3-spheres bounding rational homology 4-balls.
problem Constructing rational homology 3-spheres that bound rational homology 4-balls.
method Exploring plumbed 3-manifolds and using rational homology circles.
result Infinite families of rational homology 3-spheres that bound rational homology 4-balls.
Standardizes Dunfield-Gong's 4-sphere, solves knot sliceness problem.
problem Standardizing Dunfield-Gong's 4-sphere and solving knot sliceness problem.
method Standardization and fibered handle-ribbon disk construction.
result 18_{nh00000601} knot is slice in the standard 4-ball.
4-ball can be tiled with knotted surfaces.
problem Tiling the 4-ball with knotted surfaces.
method Using congruent knotted surfaces isotopic to the original surface.
result Tiling of the 4-ball with knotted surfaces.
New techniques in Khovanov homology help distinguish exotic surfaces in 4-ball.
problem Distinguishing exotic surfaces in the 4-ball that are not diffeomorphic.
method Developed new techniques for distinguishing cobordism maps on Khovanov homology using knot symmetries and braid factorizations.
result Distinguishes smooth surfaces in the 4-ball that are exotically knotted.
Study shows surgeries on certain knots bound rational homology 4-balls.
problem Classifying surgeries on knots that bound rational homology 4-balls.
method Used lattice embedding obstruction and Donaldson's Theorem.
result Classified surgeries on specific knots that bound rational homology 4-balls.
New surfaces in 4-ball differ topologically but not diffeomorphically.
problem Distinguishing surfaces in 4-ball that are topologically equivalent but not diffeomorphic.
method 1-twist rim surgery, sutured Floer homology, cobordism map, knot Floer homology.
result Infinitely many surfaces are topologically isotopic but not diffeomorphic.
A conformal metric on a 4-ball induces on the boundary 3-sphere a conformal metric and a trace-free second fundamental form. Conversely, such a data on the 3-sphere is the boundary of a unique selfdual conformal metric, defined in a neighborhood of the sphere. In this paper we characterize the conformal metrics and tra…
Same-genus Seifert surfaces for non-split alternating links are smoothly isotopic.
problem Understanding the uniqueness of Seifert surfaces for non-split alternating links.
method Analyzing the smooth isotopy of Seifert surfaces in the 4-ball.
result Same-genus Seifert surfaces for non-split alternating links are smoothly isotopic.
We present complete classifications of links in the 3-sphere modulo framed and twisted Whitney towers in a rational homology 4-ball. This provides a geometric characterization of the vanishing of the Milnor invariants of links in terms of Whitney towers. Our result also says that the higher order Arf invariants, which …
Paper tackles which 3-spheres bound contractible 4-manifolds or homology 4-balls.
problem Which homology 3-spheres bound contractible 4-manifolds or homology 4-balls?
method Addressed using plumbed 3-manifolds, modified Mazur's argument, and worked with Poénaru manifolds.
result Presented two new infinite families of plumbed 3-manifolds that bound contractible 4-manifolds or homology 4-balls.
Classifies knots that bound equivariant surfaces with free symmetries.
problem Classifying knots that bound equivariant surfaces with free symmetries.
method Homology cobordism classification of lens spaces using d-invariants.
result Numerical condition determining free periods for torus knots.
The paper constructs exotic surface links in 4-ball, proving their Brunnian nature.
problem Investigating exotic surface links in 4-ball.
method Two constructions of Brunnian exotic surface links, using satellite operations and covering links.
result Provided constructions of exotic Brunnian surface links with varied properties.
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. Extends Rasmussen's invariant to new surfaces in four-manifolds.
problem Computing genus bounds for surfaces in specific four-manifolds.
method Extending Khovanov-Lee homology and using Hochschild homology.
result Proves inequalities relating the invariant to surface genus.
The study classifies χ−slice pretzel links and Seifert fiber spaces.
problem Understanding χ−slice pretzel links and their properties. method Analyzing the sliceness of pretzel knots and extending results to pretzel links.
result Complete classifications of positive and negative pretzel links that are χ−slice, and partial classifications of 3-stranded and 4-stranded pretzel links. Disproves a conjecture about knots in 4D space.
problem Non-orientable slice genus of torus knots
method Found a specific knot that bounds a Möbius band in 4-ball
result Counterexample to Batson's conjecture
We found an infinite family of counterexamples to Batson's conjecture.
problem Batson's conjecture about nonorientable 4-ball genus of torus knots is false.
method We identified an infinite family of counterexamples to Batson's conjecture.
result We found an infinite family of counterexamples to Batson's conjecture.
New Seifert surfaces in 4-ball differ even when pushed in.
problem Finding distinct Seifert surfaces in 4-ball.
method Using cobordism maps on Khovanov homology.
result Examples of Seifert surfaces not isotopic in 4-ball.
A link in the 3-sphere is called (smoothly) slice if its components bound disjoint smoothly embedded disks in the 4-ball. More generally, given a 4-manifold M with a distinguished circle in its boundary, a link in the 3-sphere is called M-slice if its components bound in the 4-ball disjoint embedded copies of M. A 4-ma…
Counterexamples found for knot conjectures.
problem Knot conjectures regarding bounding Möbius bands in 4-ball.
method Examined torus knots T(2,5) and T(2,9) in 3D space.
result Found counterexamples to Allen's conjectures.
The paper constructs contractible manifolds with knotted spheres.
problem Creating contractible manifolds with non-standard boundaries.
method Using handles and constructing knotted spheres in SnimesS2. result Contractible (n+3)-manifolds with non-standard boundaries are constructed. Call a smooth knot (or smooth link) in the unit sphere in C2 analytic (respectively, smoothly analytic) if it bounds a complex curve (respectively, a smooth complex curve) in the complex ball. Let K be a smoothly analytic knot. For a small tubular neighbourhood of K we give a sharp lower bound for the 4…
We exhibit a knot P in the solid torus, representing a generator of first homology, such that for any knot K in the 3-sphere, the satellite knot with pattern P and companion K is not smoothly slice in any homology 4-ball. As a consequence, we obtain a knot in a homology 3-sphere that does not bound a piecewise-…
The paper calculates homology and intersection pairing of branched covers using disoriented homology.
problem Computing homology and intersection pairing of branched covers of 4-ball.
method Associate disoriented homology groups to projections of links and surfaces, show isomorphism to branched cover homology, define pairing on first disoriented homology of surfaces.
result Disoriented homology is isomorphic to the homology of branched cover and pairing is equal to the intersection pairing.
New surfaces in 4-ball constructed from knits, described by charts.
problem Constructing surfaces in 4-ball from knits.
method Introducing knitted surfaces, describing them with BMW charts.
result Every compact surface in 4-ball is ambiently isotopic to a knitted surface.
New spanning tree model connects knot homology, s-invariant, and exotic discs.
problem Understanding exotic discs in the 4-ball for knots.
method Explicitly defined differential in spanning tree complex, described Rasmussen's s-invariant.
result Identified new infinite family of knots bounding exotic discs.
The concordance genus of a knot K is the minimum Seifert genus of all knots smoothly concordant to K. Concordance genus is bounded below by the 4-ball genus and above by the Seifert genus. We give a lower bound for the concordance genus of K coming from the knot Floer complex of K. As an application, we prove that ther…
New 3-manifolds bound rational 4-balls through specific operations.
problem Finding rational homology 3-spheres that bound rational homology 4-balls.
method Two operations that preserve lattice embedding obstruction to bounding rational homology balls.
result Explicit examples of rational surgeries on torus knots that bound rational homology balls.
The paper quantifies how much of a 4-ball must be removed to squeeze into a cylinder, proving a lower bound on the Minkowski dimension.
problem Quantifying how much of a 4-ball must be removed to fit into a cylinder.
method Gromov's non-squeezing theorem and Minkowski dimension analysis.
result The Minkowski dimension of the removed set is at least 2, with an example showing this is optimal for certain radii.
The paper presents fundamental groups of complements of shadows in 4-balls.
problem Understanding fundamental groups of 4-manifold complements.
method Similar to Wirtinger presentation, focusing on contractible shadows.
result A presentation of fundamental groups of subpolyhedra in 4-balls.
Study shows no smooth embeddings of rational homology balls into complex projective plane.
problem Embedding rational homology balls into complex projective plane.
method Elementary arguments to prove non-existence of almost complex embeddings.
result No smooth embeddings of rational homology balls into complex projective plane.
The A-B slice problem is a reformulation of the topological 4-dimensional surgery conjecture in terms of decompositions of the 4-ball and link homotopy. We show that link groups, a recently developed invariant of 4-manifolds, provide an obstruction for the class of model decompositions, introduced by M. Freedman and X.…
The n-solvable filtration {Fn}n=0∞ of the smooth knot concordance group (denoted by C), due to Cochran-Orr-Teichner, has been instrumental in the study of knot concordance in recent years. Part of its significance is due to the fact that certain geometric characterizations of a knot …
New knots not slice in rational 4-balls found.
problem Identifying knots not slice in rational homology 4-balls.
method Using generalized Mazur patterns and immersed Heegaard Floer homology.
result Infinitely many examples of pattern knots P not slice in any rational homology 4-ball.
We construct cobordisms of small genus between torus knots and use them to determine the cobordism distance between torus knots of small braid index. In fact, the cobordisms we construct arise as the intersection of a smooth algebraic curve in C2 with the unit 4-ball from which a 4-ball of smaller radius is…
Let L be a nonunimodular definite lattice. Using a theorem of Elkies we show that whether L embeds in the standard definite lattice of the same rank is completely determined by a collection of lattice correction terms, one for each metabolizing subgroup of the discriminant group. As a topological application this gives…
Lisa Piccirillo solved the mystery of the Conway knot's sliceness.
problem Determining if the Conway knot is slice.
method The main idea of the proof is given in the title.
result The Conway knot is not slice.
We develop obstructions to a knot K in the 3-sphere bounding a smooth punctured Klein bottle in the 4-ball. The simplest of these is based on the linking form of the 2-fold branched cover of the 3-sphere branched over K. Stronger obstructions are based on the Ozsvath-Szabo correction term in Heegaard-Floer homology, al…
A link in the 3-sphere is homotopically trivial, according to Milnor, if its components bound disjoint maps of disks in the 4-ball. This paper concerns the question of what spaces give rise to the same class of homotopically trivial links when used in place of disks in an analogous definition. We show that there are 4-…
Study finds topological restrictions on 4-manifolds with uniformly positive scalar curvature.
problem Understanding which 4-manifolds can have metrics with uniformly positive scalar curvature.
method Topological obstructions and metric constructions on specific 4-manifolds.
result Existence of uncountably many exotic R4's without such metrics and topological uniqueness of certain metrics. New method classifies C-boundaries up to 6 crossings.
problem Classifying C-boundaries with up to 6 crossings. method Proposed a new construction method.
result Extended classification of C-boundaries up to 6 crossings. We prove a cabling formula for the concordance invariant ν+, defined by the author and Hom. This gives rise to a simple and effective 4-ball genus bound for many cable knots.
The first part of this paper exposits a simple geometric description of the Kirby-Siebenmann invariant of a 4--manifold in terms of a quadratic refinement of its intersection form. This is the first in a sequence of higher-order intersection invariants of Whitney towers studied by the authors, particularly for the 4--b…