arXiv research

New method distinguishes 4-manifold types using trisections.

The paper constructs trisections for fiber bundles over a circle.

Study Stein fillings of planar contact 3-manifolds with relative trisection genus 2.

Two non-diffeomorphic minimal genus trisections found for a 4-manifold.

New trisection concept for non-closed 4-manifolds connects to group theory.

Paper calculates homology and intersection form of trisected 4-manifolds with boundary.

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…

The paper proves trisection genus of Akbulut cork and constructs many corks with trisection genus 3.

In this paper, we introduce the relative $\mathcal{L}$-invariant $r\mathcal{L}(X)$ of a smooth, orientable, compact 4-manifold $X$ with boundary. This invariant is defined by measuring the lengths of certain paths in the cut complex of a trisection surface for $X$. This is motivated by the definition of the $\mathcal{L…

New exotic 4-manifolds found with small trisection genus.

Trisections are obtained by regluing surface-knots in 4-manifolds.

Standard trisection diagrams found for Mazur type 4-manifolds.

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…

Study relates trisected 4-manifolds to cork twists via γ-curves.

New method for simplifying complex 4D shapes with boundaries.

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…

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 $S^2$ equipped with a section of square …

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 an $S\cong \mathbb{R}P^2$ smoothly embedded in a 4-manifold $X^4$ with Euler number 2 or -2, the Price twist is a surgery operation on $ν(S)$ yielding (up to) three different 4-manifolds: $X^4,τ_S(X^4),Σ_S(X^4)$. This is of particular interest when $X^4=S^4$, as then $Σ_S(X^4)$ is a homotopy 4-sphere which is not…

These notes summarize and expand on a mini-course given at CIRM in February 2018 as part of Winter Braids VIII. We somewhat obsessively develop the slogan `Trisections are to 4-manifolds as Heegaard splittings are to 3-manifolds', focusing on and clarifying the distinction between three ways of thinking of things: the …

This paper constructs a Weinstein trisection for a surface bundle.

Paper studies simplified trisections and their equivalence classes.

This paper proves a conjecture about trisections with a specific length.

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.

Classifies knot traces with specific trisection genus limits.

The goal of this paper is to construct distinct trisections of the same genus on a fixed 4-manifold. For every $k \geq 2$, we construct $2^{k}-1$ 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…

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 $\mathbb{CP}^2$ and $\mathbb{CP}^1\times\mathbb{CP}^1$. We are especially interested in bridge trisections and trisections that are as simple as possible, whi…

The paper examines trisection diagrams of spun knots and shows they are standard for certain cases.

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.

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…

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…

Equivariant trisections for group actions on 4-manifolds are introduced and studied.

This paper constructs explicit trisection diagrams for elliptic surfaces.

Motivated by M. Scharlemann and A. Thompson's definition of thin position of 3-manifolds, we define the width of a handle decomposition a 4-manifold and introduce the notion of thin position of a compact smooth 4-manifold. We determine all manifolds having width equal to $\{1,\dots, 1\}$, and give a relation between th…

Algorithm converts Kirby diagrams to trisection diagrams for 4-manifolds.

The idea of studying trisections of closed smooth $4$-manifolds via (singular) triangulations, endowed with a suitable vertex-labelling by three colors, is due to Bell, Hass, Rubinstein and Tillmann, and has been applied by Spreer and Tillmann to colored triangulations associated to the so called simple crystallization…

The paper extends trisection theory to non-orientable 4-manifolds using colored triangulations.

New method shows certain group presentations are trivial.

We generalize bridge trisections to surfaces in four-manifolds, linking them to braided links.

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.

Standard trisection diagrams found for a specific type of knot.

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…

Gem theory helps estimate trisection genus of 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.