Study on deformations of polynomials and their Milnor fibration topology.
problem Understanding the topology of Milnor fibrations of deformations of polar weighted homogeneous polynomials.
method Round handle decomposition of the Milnor fibration, calculation of characteristic polynomials.
result Description of the number of round handles and a formula for characteristic polynomials.
Classifies symplectic fillings of specific torus bundles.
problem Classifying strong and exact symplectic fillings of virtually overtwisted torus bundles.
method Using Menke's JSJ-type decomposition theorem and round symplectic 1-handle attachment.
result Conditions for distinct tight lens space fillings to yield the same torus bundle filling.
New findings show infinitely many knots cannot be smoothly round handle slices.
problem Understanding the smoothability of knots in 4-dimensional space.
method Analyzing the properties of knots under surgery conjectures and cobordism conjectures.
result Infinitely many knots fail to be smoothly round handle slices.
The Round Handle Problem asks if certain links are slice in a 4-manifold.
problem Whether a specific collection of links is slice in a 4-manifold constructed from round handles.
method Analyzes the slice property of links in a 4-manifold constructed by adding round handles.
result A negative answer would contradict major conjectures in 4-manifold theory.
We develop a construction of Engel stuctures on 4-manifolds based on decompositions of manifolds into round handles. This allows us to show that all parallelizable 4-manifolds admit an Engel structure. We also show that, given two Engel manifolds M_1,M_2 satisfying a certain condition on the characteristic foliation, t…
The topology of broken Lefschetz fibrations is studied by means of handle decompositions. We consider a slight generalization of round handles, and describe the handle diagrams for all that appear in dimension four. We establish simplified handlebody and monodromy representations for a certain subclass of broken Lefsch…
Elliptic surfaces without 1-handles proven for specific cases.
problem Proving elliptic surfaces have no 1-handles.
method Analyzing handle decompositions of elliptic surfaces E(n)p,q for specific n. result Proven existence of handle decompositions without 1-handles for specific elliptic surfaces.
Handles decompositions reveal new open book structures.
problem Understanding and manipulating open book structures on manifolds.
method Build handle decompositions and describe handle slides.
result Stabilization of open books to trivial monodromy.
The paper describes handle decompositions and Kirby diagrams for plane algebraic curves.
problem Understanding the topology of the complement of plane algebraic curves.
method Using braid monodromy to refine handle decompositions and Kirby diagrams.
result Explicit handle decompositions and Kirby diagrams for plane algebraic curves are provided.
The study shows conditions for elliptic surfaces without 1-handles.
problem Finding conditions for elliptic surfaces without 1-handles.
method Analyzing knot surgeries and log-transformations of elliptic surfaces.
result Conditions for elliptic surfaces without 1-handles are established.
A theorem on the existence of the unique minimal topologic handle decomposition of differentiable simply connected five-dimensional manifolds is proved. For a decomposition of this sort, the number of handles of each index is given.
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…
Study handles in 4-manifolds with cyclic fundamental group.
problem Understanding handle decompositions for specific 4-manifolds.
method Constructed handle decompositions based on fundamental group and Betti numbers.
result Handle decomposition formula for specific 4-manifolds.
The study describes handle decompositions and Kirby diagrams for line arrangements.
problem Understanding handle decompositions and Kirby diagrams for line arrangements.
method Introduced the divide with cusps and used Lefschetz hyperplane section theorem.
result Described the Kirby diagram for line arrangements.
Round handles are affiliated with smooth 4-manifolds in two major ways: 5-dimensional round handles appear extensively as the building blocks in cobordisms between 4-manifolds, whereas 4-dimensional round handles are the building blocks of broken Lefschetz fibrations on them. The purpose of this article is to shed more…
Study compact PL 4-manifolds with special handle decompositions.
problem Existence of special handlebody decompositions for simply-connected closed PL 4-manifolds.
method Investigate colored triangulations inducing handle decompositions without 1-handles or 1- and 3-handles.
result Detect a class of compact simply-connected PL 4-manifolds with empty or connected boundary that admit such decompositions.
Harer-Kas-Kirby conjectured that every handle decomposition of the elliptic surface E(1)_{2,3} requires both 1- and 3-handles. We prove that the elliptic surface E(n)_{p,q} has a handle decomposition without 1-handles for n≥1 and (p,q)=(2,3),(2,5),(3,4),(4,5).
Study shows knots surgered elliptic surfaces admit handle decompositions without 1- and 3-handles.
problem Understanding handle decompositions of knots surgered elliptic surfaces.
method Using Kirby diagrams derived from Lefschetz fibrations.
result Knot surgered elliptic surfaces admit handle decompositions without 1- and 3-handles.
A double pants decomposition of a 2-dimensional surface is a collection of two pants decomposition of this surface introduced in arXiv:1005.0073v2. There are two natural operations acting on double pants decompositions: flips and handle twists. It is shown in arXiv:1005.0073v2 that the groupoid generated by flips and h…
Skein lasagna module calculates 4-manifold invariants using handle decompositions.
problem Calculating invariants of 4-manifolds and links.
method Express skein lasagna module in terms of handle decompositions.
result Skein lasagna module can be locally infinite dimensional.
Study rational homology balls using Casson-Gordon invariants to measure complexity.
problem Measuring complexity of rational homology balls.
method Use Casson-Gordon invariants and Levine-Tristram signatures.
result Obtain lower bounds on the number of 1-handles in handle decompositions.
Trisection method for 4-manifolds with boundary.
problem Creating a trisection diagram for 4-manifolds with boundary.
method Explicit proof of relative trisections, extending to multiple boundary components.
result Existence of trisections for 4-manifolds with boundary and multiple components.
The paper calculates a new invariant for 4-manifolds using handle decompositions and skein relations.
problem Computing invariants for 4-manifolds built from handles.
method Handle attachment formulas, cabled colimits, lasso relation.
result Explicit calculations and partial vanishing results for specific 4-manifolds.
New insights into knot fusion numbers via cabling.
problem Understanding fusion numbers of ribbon knots and their behavior under cabling.
method Utilizing knot Floer homology and cabling formulas to analyze fusion numbers.
result The fusion number and strong homotopy fusion number of (p,1)-cable knots are preserved.
We determine the structure of the circular handle decompositions of the family of free genus one knots. Namely, if k is a free genus one knot, then the handle number h(k)= 0, 1 or 2, and, if k is not fibered (that is, if h(k)>0), then k is almost fibered. For this, we develop practical techniques to construct circular …
New 4-manifolds without 1- and 3-handles are created from knots.
problem Creating 4-manifolds without specific handles. method Knot surgery with specific knots.
result Knot surgery 4-manifolds E(n)K have a handle decomposition without 1- and 3-handles. New 4-manifolds found without 1- and 3-handles.
problem Finding 4-manifolds without specific handles.
method Perfect Morse functions and handle decompositions.
result Infinitely many 4-manifolds homeomorphic but not diffeomorphic to standard manifolds.
We present an approach to constructing broken Lefschetz fibrations (BLFs) f:X→S2 from a handle decomposition of a 4-manifold X. Given a handle decomposition as input these techniques yield explicit descriptions of the BLFs, and do not rely on classification results from contact topology or the choice of…
Infinite order corks are described with detailed handle decompositions.
problem Understanding the structure of infinite order corks in 4-manifolds.
method Detailed handle decompositions and embeddings of corks in E(n). result The twisted doubles of infinite order corks are Gluck twists and log transforms of S4. D.Nash defined a family of homotopy 4-spheres in [11]. Proving that his manifolds Sm,n,m′,n′ are all real S4, we find that they have handle decomposition with no 1-handles, two 2-handles and two 3-handles. The handle structures give new potential counterexamples of Property 2R conjecture.
Geometric proof shows topological invariance of handle homology.
problem Topological invariance of handle homology in manifolds.
method Entirely geometric proof using Cerf theory.
result Proof of ∂2=0 in chain complex defined by handle decomposition. New tensor decomposition method handles more noise and higher orders.
problem Efficient tensor decomposition for noisy data.
method Two-mode higher-order SVD (HOSVD) with Kruskal's theorem.
result Proves higher noise tolerance and high accuracy.
Solves Inverse Invariant Theory for sphere partitions.
problem Inverse Invariant Theory for manifold submetries of the round sphere.
method One-to-one correspondence between manifold submetries and maximal Laplacian algebras.
result Solves the Inverse Invariant Theory problem for manifold submetries of the round sphere.
The study solves a 2008 problem by Kalai about infinitely many homology-spheres.
problem Proving infinitely many homology-spheres with g3=0 in dimensions higher than four. method Using handlebody decompositions and PL manifolds with specific triangulations.
result There are infinitely many homology-spheres with g3=0 in dimensions higher than four. The paper bounds the handle number of sutured manifolds using Morse-Novikov numbers and tunnel numbers.
problem Bounding the handle number of sutured manifolds.
method Developed bounds on the Morse-Novikov number of a link in terms of its tunnel number, and used these to bound the handle number of Heegaard splittings.
result The handle number function is bounded, constant on rays from the origin, and locally maximal.
New method uses Khovanov homology to distinguish exotic 4-manifolds.
problem Distinguishing exotic compact, orientable 4-manifolds.
method Introduces a new, simple way of using Khovanov homology.
result First analysis-free proof of existence of exotic Mazur manifolds.
We give a method for obtaining a handle decomposition of an n-manifold if the manifold is given by isometric side-pairings of a polyhedron in $\en$, $\sn$ or $\hn$. Every cycle of k-faces on the polyhedron corresponds to an (n−k)-handle of the manifold. Two applications of the method are given. One helps recogniz…
4-manifolds with specific trisections are 3-fold covers of S^4.
problem Understanding 4-manifolds with (g;k1,k2,0)-trisections. method Proving 4-manifolds with certain trisections are irregular 3-fold covers of S4. result Any 4-manifold with a (g;k1,k2,0)-trisection is an irregular 3-fold cover of S4. Review of gem theory's interactions with Kirby diagrams and trisections.
problem Representing and understanding PL 4-manifolds.
method Combining gem theory with Kirby diagrams and trisections.
result New gems representing 4-manifolds requiring 3-handles.
Bordered Heegaard Floer homology is an invariant for three-manifolds with boundary. In particular, this invariant associates to a handle decomposition of a surface F a differential graded algebra, and to an arc slide between two handle decompositions, a bimodule over the two algebras. In this paper, we describe these b…
Proves complex homology three-spheres can be bounded by many handles.
problem Complex homology three-spheres and their bounding four-manifolds.
method Uses homology cobordism invariant Γ from instanton Floer homology.
result Any bounding four-manifold must be built out of many handles.
Two codimension-one submanifolds are cobordant if they have the same homology class.
problem Understanding cobordism equivalence of codimension-one submanifolds.
method Using handle decompositions and surgeries, we relate cobordisms to homology classes and prove equivalence for Seifert surfaces.
result Seifert surfaces are related by tube attachments and removals, providing a conceptual proof.
Simple branched coverings established between certain 4-manifolds.
problem Establishing simple branched coverings between specific 4-manifolds.
method Using handle decompositions and isometric embeddings of intersection lattices.
result Existence and construction of d-fold simple branched coverings. New hyperbolic 4-manifolds found with special functions.
problem Finding hyperbolic 4-manifolds with specific circle-valued Morse functions.
method Constructing hyperbolic 4-manifolds with only index 2 critical points.
result Existence of infinitely many hyperbolic 4-manifolds with bounded Betti numbers.
We construct an orientable ribbon surface F in B^4, which is universal in the following sense: any compact orientable pl 4-manifold having a handle decomposition with 0-, 1- and 2-handles can be represented as a cover of B^4 branched over F.
Starting with the Dolgachev surface E(1)_{2,3} we construct an infinite family of distinct exotic copies of the rational surface E(1), each of which admits a handlebody decomposition without 1- and 3- handles, and we draw these handlebodies.
Concerns decompositions of smooth 4-manifolds as the union of two handlebodies, each with handles of index <=2 (``Heegard'' decompositions).Sample result: Two 2-complexes are (up to 2-deformation) dual spines of a Heegard decomposition of the 4-sphere if and only if they satisfy the conclusions of the Alexander-Lefshet…
We give new rational blowdown constructions of exotic CP^2#n(-CP^2) (5\leq n\leq 9) without using elliptic fibrations. We also show that our 4-manifolds admit handle decompositions without 1- and 3-handles, for 7\leq n\leq 9. A strategy for rational blowdown constructions of exotic CP^2#n(-CP^2) (1\leq n\leq 4) is also…