arXiv research

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…

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

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…

Bridge trisections and knotted surfaces connected via tri-plane diagrams.

New bounds found for complexity of spun knots.

New method for simplifying complex 4D shapes with boundaries.

Every cubic graph is a bridge trisection's 1-skeleton for a knotted surface.

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 constructs Seifert solids from bridge trisections.

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

Meier and Zupan showed that every surface in the four-sphere admits a bridge trisection and can therefore be represented by three simple tangles. This raises the possibility of applying methods from link homology to knotted surfaces. We use link homology to construct an invariant of knotted surfaces (up to isotopy) whi…

2-twist trefoil has 6 crossings, proving non-trivial knotted surface.

We adapt work of Kirby-Thompson and Zupan to define an integer invariant $\mathcal{L}(\mathcal{T})$ of a bridge trisection $\mathcal{T}$ of a smooth surface $\mathcal{K}$ in $S^4$ or $B^4$. We show that when $\mathcal{L}(\mathcal{T})=0$, then the surface $\mathcal{K}$ is unknotted. We also show show that for a trisecti…

In this paper, we develop new techniques for understanding surfaces in $\mathbb{CP}^2$ via bridge trisections. Trisections are a novel approach to smooth 4-manifold topology, introduced by Gay and Kirby, that provide an avenue to apply 3-dimensional tools to 4-dimensional problems. Meier and Zupan subsequently develope…

Symplectic 4-manifolds can be divided into three parts with a special structure.

We give a new characterization of symplectic surfaces in CP^2 via bridge trisections. Specifically, a minimal genus surface in CP^2 is smoothly isotopic to a symplectic surface if and only if it is smoothly isotopic to a surface in transverse bridge position. We discuss several potential applications, including the cla…

Hexagonal diagrams link complex curves in $\mathbb{CP}^2$ to minimal genus surfaces.

Study surfaces in 4-manifolds using banded unlink diagrams.

Paper calculates L-invariant and L*-invariant for complex surface sums.

New method finds infinitely many surface knots with specific bridge numbers.

Algorithms compute invariants of 4-manifolds as branched covers.

Study of 3-manifolds in 5-sphere using bridge decompositions.

In this paper, we study surfaces embedded in $4$-manifolds. We give a complete set of moves relating banded unlink diagrams of isotopic surfaces in an arbitrary $4$-manifold. This extends work of Swenton and Kearton-Kurlin in $S^4$. As an application, we show that bridge trisections of isotopic surfaces in a trisected …

The paper explores meridional ranks of knotted surfaces and welded knots, proving equalities and relationships.

New method distinguishes 4-manifold types using trisections.

This paper constructs a Weinstein trisection for a surface bundle.

The paper constructs trisections for fiber bundles over a circle.

Paper studies simplified trisections and their equivalence classes.

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

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

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.

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…

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…

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

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

Standard trisection diagrams found for Mazur type 4-manifolds.

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

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

New exotic 4-manifolds found with small trisection genus.

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…

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

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.

This paper constructs explicit trisection diagrams for elliptic surfaces.