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.
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 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. 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 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.
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.
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 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…
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. 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.
We study embedded spheres in 4-manifolds (2-knots) via doubly pointed trisection diagrams, showing that such descriptions are unique up to stabilization and handleslides, and we describe how to obtain trisection diagrams for certain cut-and-paste operations along 2-knots directly from doubly pointed trisection diagrams…
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 …
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.
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.
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. 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…
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.
In this note, we provide a generalization for the definition of a trisection of a 4-manifold with boundary. We demonstrate the utility of this more general definition by finding a trisection diagram for the Cacime Surface, and also by finding a trisection-theoretic way to perform logarithmic surgery. In addition, we de…
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.
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.
New 4-manifold invariant defined from trisection diagrams.
problem Defining a new 4-manifold invariant from trisection diagrams.
method Algebraic data from bimodule categories and spherical fusion categories, described diagrammatically.
result Includes Hopf algebraic invariants and modular fusion category invariants.
New method constructs Seifert solids from bridge trisections.
problem Constructing Seifert solids from bridge trisections.
method Adapting Seifert's algorithm to tri-plane diagrams.
result Classification results on surface decomposability and unknottedness.
We establish a correspondence between trisections of smooth, compact, oriented 4--manifolds with connected boundary and diagrams describing these trisected 4--manifolds. Such a diagram comes in the form of a compact, oriented surface with boundary together with three tuples of simple closed curves, with possibly fe…
Given a diagram for a trisection of a 4-manifold X, we describe the homology and the intersection form of X in terms of the three subgroups of H1(Σ;Z) generated by the three sets of curves and the intersection pairing on the diagram surface Σ. This includes explicit formulas for the second and third h…
A theorem of Katanaga, Saeki, Teragaito, and Yamada relates Gluck and Price twists of 4-manifolds. Using trisection diagrams, we give a purely diagrammatic proof of this theorem, and answer a question of Kim and Miller.
New invariant calculates 4-manifolds using trisection diagrams and combings.
problem Calculating non-semisimple 4-manifold invariants.
method Using trisection diagrams and combings of the trisection surface.
result Invariant calculated for Stein nuclei, generalizing earlier semisimple version.
Study surfaces in 4-manifolds using banded unlink diagrams.
problem Representing and manipulating immersed surfaces in 4-manifolds.
method Singular banded unlink diagrams and associated moves.
result Every immersed surface can be represented uniquely by a singular banded unlink diagram.
Multisections generalize Heegaard splittings and trisections to higher dimensions.
problem Decomposing higher-dimensional manifolds into 1-handlebodies with specific intersection properties.
method Considering multisections of higher-dimensional smooth or PL closed orientable manifolds, and associating diagrams to them.
result Multisection diagrams uniquely determine manifolds in all dimensions up to 6.
Let X be a bundle over S1 with fiber a 3--manifold M and with monodromy φ. Gay and Kirby showed that if φ fixes a genus g Heegaard splitting of M then X has a genus 6g+1 trisection. Genus 3g+1 trisections have been found in certain special cases, such as the case where φ is triv…
Every cubic graph is a bridge trisection's 1-skeleton for a knotted surface.
problem Understanding cubic graphs and their relation to bridge trisections.
method Proving every Tait-colored cubic graph is a 1-skeleton of a bridge trisection.
result Every Tait-colored cubic graph corresponds to a bridge trisection of a knotted surface.
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 prove that every smoothly embedded surface in a 4--manifold can be isotoped to be in bridge position with respect to a given trisection of the ambient 4--manifold; that is, after isotopy, the surface meets components of the trisection in trivial disks or arcs. Such a decomposition, which we call a \emph{generalized …
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.
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…
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. 2-twist trefoil has 6 crossings, proving non-trivial knotted surface.
problem Computing the crossing number of non-trivial knotted surfaces.
method Bridge trisection and tri-plane diagrams to minimize crossings.
result 2-twist trefoil has crossing number 6.
Involutory Hopf group-coalgebras provide new invariants for 4-manifold bundles.
problem Developing invariants for flat bundles over 4-manifolds.
method Utilizing Hopf G-triplets and colored trisection diagrams. result Involutory Hopf G-triplets yield well-defined invariants of G-colored trisection diagrams. 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 smooth isotopy classes of complex curves in complex surfaces from the perspective of the theory of bridge trisections, with a special focus on curves in CP2 and CP1×CP1. We are especially interested in bridge trisections and trisections that are as simple as possible, whi…
Gay and Kirby introduced trisections which describe any closed oriented smooth 4-manifold X as a union of three four-dimensional handlebodies. A trisection is encoded in a diagram, namely three collections of curves in a closed oriented surface Σ, guiding the gluing of the handlebodies. Any morphism φ from …
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…
Algorithm trisects 4D book into three chapters.
problem Trisecting 4D book into chapters.
method Algorithm for open book decompositions of 4-manifolds.
result Algorithm works for various types of manifolds.
Multisections generalize trisections for 4-manifolds, allowing complex operations and explicit diagrams.
problem Decomposing arbitrary smooth 4-manifolds into simpler pieces.
method Introducing multisections as a sequence of cut systems on a surface, enabling smooth operations and explicit diagrams.
result Elliptic fibrations admit genus 3 multisections, with explicit diagrams.
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…
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. We show that the only closed 4-manifolds admitting genus two trisections are S2×S2 and connected sums of S1×S3, CP2, and CP2 with two summands. Moreover, each of these manifolds admits a unique genus two trisection up to diffeomorphism. The proof relies heavily o…
Algorithms compute invariants of 4-manifolds as branched covers.
problem Computing invariants of 4-manifolds as branched covers.
method Diagrammatic algorithms using tri-plane diagrams and permutations.
result Automated algorithm for computing Kjuchukova's homotopy-ribbon obstruction.
Nonorientable 4-manifolds can be fibred over 2-disks with nonorientable fibers.
problem Fibering nonorientable 4-manifolds over 2-disks.
method Constructing Lefschetz fibrations over 2-disks with nonorientable fibers.
result Every nonorientable closed 3-manifold admits an open book decomposition.