Corpus ID: 220525431

Cubic graphs induced by bridge trisections

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


Bridge trisections of knotted surfaces in $S^4$
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 ofExpand
Every Planar Map Is Four Colorable
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 fourExpand
Bridge trisections of knotted surfaces in 4-manifolds
  • J. Meier, A. Zupan
  • 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
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 embeddedExpand
Cords and 1-handles attached to surface-knots
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 areExpand
Tait colorings, and an instanton homology for webs and foams
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-vanishingExpand
Surfaces In 4 Space
Proof of a conjecture of Whitney.