Analytic knots have lower 4-ball genus bounds.
problem Understanding the 4-ball genus of analytic knots and links.
method Sharp lower bounds for 4-ball genus of analytic links contained in a tubular neighborhood of smoothly analytic knots.
result Sharp lower bounds for the 4-ball genus of analytic links.
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.
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…
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.
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
New knots found without Lagrangian fillings.
problem Existence of Legendrian knots without Lagrangian fillings.
method Provided a family of Legendrian knots with augmentations not induced by any exact Lagrangian filling.
result Negative answer to the converse of the original statement.
We define Casson-Gordon sigma-invariants for links and give a lower bound of the slice genus of a link in terms of these invariants. We study as an example a family of two component links of genus h and show that their slice genus is h, whereas the Murasugi-Tristram inequality does not obstruct this link from bounding …
Based on work of Rasmussen, we construct a concordance invariant associated to the knot Floer complex, and exhibit examples in which this invariant gives arbitrarily better bounds on the 4-ball genus than the Ozsvath-Szabo tau invariant.
For a closed 4-manifold X and a knot K in the boundary of punctured X, we define γX0(K) to be the smallest first Betti number of non-orientable and null-homologous surfaces in punctured X with boundary K. Note that γS40 is equal to the non-orientable 4-ball genus and hence γX0 is a generalizati…
We give a method for obtaining infinitely many framed knots which represent a diffeomorphic 4-manifold. We also study a relationship between the n-shake genus and the 4-ball genus of a knot. Furthermore we give a construction of homotopy 4-spheres from a slice knot with unknotting number one.
This paper calculates the non-orientable 4-genus for knots with 10 crossings.
problem Determining the non-orientable 4-genus for knots with a specific number of crossings.
method Calculating the minimal first Betti number of non-orientable surfaces smoothly embedded in a 4-ball with boundary the knot.
result The non-orientable 4-genus for knots with 10 crossings has been calculated.
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.
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.
New knot Floer homology gives bounds on 4-ball genus.
problem Calculating the smooth 4-ball genus of knots.
method Elementary methods based on grid diagrams and surface cobordisms.
result Shows unoriented knot Floer homology gives a lower bound for 4-ball genus.
New knots found that are 4-genus minimal.
problem Finding knots with minimal 4-genus.
method Constructing infinitely many amphichiral knots with specific properties.
result Knots with 4-genus minimal for each g>0. Alexander polynomial degree bounds twice a knot's topological slice genus.
problem Determining the topological slice genus of knots.
method Using Freedman's disc theorem and Alexander polynomial properties.
result The degree of the Alexander polynomial is an upper bound for twice the topological slice genus.
The paper proves trisection genus of Akbulut cork and constructs many corks with trisection genus 3.
problem Determining the trisection genus of contractible 4-manifolds and exotic pairs.
method Proved trisection genus of Akbulut cork and constructed corks with trisection genus 3.
result First examples of contractible 4-manifolds with determined trisection genus except for 4-ball.
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.
Researchers calculate the non-orientable 4-genus for knots with 8 or 9 crossings.
problem Determining the minimum non-orientable surface complexity for knots with specific crossing numbers.
method Computed the non-orientable 4-genus for knots with 8 or 9 crossings using surface embeddings in 4-ball.
result Computed the non-orientable 4-genus for all knots with 8 or 9 crossings, proving conjectures and providing new slicing number bounds.
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…
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…
The paper defines new invariants for surfaces in 4-ball and proves bounds on stabilization and double point distances.
problem Bounding distances between surfaces in 4-ball with fixed boundary.
method Link Floer homology and Heegaard Floer homology to construct invariants and prove bounds.
result Lower bounds on stabilization and double point distances for surfaces in 4-ball.
We compute the non-orientable 4-ball genus for a new family of torus knots.
problem Computing the non-orientable 4-ball genus for a new family of torus knots.
method Combination of band surgeries and known bounds.
result Computed the non-orientable 4-ball genus for the knots in the family.
New proof of link 4-genus bounds and satellite link invariants.
problem Determining the 4-genus of links and its behavior under satellite operations.
method New proof using Levine-Tristram signatures and satellite constructions.
result The 4-genus of a link does not increase under certain satellite operations.
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. The paper shows how to unknot certain nonorientable surfaces in 4-dimensional spaces.
problem Tackles the unknotting of nonorientable surfaces in 4-spheres and 4-balls.
method Uses topological isotopy and properties of knot groups and normal Euler numbers.
result Proves that certain nonorientable surfaces are topologically unknotted under specific conditions.
New method uses binary quadratic forms to classify Seifert surfaces in 4-ball.
problem Classifying non-isotopic Seifert surfaces in 4-ball.
method Composition of binary quadratic forms and number-theoretic approach.
result Established a new connection between Bhargava cube and Gauss composition.
Study of knots in 4-manifolds, proving slice genus is not always an invariant.
problem Slice genus is not always an invariant of X0(K). method Analyzing 0-shake genus and using satellite operations. result Infinitely many knots with 0-shake genus strictly less than slice genus. Study on nonorientable analogue of Milnor's conjecture for torus knots.
problem Prove a conjecture about the nonorientable 4-genus of torus knots.
method Relied on Ozsváth, Stipsicz, and Szabó's lower bound for γ4 and new closed formulas for the signature of torus knots. result Proved the conjecture for many infinite families of torus knots.
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.
Study on non-orientable surfaces and inscribed rectangles in 4-manifolds.
problem Understanding non-orientable surfaces and their inscribed rectangles.
method Analyzing smooth and locally-flat non-orientable surfaces in 4-ball with specific knots, comparing results.
result Established differences between smooth and locally-flat non-orientable 4-genus of torus knots.
New findings on Stein fillings and braids in 3D and 4D spaces.
problem Understanding Stein fillings and braids in contact 3-manifolds and 4-ball.
method Analyzing Stein fillings and Lefschetz fibrations, studying braids and their factorizations.
result Existence of infinitely many Stein fillings with distinct homology groups for certain contact 3-manifolds.
Ozsvath and Szabo have defined a knot concordance invariant tau that bounds the 4-ball genus of a knot. Here we discuss shortcuts to its computation. We include examples of Alexander polynomial one knots for which the invariant is nontrivial, including all iterated untwisted positive doubles of knots with nonnegative T…
In knot concordance three genera arise naturally, g(K), g_4(K), and g_c(K): these are the classical genus, the 4-ball genus, and the concordance genus, defined to be the minimum genus among all knots concordant to K. Clearly 0 <= g_4(K) <= g_c(K) <= g(K). Casson and Nakanishi gave examples to show that g_4(K) need not …
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.
Classifies links in 3-sphere using Whitney towers in rational homology 4-ball.
problem Classifying links in 3-sphere using geometric methods.
method Complete classifications of links using Whitney towers in rational homology 4-ball.
result Geometric characterization of Milnor invariants and higher order Arf invariants.
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.
Invariants from surface Khovanov-Jacobsson classes help detect knots and slices.
problem Detecting and obstructing sliceness of knots using surface invariants.
method Defining an invariant from Khovanov homology of boundary links.
result The invariant can obstruct sliceness of knots and detect slices for specific knots.
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.
Refines knot defect measurement in 3D and 4D.
problem Measuring how far knots are from being alternating.
method Extends spanning surface defect to 4-ball, making comparisons and proving formulas.
result Connected sum formula proven.
The article proves properties of Seifert links and their cyclic branched covers.
problem Properties of Seifert links and their cyclic branched covers.
method Left-orderability, co-oriented taut foliations, and L-space properties. result Proves the ADE link conjecture for Seifert links. The paper characterizes and contrasts knots with high 4D clasp numbers.
problem Characterizing knots with high 4-dimensional clasp numbers.
method Topological category analysis and construction of counterexamples.
result Characterization and contrast of knots with high 4D clasp numbers.
Let νbe any integer-valued additive knot invariant that bounds the smooth 4-genus of a knot K, |ν(K)| <= g_4(K), and determines the 4-ball genus of positive torus knots, ν(T_{p,q}) = (p-1)(q-1)/2. Either of the knot concordance invariants of Ozsvath-Szabo or Rasmussen, suitably normalized, have these properties. Let D_…
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.