# Cubic graphs induced by bridge trisections

@article{Meier2020CubicGI, title={Cubic graphs induced by bridge trisections}, author={J. Meier and A. Thompson and A. Zupan}, journal={arXiv: Geometric Topology}, year={2020} }

Every embedded surface $\mathcal{K}$ in the 4-sphere admits a bridge trisection, a decomposition of $(S^4,\mathcal{K})$ into three simple pieces. In this case, the surface $\mathcal{K}$ is determined by an embedded 1-complex, called the $\textit{1-skeleton}$ of the bridge trisection. As an abstract graph, the 1-skeleton is a cubic graph $\Gamma$ that inherits a natural Tait coloring, a 3-coloring of the edge set of $\Gamma$ such that each vertex is incident to edges of all three colors. In this… Expand

#### Figures from this paper

#### References

SHOWING 1-10 OF 12 REFERENCES

Bridge trisections of knotted surfaces in $S^4$

- Mathematics
- 2015

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… Expand

Every Planar Map Is Four Colorable

- Mathematics
- 1989

As has become standard, the four color map problem will be considered in the dual sense as the problem of whether the vertices of every planar graph (without loops) can be colored with at most four… Expand

Bridge trisections of knotted surfaces in 4-manifolds

- Mathematics, Medicine
- Proceedings of the National Academy of Sciences
- 2018

This paper defines generalized bridge trisections for knotted surfaces in more complicated four-dimensional spaces, offering a different approach to knotted surface theory, and proves 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. Expand

Knotted surfaces in 4-manifolds and stabilizations

- Mathematics
- 2015

In this paper, we study stable equivalence of exotically knotted surfaces in 4-manifolds, surfaces that are topologically isotopic but not smoothly isotopic. We prove that any pair of embedded… Expand

Cords and 1-handles attached to surface-knots

- Mathematics
- 2014

Boyle classified 1-handles attached to surface-knots, that are closed and connected surfaces embedded in the Euclidean $$4$$4-space, in the case that the surfaces are oriented and 1-handles are… Expand

Tait colorings, and an instanton homology for webs and foams

- Mathematics
- 2015

We use SO(3) gauge theory to define a functor from a category of unoriented webs and foams to the category of finite-dimensional vector spaces over the field of two elements. We prove a non-vanishing… Expand