New families of Brieskorn spheres bound rational homology balls.
problem Identifying new Brieskorn spheres that bound rational homology balls.
method Using techniques from Akbulut and Larson's work, we present new families of Brieskorn spheres.
result We discover new infinite families of Brieskorn spheres that non-trivially bound rational homology balls.
Classifies surgeries on torus knots and cables that bound rational homology balls.
problem Which surgeries on torus knots and cables bound rational homology balls?
method Classification based on integral surgeries and rational numbers q/p for cables.
result Set of rational numbers q/p for cables of a given knot K is bounded.
Fintushel and Stern showed that the Brieskorn sphere Σ(2,3,7) bounds a rational homology ball, while its non-trivial Rokhlin invariant obstructs it from bounding an integral homology ball. It is known that their argument can be modified to show that the figure-eight knot is rationally slice, and we use this fact to p…
We consider the question of which Dehn surgeries along a given knot bound rational homology balls. We use Ozsváth and Szabó's correction terms in Heegaard Floer homology to obtain general constraints on the surgery coefficients. We then turn our attention to the case of integral surgeries, with particular emphasis on p…
Determines surgeries on chain links bounding rational homology balls using lattice-theoretic methods.
problem Integral surgeries on chain links bounding rational homology balls.
method Lattice-theoretic cubiquity obstruction and practical computation methods.
result Proves slice-ribbon conjecture for quasi-alternating 3-braid links, extending previous results.
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.
Classifies torus bundles bounding 4-manifolds with rational homology.
problem Classifying torus bundles over the circle that bound 4-manifolds with rational homology.
method Completely classified torus bundles over the circle that bound 4-manifolds with rational homology.
result Completely classified torus bundles over the circle that bound 4-manifolds with rational homology.
Generalizes Seifert algorithm to integral homology spheres.
problem Finding Seifert circles for integral homology spheres.
method Planar diagram representation of Heegaard splitting, tangle projection.
result Natural construction of Seifert circles and bands for integral homology spheres.
We relate the tree part of the Aarhus integral to Milnor's mu-invariants of string-links in homology balls thus generalizing results of Habegger and Masbaum.
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.
New examples show non-locally-flat PL-disk bounds in rational homology balls but not in integer homology balls.
problem Characterizing knots that bound PL-disks in integer homology balls.
method Involutive Heegaard Floer homology formal properties.
result Found infinitely many manifold-knot pairs (Y, J) where J does not bound a PL-disk in an integer homology ball but does in a rational homology ball.
Large PL surfaces in homology balls can have arbitrarily high genus.
problem Finding the minimum genus of PL surfaces in homology balls.
method Utilizes Heegaard Floer homology.
result The minimum genus can be arbitrarily large.
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.
The Euler class conjecture links geometric structures to integral points on the Thurston norm ball.
problem Determining if integral points on the Thurston norm dual ball correspond to geometric structures.
method Examining various geometric, topological, and dynamical structures on 3-manifolds.
result Integral points on the Thurston norm dual ball correspond to the Euler class of taut foliations and other structures.
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.
Study shows Seifert fibered spaces don't bound rational homology balls.
problem Understanding when Seifert fibered spaces bound rational homology balls.
method Analyzes Seifert fibered spaces with different conditions and orientations.
result Characterizes conditions for Seifert fibered spaces to bound rational homology balls.
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 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.
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.
We give a complete classification of the spherical 3-manifolds that bound smooth rational homology 4-balls. Furthermore, we determine the order of spherical 3-manifolds in the rational homology cobordism group of rational homology 3-spheres. To this end, we use constraints for 3-manifolds to bound rational homology bal…
New knots bound rational homology balls, using Alexander polynomials.
problem Finding sliceness obstructions for knots.
method Computing twisted Alexander polynomials and simplifying their calculation.
result New non-slice knots with rational homology ball bounds.
The Farey tree helps embed rational balls and lens spaces into complex projective space.
problem Embedding rational homology balls and lens spaces into complex projective space.
method Recursive Kirby calculus argument using the Farey tree.
result Explicit constructions of embeddings of triples of rational homology balls into homotopy CP2. Knots generating infinite subgroup bound rational homology balls.
problem Understanding knots that bound rational homology balls.
method Cyclic branched covers and rational homology balls.
result Infinite two-torsion subgroup in knot concordance group.
Characterizes symplectic rational homology ball fillings of Seifert fibered spaces.
problem Understanding which Seifert fibered spaces can be boundaries of symplectic rational homology balls.
method Analyzes convex boundaries and Lagrangian disk fillings of Legendrian knots.
result Strong restrictions on which Seifert fibered spaces can bound symplectic rational homology 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.
New balls smoothly fit in CP² but not symplectically.
problem Embedding Stein rational homology balls in CP².
method Constructing a family of smooth but not symplectic embeddings.
result Existence of a doubly infinite family of such embeddings.
Using the Heegaard Floer homology of Ozsvath and Szabo we investigate obstructions to definite intersection pairings bounded by rational homology spheres. As an application we obtain new lower bounds for the four-ball genus of Montesinos links.
We prove that there are rational homology balls Bp smoothly embedded in the 2-handlebodies associated to certain knots. Furthermore we show that, if we rationally blow up the 2-handlebody along the embedded rational homology ball Bp, then the resulting 4-manifold cannot be obtained just by a sequence of ord…
We generalise theorems of Khodorovskiy and Park-Park-Shin, and give new topological proofs of those theorems, using embedded surfaces in the 4-ball and branched double covers. These theorems exhibit smooth codimension-zero embeddings of certain rational homology balls bounded by lens spaces.
Alternating links bound rational homology balls if their chessboard lattice is cubiquitous.
problem When do alternating links bound rational homology balls?
method Heegaard Floer homology and flows on planar graphs.
result The normalized determinant of the link's chessboard lattice is a necessary and sufficient condition for the link to bound a rational homology ball.
The study explores pinwheels in symplectic surfaces and non-squeezing of rational homology balls.
problem Understanding when Lagrangian pinwheels embed in symplectic rational and ruled surfaces.
method Almost toric fibrations and symplectic rational blow-up.
result A rational homology ball embeds into a rational homology cylinder if and only if the parameter is greater than or equal to 1.
We classify connected sums of three-dimensional lens spaces which smoothly bound rational homology balls. We use this result to determine the order of each lens space in the group of rational homology 3-spheres up to rational homology cobordisms, and to determine the concordance order of each 2-bridge knot.
The study shows a 3D manifold's macroscopic dimension is 1 under specific curvature constraints.
problem Understanding the macroscopic dimension of 3D Riemannian manifolds with curvature restrictions.
method Analyzing the volume and homology of balls in the manifold.
result A 3D manifold with the specified curvature constraints has macroscopic dimension 1.
We construct homology theories with coefficients in L-spectra on the category of ball complexes and we define products in this setting. We also obtain signatures of geometric situations in these homology groups and prove product formulae which we hope will clarify products used in the theory of the total surgery obstru…
In this paper we prove the existence of rational homology balls smoothly embedded in regular neighborhoods of certain linear chains of smooth 2-spheres by using techniques from minimal model program for 3-dimensional complex algebraic variety.
The rational homology balls Bn appeared in Fintushel and Stern's rational blow-down construction [FS] and were subsequently used (e.g. Fintushel-Stern[FS4], Park[Pa2]) to construct exotic smooth manifolds with small Euler numbers. We show that a large class of smooth 4-manifolds have all of the Bn's for odd $n \g…
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.
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 study confirms the non-existence of rational homology ball symplectic fillings for certain Brieskorn spheres.
problem Non-existence of rational homology ball symplectic fillings for specific Brieskorn spheres.
method Analyzing contact structures and using obstruction techniques.
result Confirmation of non-existence for certain Brieskorn spheres.
The smooth rational homology cobordism group of rational homology three spheres, T, contains subgroups T_p generated by 3-manifolds with first homology p-torsion, where p is a prime. Rochlin's theorem and gauge theoretic methods show that the inclusion of the direct sum of the T_p into T has infinitely generated kernel…
We study two homomorphisms to the rational homology sphere group. If ψ denotes the inclusion homomorphism from the integral homology sphere group, then using work of Lisca we show that the image of ψ intersects trivially with the subgroup of the rational homology sphere group generated by lens spaces. As corollarie…
Given a rational homology sphere which bounds rational homology balls, we investigate the complexity of these balls as measured by the number of 1-handles in a handle decomposition. We use Casson-Gordon invariants to obtain lower bounds which also lead to lower bounds on the fusion number of ribbon knots. We use Levine…
Flexible knot construction for low genus surfaces.
problem Finding knots with unexpectedly low genus surfaces.
method Flexible construction of knots in 3-sphere that bound surfaces of low genus in punctured open books.
result First examples of knots with differing genus in different homology balls.
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.
New bounds on nonorientable four-ball genus for torus knots.
problem Finding bounds on the nonorientable four-ball genus of torus knots.
method Combining knot Floer homology techniques to derive lower bounds.
result Sharp bounds for several families of torus knots, including T4n,(2n±1)2. In an earlier paper, we used the absolute grading on Heegaard Floer homology to give restrictions on knots in S3 which admit lens space surgeries. The aim of the present article is to exhibit stronger restrictions on such knots, arising from knot Floer homology. One consequence is that all the non-zero coefficients …
New symplectic caps and embeddings found in complex projective plane.
problem Embeddings of homology balls in complex projective plane.
method Handlebody construction of symplectic caps and embeddings.
result First examples of symplectic handlebody decompositions of a closed symplectic 4-manifold.
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.