Alex Zupan

In 2016, Jeff Meier and I adapted the theory of 4-manifold trisections to the setting of embedded 2-dimensional surfaces in 4-dimensional space, producing a new type of decomposition called a bridge trisection.  A bridge trisection of a surface S induces a graph G contained in S with the following properties:  Every vertex of G has valence three, and there is a 3-coloring of the edges of G such that every vertex is incident to edges of each color.  Such a graph is called a cubic Tait colored graph.  A natural question is whether this process can be reversed; namely, for every cubic Tait colored graph, is there a bridge trisection of an embedded surface S inducing it?  We answer this question affirmatively, proving further that the surface S can be chosen to be as simple as possible.  This is joint work with Jeff Meier and Abby Thompson.