Study complex curves in rational surfaces using bridge trisections.
problem Classifying and understanding complex curves in rational surfaces.
method Bridge trisections and branched covering operations.
result Any curve in CP2 or CP1imesCP1 admits an efficient bridge trisection. Algorithm converts Kirby diagrams to trisection diagrams for 4-manifolds.
problem Creating efficient trisection diagrams for 4-manifolds.
method Algorithm converting Kirby diagrams to trisection diagrams.
result Provides examples of trisection diagrams for 4-manifolds.
The paper generalizes trisection definitions and applies them to specific cases.
problem Defining and applying trisections to 4-manifolds and surfaces.
method Generalized trisection definitions and their application to specific cases.
result Classification of an infinite family of genus three trisections.
New method distinguishes 4-manifold types using trisections.
problem Distinguishing different 4-manifold types.
method Capping operation to transform relative trisections into closed 4-manifold diagrams.
result Examples of non-diffeomorphic relative trisections of the same 4-manifold.
This paper constructs a Weinstein trisection for a surface bundle.
problem Constructing a trisection for a surface bundle.
method Weinstein trisection approach applied to ΣgimesΣh. result Explicit construction of a Weinstein trisection for S2imesS2. Paper studies simplified trisections and their equivalence classes.
problem Understanding right-left equivalence of simplified (2,0)-trisections. method Analyzes simplified trisection diagrams and upper-triangular handle-slides.
result At least two simplified (2,0)-trisections can be right-left equivalent without being related by automorphisms or handle-slides. The paper constructs trisections for fiber bundles over a circle.
problem Building trisections for fiber bundles over the circle.
method Using sutured Heegaard splittings and diagrams, the paper provides an algorithm to construct relative trisection diagrams.
result Explicit construction of relative trisection diagrams for fiber bundles over the circle.
This paper proves a conjecture about trisections with a specific length.
problem Proving a conjecture about trisections with a specific length.
method Examining trisections with Kirby-Thompson length 2 and proving the conjecture.
result Proves the conjecture about length 2 trisection being a 4-manifold with length 0.
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.
We define a trisection of a closed, orientable three dimensional manifold into three handlebodies, and a notion of stabilization for these trisections. Several examples of trisections are described in detail. We define the trisection genus t(M) of a 3-manifold, and relate it to the Heegaard genus g(M), showing that…
Study trisections of non-orientable 4-manifolds with boundary.
problem Understanding trisections in non-orientable 4-manifolds.
method Introduced trisections of non-orientable 4-manifolds with boundary, proved a non-orientable analogue of a theorem, and discussed adaptation of trisection theory.
result Existence of trisection diagrams and Kirby diagrams for closed non-orientable 4-manifolds.
Hexagonal diagrams link complex curves in CP2 to minimal genus surfaces.
problem Understanding the relationship between complex curves and surfaces in CP2. method Hexagonal lattice diagrams and trisection of CP2. result Positive genus surfaces in CP2 are isotopic to complex curves if they admit hexagonal lattice diagrams. New method for simplifying complex 4D shapes with boundaries.
problem Complexity reduction in 4D shape analysis.
method Introducing pseudo-trisections and pseudo-bridge trisections.
result Existence and uniqueness of pseudo-trisections established.
Classifies knot traces with specific trisection genus limits.
problem Classifying knot traces with specific trisection genus limits.
method Classifying knot traces with specific trisection genus limits.
result Infinitely many knots have traces with trisection genus 3 and 4, and arbitrarily large trisection genus.
The goal of this paper is to construct distinct trisections of the same genus on a fixed 4-manifold. For every k≥2, we construct 2k−1 non-diffeomorphic (3k,k)-trisections on infinitely many 4-manifolds. Here, the manifolds are spun Seifert fiber spaces and the trisections come from Meier's spun trisection…
Paper calculates homology and intersection form of trisected 4-manifolds with boundary.
problem Calculating homology and intersection form of trisected 4-manifolds with boundary.
method Uses relative trisection diagrams to calculate homology and intersection form.
result Describes a representative of the second Stiefel-Whitney class using relative trisection diagrams.
New trisection concept for non-closed 4-manifolds connects to group theory.
problem Defining and analyzing trisections of non-closed 4-manifolds.
method Generalized group trisections to non-closed cases and established a one-to-one correspondence.
result Established a functorial relationship between relative trisections and groups.
The paper examines trisection diagrams of spun knots and shows they are standard for certain cases.
problem Whether trisection diagrams induced by the Gluck surgery on specific knots are standard.
method Explicit depiction and analysis of trisection diagrams for spun (2n+1,−2)-torus knots. result Trisection diagrams are standard for spun (2n+1,−2)-torus knots when n=1 and homologically standard for all n. Two non-diffeomorphic minimal genus trisections found for a 4-manifold.
problem Existence of non-diffeomorphic minimal genus trisections of the same 4-manifold.
method Introduced a simple operation to create a trisection diagram from a relative trisection diagram.
result Existence of two non-diffeomorphic minimal genus trisections of the same (g,k;p,b)-type 4-manifold. Standard trisection diagrams found for Mazur type 4-manifolds.
problem Finding standard trisection diagrams for Mazur type 4-manifolds.
method Doubling a relative trisection diagram and using an algorithm from Kirby diagrams to trisection diagrams.
result Certain trisection diagrams of Mazur type 4-manifolds are standard.
The paper generalizes trisections to compact PL 4-manifolds with boundary and introduces gem-induced trisections.
problem Generalizing trisections to compact PL 4-manifolds with boundary and any 5-colored graph encoding simply-connected 4-manifolds.
method Introducing gem-induced trisections and analyzing their properties for compact PL 4-manifolds with and without boundary.
result Conditions for gem-induced trisections to realize the G-trisection genus and direct determination of the genus from the graph.
New exotic 4-manifolds found with small trisection genus.
problem Finding exotic 4-manifolds with small trisection genus.
method Constructing genus-3 relative trisections for contractible 4-manifolds.
result Exotic 4-manifolds with boundary have trisection genus of 4.
A simplified trisection is a trisection map on a 4-manifold such that, in its critical value set, there is no double point and cusps only appear in triples on innermost fold circles. We give a necessary and sufficient condition for a 3-tuple of systems of simple closed curves in a surface to be a diagram of a simplifie…
The paper extends trisection construction for Lefschetz fibrations with (−n)-sections.
problem Constructing trisections for 4-manifolds with specific Lefschetz fibrations.
method Explicit construction of trisections using monodromies of Lefschetz fibrations.
result Trisections for various 4-manifolds constructed from Lefschetz fibrations.
A trisection of a smooth, closed, oriented 4-manifold is a decomposition into three 4-dimensional 1-handlebodies meeting pairwise in 3-dimensional 1-handlebodies, with triple intersection a closed surface. The fundamental groups of the surface, the 3-dimensional handlebodies, the 4-dimensional handlebodies, and the clo…
Study Stein fillings of planar contact 3-manifolds with relative trisection genus 2.
problem Classify Stein fillings of planar contact 3-manifolds under constraints on their relative trisections.
method Partial classification of diffeomorphism types of fillings with relative trisections of genus at most 2.
result Partially classify the diffeomorphism types of Stein fillings with relative trisections of genus at most 2.
Equivariant trisections for group actions on 4-manifolds are introduced and studied.
problem Understanding the equivariant topology of G-manifolds and their quotients. method Introducing G-equivariant trisections and bridge trisections, and establishing their existence for G-manifolds. result Any G-manifold X admits a G-equivariant trisection such that a G-invariant surface S is in equivariant bridge trisection position. Gay and Kirby recently introduced the concept of a trisection for arbitrary smooth, oriented closed 4-manifolds, and with it a new topological invariant, called the trisection genus. This paper improves and implements an algorithm due to Bell, Hass, Rubinstein and Tillmann to compute trisections using triangulations, a…
Trisections are obtained by regluing surface-knots in 4-manifolds.
problem Understanding trisections in 4-manifolds using surface-knots.
method Trivial gluing of relative trisections of ν(S) and X−ν(S). result Trisection obtained is diffeomorphic to a stabilization of the original trisection.
This paper constructs explicit trisection diagrams for elliptic surfaces.
problem Constructing explicit trisection diagrams for elliptic surfaces.
method Using handle diagrams from Lefschetz fibrations to create trisection diagrams.
result Explicit (12n−2,0)-trisection diagrams of elliptic surfaces E(n) are constructed. The paper extends trisection theory to non-orientable 4-manifolds using colored triangulations.
problem Extending trisection theory to non-orientable 4-manifolds.
method Using colored triangulations and gems to induce trisections.
result Gem-induced trisections naturally give rise to trisections of closed 4-manifolds.
We extend the theory of relative trisections of smooth, compact, oriented 4-manifolds with connected boundary given by Gay and Kirby to include 4-manifolds with an arbitrary number of boundary components. Additionally, we provide sufficient conditions under which relatively trisected 4-manifolds can be glued to o…
We generalize bridge trisections to surfaces in four-manifolds, linking them to braided links.
problem Embedding surfaces in four-manifolds and understanding their braiding.
method Introducing bridge trisections, isotoping surfaces, and using trisections and open-book decompositions.
result Any neatly embedded surface can be isotoped to lie in bridge trisection position with respect to any trisection of the four-manifold.
We classify a large class of "unbalanced" 4-manifold GK-trisections, which are a slight generalization of 4-manifold trisections defined by Gay and Kirby.
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.
Standard trisection diagrams found for a specific type of knot.
problem Finding standard trisection diagrams for Gluck twists.
method Using a specific method to obtain trisection diagrams for spun (p+1,p)-torus knots. result Standard trisection diagrams were found for the spun (p+1,p)-torus knots. We summarize and expand known connections between the study of Dehn surgery on links and the study of trisections of closed, smooth 4-manifolds. In addition, we describe how the potential counterexamples to the Generalized Property R Conjecture given by Gompf, Scharlemann, and Thompson yield genus four trisections of t…
Given a handle decomposition of a 4-manifold with boundary, and an open book decomposition of the boundary, we show how to produce a trisection diagram of a trisection of the 4-manifold inducing the given open book. We do this by making the original proof of the existence of relative trisections more explicit, in terms…
Gem theory helps estimate trisection genus of 4-manifolds.
problem Estimating the trisection genus of 4-manifolds.
method Using gem theory, a type of edge-colored graphs dual to colored triangulations.
result Regular genus is an upper bound for trisection genus of closed 4-manifolds.
We study trisections of 4-manifolds obtained by spinning and twist-spinning 3-manifolds, and we show that, given a (suitable) Heegaard diagram for the 3-manifold, one can perform simple local modifications to obtain a trisection diagram for the 4-manifold. We also show that this local modification can be used to conver…
Notes describe geometric interpretations of cohomology in trisected 4-manifolds.
problem Understanding geometric interpretations of cohomology classes in trisected 4-manifolds.
method Analogy with Hodge theory and sheaf cohomology in algebraic geometry.
result Classes in H2(X) can be interpreted as (1,1)-classes. Explains Kirby's work on 4-manifold trisections.
problem Understanding trisections of 4-manifolds.
method Near-symplectic constructions.
result Developed a theory of trisections.
We show that the members of a large class of unbalanced four-manifold trisections are standard, and we present a family of trisections that is likely to include non-standard trisections of the four-sphere. As an application, we prove a stable version of the Generalized Property R Conjecture for c-component links with…
We develop a technique for gluing relative trisection diagrams of 4-manifolds with nonempty connected boundary to obtain trisection diagrams for closed 4-manifolds. As an application, we describe a trisection of any closed 4-manifold which admits a Lefschetz fibration over S2 equipped with a section of square …
Trisections help compute Heegaard Floer homology invariants.
problem Computing Heegaard Floer homology invariants for four-manifolds.
method Using trisection maps to induce cobordism maps on Heegaard Floer homology.
result Trisections provide a method to compute cobordism invariants.
We introduce bridge trisections of knotted surfaces in the four-sphere. This description is inspired by the work of Gay and Kirby on trisections of four-manifolds and extends the classical concept of bridge splittings of links in the three-sphere to four dimensions. We prove that every knotted surface in the four-spher…
Efficient multisections found for odd-dimensional tori.
problem Finding optimal multisections of odd-dimensional tori.
method Constructing multisections with specific properties (genus n, symmetry) for odd-dimensional tori. result Optimal multisections of genus n found for odd-dimensional tori. Smooth 5-manifolds can be decomposed into three pieces.
problem Decomposing smooth 5-manifolds into three pieces.
method Introduced a trisection of a 5-manifold, compatible with any desired trisection of its boundary.
result Every smooth, oriented, compact 5-manifold admits a smooth trisection.