arXiv subjects

Martín Eduardo Frías-Armenta

Publications and source records attributed to Martín Eduardo Frías-Armenta.

2 recordsLinked to original sources

arXiv2022

Minimal graphs for contractible and dismantlable properties

The notion of a contractible transformation on a graph was introduced by Ivashchenko as a means to study molecular spaces arising from digital topology and computer image analysis, and more recently has been applied to topological data analysis. Contractible transformations involve a list of four elementary moves that can be performed on the vertices and edges of a graph, and it has been shown by Chen, Yau, and Yeh that these moves preserve the simple homotopy type of the underlying clique complex. A graph is said to be ${\mathcal I}$-contractible if one can reduce it to a single isolated vertex via a sequence of contractible transformations. Inspired by the notions of collapsible and non-evasive simplicial complexes, in this paper we study certain subclasses of ${\mathcal I}$-contractible graphs where one can collapse to a vertex using only a subset of these moves. Our main results involve constructions of minimal examples of graphs for which the resulting classes differ. We also relate these classes of graphs to the notion of $k$-dismantlable graphs and $k$-collapsible complexes, which also leads to a minimal counterexample to an erroneous claim of Ivashchenko from the literature. We end with some open questions.

math.CO↗

arXiv2018

Classification of rational differential forms on the Riemann sphere, via their isotropy group

We classify the rational differential 1-forms with simple poles and simple zeros on the Riemann sphere according to their isotropy group; when the 1-form has exactly two poles the isotropy group is isomorphic to $\mathbb{C}^{*}$, namely $\{z\mapsto az\ \vert\ a\in\mathbb{C}, a\neq0\}$, and when the 1-form has $k\geq 3$ poles the isotropy group is finite. In particular we show that all the finite subgroups of $PSL(2,\mathbb{C})$ are realizable as isotropy groups for a rational 1-form on $\widehat{\mathbb{C}}$. We also present local and global geometrical conditions for their classification. The classification result enables us to describe the moduli space of rational 1-forms with finite isotropy that have exactly $k$ simple poles and $k-2$ simple zeros on the Riemann sphere. Moreover, we provide sufficient (geometrical) conditions for when the 1-forms are isochronous. Concerning the recent work of J.C.~Langer, we reflect on the strong relationship between our work and his and provide a partial answer regarding polyhedral geometries that arise from rational quadratic differentials on the Riemann sphere.

math.GT↗