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**Chapter 7**

**Abstract**

Application of Spin-Orbit

Structures and Biology

*Richard Pinčák and Erik Bartoš*

graphene wormhole, genetic code

a radius of the nanotube is larger than its length. The metric tensor of the wormhole is given by

> 1 0 0 *r*<sup>2</sup> �

*,* <sup>Λ</sup>ð Þ¼ *<sup>r</sup>*� ð Þ *<sup>a</sup>=r*�

**1. Graphitic wormhole**

*<sup>g</sup>μν* <sup>¼</sup> <sup>Λ</sup><sup>2</sup>

**95**

ð Þ *r*�

Coupling in Exotic Graphene

An important measurable quantity in the carbon nanostructures, including the nanotubular part of the graphitic wormhole, is the spin-orbit coupling. We will present in this chapter spin-orbit coupling for the fermions located in exotic graphene structures as is graphene wormhole and also in biological systems. Considering this influence, the two-component Dirac equation is changed into the usual four-component form. As a consequence, the chiral fermions should be detected close to the wormhole bridge. We will show that the smaller is the radius of the wormhole bridge, the stronger this effect should be. Finally, we will describe the role of spinor fields in the time series of genetic code. The reversed transcription process of the gene expression could be defined by a moduli state space model of a coupling spinor field between the gene of a viral particle and the host cell. As a general result, all states of codon can be computed by the Chern-Simons 3-forms.

**Keywords:** spinor network structure, spin orbit coupling, Chern-Simons fields,

The investigation of unique chemical and mechanical properties of

nanostructures, e.g., fullerene, graphene, and nanotubes, promises a wide application in many technical areas. The electronic properties of the nanostructures are basically defined by their hexagonal carbon lattice structure and its variations. New promising results are expected with the preparation of more complicated forms as a wormhole. The wormhole is usually composed of two different kinds of nanostructure: two graphene sheets are connected together with the help of a connecting nanotube [1] (**Figure 1**). This is achieved by a supply of two sets of six heptagonal defects onto both sides of the given nanotube. There exists the restrictions on the form of the nanotube—the chirality must be 6ð Þ *n;* 6*n* armchair or 6ð Þ *n;* 0 zigzag and

2

*θ*ð Þþ *a* � *r*� *θ*ð Þ *r*� � *a ,* (1)

## **Chapter 7**
