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.
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 C P 2 \mathbb{CP}^2 CP 2 or C P 1 i m e s C P 1 \mathbb{CP}^1 imes\mathbb{CP}^1 CP 1 im es CP 1 admits an efficient bridge trisection. This paper constructs a Weinstein trisection for a surface bundle.
problem Constructing a trisection for a surface bundle.
method Weinstein trisection approach applied to Σ g i m e s Σ h Σ_g imes Σ_h Σ g im es Σ h . result Explicit construction of a Weinstein trisection for S 2 i m e s S 2 S^2 imes S^2 S 2 im es S 2 . The paper constructs non-diffeomorphic trisections of 4-manifolds.
problem Constructing distinct trisections of the same genus on a fixed 4-manifold.
method Using spun Seifert fiber spaces and Meier's spun trisections, the paper constructs 2 k − 1 2^{k}-1 2 k − 1 non-diffeomorphic ( 3 k , k ) (3k,k) ( 3 k , k ) -trisections. result The Nielsen classes of the generators of the fundamental group are isotopy invariants of the trisection.
Trisections divide 3-manifolds into three parts, with genus bounds and stabilizations.
problem Understanding the structure of 3-manifolds through trisections.
method Defining trisections, stabilization, and genus bounds for 3-manifolds.
result Stable equivalence of all trisections of a 3-manifold, with tight genus bounds.
Paper studies simplified trisections and their equivalence classes.
problem Understanding right-left equivalence of simplified ( 2 , 0 ) (2, 0) ( 2 , 0 ) -trisections. method Analyzes simplified trisection diagrams and upper-triangular handle-slides.
result At least two simplified ( 2 , 0 ) (2, 0) ( 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.
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.
The paper defines simplified trisections and provides a mapping class group criterion for their diagrams.
problem Characterizing diagrams of simplified trisections in terms of mapping class groups.
method Developed a necessary and sufficient condition for diagrams of simplified trisections using mapping class groups.
result Established that trisections of spun 4-manifolds are diffeomorphic to simplified ones.
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.
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.
Optimal policy for dynamic assortment planning under MNL model with O ( T ) O(\sqrt{T}) O ( T ) regret.
problem Maximizing revenue in dynamic assortment planning under MNL model with unknown parameters.
method Trisection-based policy with adaptive confidence bounds.
result Achieves O ( T ) O(\sqrt{T}) O ( T ) regret bound, independent of the number of products. 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 ( 2 n + 1 , − 2 ) (2n + 1, -2) ( 2 n + 1 , − 2 ) -torus knots. result Trisection diagrams are standard for spun ( 2 n + 1 , − 2 ) (2n + 1, -2) ( 2 n + 1 , − 2 ) -torus knots when n = 1 n = 1 n = 1 and homologically standard for all n n n . 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.
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 ) (g,k;p,b) ( 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.
This paper computes trisection genus for PL 4-manifolds using triangulations.
problem Computing the trisection genus for PL 4-manifolds.
method Algorithm implementation and triangulation-based approach.
result Lower bounds on trisection genus and construction of minimal genus trisections.
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.
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 extends trisection construction for Lefschetz fibrations with ( − n ) (-n) ( − 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.
The study connects Dehn surgeries on links to trisections of 4-manifolds.
problem Understanding Dehn surgeries on links via trisections.
method Expanding known connections and introducing new trisection conditions.
result New trisection conditions can obstruct reducibility of trisections.
Study trisections of spun 4-manifolds from 3-manifolds.
problem Understanding trisections of 4-manifolds from 3-manifolds.
method Local modifications to Heegaard diagrams to create trisection diagrams.
result New 4-manifolds with trisection genus three identified.
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 G G -manifolds and their quotients. method Introducing G G G -equivariant trisections and bridge trisections, and establishing their existence for G G G -manifolds. result Any G G G -manifold X X X admits a G G G -equivariant trisection such that a G G G -invariant surface S \mathcal{S} S is in equivariant bridge trisection position. 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 ) ‾ \overline{ν(S)} ν ( S ) and X − ν ( S ) X-ν(S) 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 ( 12 n − 2 , 0 ) (12n-2,0) ( 12 n − 2 , 0 ) -trisection diagrams of elliptic surfaces E ( n ) E(n) E ( n ) are constructed. 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.
Study of 2-knots via trisection diagrams, showing unique descriptions and operations.
problem Understanding and describing 2-knots via trisection diagrams.
method Use of doubly pointed trisection diagrams to describe and manipulate 2-knots.
result Unique descriptions of 2-knots via trisection diagrams, and operations on these diagrams.
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 4 4 -manifolds with connected boundary given by Gay and Kirby to include 4 4 4 -manifolds with an arbitrary number of boundary components. Additionally, we provide sufficient conditions under which relatively trisected 4 4 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 find trisections for 3-manifold bundles with specific Heegaard surfaces.
problem Finding trisections for 3-manifold bundles over S^1.
method Generalized the results of Gay and Kirby to find trisections of genus 3g+1.
result Existence of trisections of genus 3g+1 for bundles with specific Heegaard surfaces.
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 ) (p+1,p) ( p + 1 , p ) -torus knots. result Standard trisection diagrams were found for the spun ( p + 1 , p ) (p+1,p) ( p + 1 , p ) -torus knots. 3-manifolds can be embedded almost in the spine of a trisected 4-manifold.
problem Embedding 3-manifolds in trisected 4-manifolds.
method Defining and using the spine of a trisected 4-manifold, isotoping 3-manifolds to lie in the spine, and calculating bounds based on graph distances.
result Every 3-manifold can be embedded almost in the spine of a minimal genus trisection of connect sums of S 2 i l d e i m e s S 2 S^2 ilde imes S^2 S 2 i l d e im es S 2 . 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.
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 H 2 ( X ) H^2(X) H 2 ( X ) can be interpreted as ( 1 , 1 ) (1,1) ( 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 develop a technique for gluing relative trisection diagrams of 4 4 4 -manifolds with nonempty connected boundary to obtain trisection diagrams for closed 4 4 4 -manifolds. As an application, we describe a trisection of any closed 4 4 4 -manifold which admits a Lefschetz fibration over S 2 S^2 S 2 equipped with a section of square …
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 c c -component links with…
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.