Topological Surgery, Dynamics and Applications To Natural Processes

TL;DR

This paper introduces a dynamic, continuous model of 2-dimensional 0-surgery, including a solid version for 3D phenomena, linking topology with natural processes like tornado formation.

Contribution

It extends formal 2D 0-surgery to include dynamics and interior space, providing a new theoretical framework for modeling natural phenomena.

Findings

Extended 2D 0-surgery to a continuous process influenced by local forces.

Defined solid 2D 0-surgery for 3D phenomena.

Connected the models with dynamical systems to simulate natural processes.

Abstract

In this paper we observe that 2-dimensional 0-surgery occurs in natural processes, such as tornado formation and other phenomena reminiscent of hole drilling. Inspired by such phenomena, we introduce new theoretical concepts which enhance the formal definition of 2-dimensional 0-surgery with the observed dynamics. To do this, we first present a schematic model which extends the formal definition to a continuous process caused by local forces. Next, for modeling phenomena which do not happen on surfaces but are three-dimensional, we fill the interior space by defining the notion of solid 2-dimensional 0-surgery. Finally, we connect these new theoretical concepts with a dynamical system and present it as a model for both 2-dimensional 0-surgery and natural phenomena exhibiting it. We hope that through this study, topology and dynamics of many natural phenomena will be better understood.