Graph and string representation of trivalent 2-stratifolds with trivial fundamental group

Abstract

On topology, once a new family of mathematical objects is defined, it is common to look for characterizations or invariants that allow us to classify its members by equivalence classes. This is important because, in some cases, the family is compound by an infinite number of members, but with the use of class representatives the work is narrowed (even when the number of classes is infinite). This is the case of the trivalent 2-stratifolds, defined as a compact, simple connected, Hausdorff space X that contains a 1-manifold M such that the closure of X-M is the union of surfaces and every point of M has a regular neighborhood homeomorphic to the product of an Interval and the open cone of 3 points. In this thesis, we studied the trivalent 2-stratifolds with trivial fundamental group. Inspired by the work published by J. C. Gómez- Larrañaga, F. González-Acuña, and W. Heil, who proved that the trivalent 2-stratifolds with trivial fundamental group can be characterized by a bipartite graph, we developed an invariant that can identify if two graphs come from the same trivalent 2-stratifolds. In collaboration with Dr. Jesús Rodríguez-Viorato, we implemented a Python program that not only calculates this invariant but also builds and draws the graphs from “small” trivalent 2-stratifolds.

Premio “Sotero Prieto” 2023, por la Mejor Tesis de Licenciatura, teniendo como marco el comienzo del Quincuagésimo Sexto Congreso Nacional de la Sociedad Matemática Mexicana