Stathis Antoniou

Extending topological surgery to natural processes and dynamical systems

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 S3 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.

Topological surgery, dynamics and applications to natural processes

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.

A Dynamical System Modeling Solid 2-Dimensional 0-Surgery

So far, inspired by natural processes undergoing surgery, we have extended the formal definition of topological surgery by introducing new notions such as continuity and solid surgery and presented a model showing where the observed forces act. However, in our model, time and dynamics were not introduced by differential equations. In this chapter we connect topological surgery, enhanced with these notions, with a dynamical system. We will see that, with a small change in parameters, the trajectories of its solutions are performing solid 2-dimensional 0-surgery. Therefore, this dynamical system constitutes a specific set of equations modeling natural phenomena undergoing solid 2-dimensional 0-surgery.

3-Dimensional Surgery

In this chapter we present a novel way of visualizing 3-dimensional surgery as well as a phenomenon exhibiting it. In Sect. 10.1, we introduce the notion of decompactified 2-dimensional surgery which allows us to visualize the process of 2-dimensional surgery in \({\mathbb R}^2\) instead of \({\mathbb R}^3\). Using this new notion and rotation, in Sect. 10.2, we present a way to visualize the 4-dimensional process of 3-dimensional surgery in \({\mathbb R}^3\). In Sect. 10.3, we analyze the concept of continuity introduced in Chap. 4 in the case of 3-dimensional surgery by looking at the local process inside the 4-dimensional handle. Finally, in Sect. 10.4, we model a phenomenon exhibiting 3-dimensional surgery: the formation of black holes from cosmic strings.

Useful Mathematical Notions

In this chapter we introduce the basic notions related to topological surgery. Readers that are familiar with the formalism of the topic can directly move to the formal definition in Chap. 3.

The Ambient Space \(S^{3}\)

All natural phenomena exhibiting surgery (1- or 2-dimensional, solid or usual) take place in the ambient 3-space. Moreover, as mentioned in Sect. 5.4, the ambient space can play an important role in the process of surgery. This will be detailed in next chapter where the notion of embedded surgery in 3-space is introduced. By 3-space we mean here the compactification of \({\mathbb R}^3\) which is the 3-sphere \(S^3\). This choice, as opposed to \({\mathbb R}^3\), takes advantage of the duality of the descriptions of \(S^3\). In this section we present the three most common descriptions of \(S^3\) (see Sect. 8.1) in which this duality is apparent and which will set the ground for defining the notion of embedded surgery in \(S^3\) (see Chap. 9). Beyond that, in Sect. 8.2, we also demonstrate how these descriptions are interrelated. Finally, in Sect. 8.3, we pin down how the trajectories of the dynamical system (\(\Sigma \)) presented in Chap. 7 are related to the descriptions of \(S^3\) and further introduce a Hamiltonian system exhibiting the topology of \(S^3\).

Topological Surgery in Nature

In this paper, we extend the formal definition of topological surgery by introducing new notions in order to model natural phenomena exhibiting it. On the one hand, the common features of the presented natural processes are captured by our schematic models and, on the other hand, our new definitions provide the theoretical setting for examining the topological changes involved in these processes.