Extending Topological Surgery to Natural Processes and Dynamical Systems

TL;DR

This paper extends topological surgery to model natural phenomena and dynamical systems, incorporating forces and dynamics to better understand processes like DNA recombination, tornado formation, and black hole creation.

Contribution

It introduces continuous, solid, and embedded topological surgery concepts, linking them with dynamical systems to model complex natural processes.

Findings

Extended formal definition to continuous processes caused by local forces

Defined solid topological surgery for higher-dimensional phenomena

Connected topological surgery with dynamical systems models

Abstract

Topological surgery is a mathematical technique used for creating new manifolds out of known ones. We observe that it occurs in natural phenomena where a sphere of dimension 0 or 1 is selected, forces are applied and the manifold in which they occur changes type. For example, 1-dimensional surgery happens during chromosomal crossover, DNA recombination and when cosmic magnetic lines reconnect, while 2-dimensional surgery happens in the formation of tornadoes, in the phenomenon of Falaco solitons, in drop coalescence and in the cell mitosis. Inspired by such phenomena, we introduce new theoretical concepts which enhance topological surgery with the observed forces and dynamics. To do this, we first extend the formal definition to a continuous process caused by local forces. Next, for modeling phenomena which do not happen on arcs or surfaces but are 2-dimensional or 3-dimensional, we fill in the interior space by defining the notion of solid topological surgery. We further introduce the notion of embedded surgery in $S^3$ for modeling phenomena which involve more intrinsically the ambient space, such as the appearance of knotting in DNA and phenomena where the causes and effect of the process lies beyond the initial manifold, such as the formation of black holes. Finally, we connect these new theoretical concepts with a dynamical system and we present it as a model for both 2-dimensional 0-surgery and natural phenomena exhibiting a `hole drilling' behavior. We hope that through this study, topology and dynamics of many natural phenomena, as well as topological surgery itself, will be better understood.