**3. Circular Artin braid group representation for spinor field in genetic code**

In each cell division, the telomeres are shortened [31], and total length of DNA is changing. As the result of shorter biological clock from cell division, the living things die. In order to understand cell cocycle and division mechanism of telomerase aging, one can explain the source of cancer as a source of age acceleration and its relationship to telomere shortening mechanism. It is a source of braid group operation [32] so-called self-diffeomorphism in the genetic code. The age acceleration is a relative measurement between the chronical clock and the biological clock in telomere. Up to now, scientists understand that a telomere and telomerase are the locations of ancient viruses that rely on DNA in the chromosomes of living organisms. Telomere is composed of the repeated sequence of ð Þ *TTAGGG dt* <sup>∗</sup> where

1000 ≤*dt* <sup>∗</sup> ≤2000. The size of the duplicate sequence at the end of this open chromosome is amplified by six braids caused by six superspaces in time series data of organisms. The G alphabet might be suitable to be chosen as hidden time scale in the biological clock.

Here we assume that all genetic code cannot be completely separated and biological clock in telomere length is parametrized by a hidden state of the number of *dt* <sup>∗</sup> <sup>≔</sup> ½ � *<sup>G</sup>* alphabet in ð Þ *TTAGGG <sup>n</sup>*¼*dt* <sup>∗</sup> repeated pattern of the telomere. The element is Grothendieck topology over an adjoint cofunctor; it is a selfdiffeomorphism *ξ* : *Bc* <sup>3</sup> ! *Bc* 3. The loop braid generator for *B<sup>c</sup>* <sup>3</sup> is a quaternionic field in genetic code. We define their explicit forms and their permutations over the symmetric group by a chosen basis in Clifford algebra as

$$
\sigma\_D \to \begin{bmatrix} \mathbf{1}, \mathbf{1}, -\mathbf{1} \\ \mathbf{2}, \mathbf{7}, -\mathbf{2} \end{bmatrix}, \quad \sigma\_R \to \begin{bmatrix} -\mathbf{1}, \mathbf{1}, \mathbf{1} \\ \mathbf{2}, \mathbf{7}, \mathbf{2} \end{bmatrix}, \quad \sigma\_P \to \begin{bmatrix} \mathbf{1}, -\mathbf{1}, \mathbf{1} \\ \mathbf{2}, -\mathbf{2}, \mathbf{7} \end{bmatrix}, \tag{46}
$$

and *<sup>σ</sup>D*<sup>∗</sup> <sup>¼</sup> *<sup>σ</sup>*�<sup>1</sup> *<sup>D</sup>* , *<sup>σ</sup>R*<sup>∗</sup> <sup>¼</sup> *<sup>σ</sup>*�<sup>1</sup> *<sup>R</sup>* , and *<sup>σ</sup>P*<sup>∗</sup> <sup>¼</sup> *<sup>σ</sup>*�<sup>1</sup> *<sup>P</sup>* . We have

$$
\sigma\_{D^\*} \to \left[ -\frac{1}{2}, -\frac{1}{2}, \frac{1}{2} \right], \quad \sigma\_{R^\*} \to \left[ \frac{1}{2}, -\frac{1}{2}, -\frac{1}{2} \right], \quad \sigma\_{P^\*} \to \left[ -\frac{1}{2}, \frac{1}{2}, -\frac{1}{2} \right], \tag{47}
$$

therefore one can write eight bases for spinor field in the genetic code in braid form as follows

$$\begin{aligned} \sigma^{[G]} &:= [0, 0, 0] \sigma\_R^{-1} \sigma\_D = \sigma\_R^{-1} \sigma\_D, \\ \sigma^{[A]} &:= [0, 0, 1] \sigma\_R^{-1} \sigma\_D = (\sigma\_D \sigma\_R) \left(\sigma\_R^{-1} \sigma\_D\right), \\ \sigma^{[U]} &:= [0, 1, 0] \sigma\_R^{-1} \sigma\_D = (\sigma\_P \sigma\_D) \left(\sigma\_R^{-1} \sigma\_D\right), \\ \sigma^{[C]} &:= [1, 0, 0] \sigma\_R^{-1} \sigma\_D = (\sigma\_R \sigma\_P) \left(\sigma\_R^{-1} \sigma\_D\right), \\ \sigma^{[NU]} &:= [0, 1, 1] \sigma\_R^{-1} \sigma\_D = (\sigma\_R \sigma\_P) (\sigma\_P \sigma\_D) \left(\sigma\_R^{-1} \sigma\_D\right), \\ \sigma^{[NU]} &:= [1, 1, 0] \sigma\_R^{-1} \sigma\_D = (\sigma\_D \sigma\_D) (\sigma\_R \sigma\_P) \sigma\_R^{-1} \sigma\_D, \\ \sigma^{[NU]} &:= [1, 0, 1] \sigma\_R^{-1} \sigma\_D = (\sigma\_R \sigma\_D) (\sigma\_D \sigma\_D) (\sigma\_R^{-1} \sigma\_D), \\ \sigma^{[NG]} &:= [1, 1, 1] \sigma\_R^{-1} \sigma\_D = (\sigma\_R \sigma\_P) (\sigma\_P \sigma\_D) (\sigma\_D \sigma\_R) (\sigma\_R^{-1} \sigma\_D). \end{aligned}$$

We may also use *θ* ¼ 2*πs* with spin quantum number *s* being an integer for retrotransposon and half-integer for geneon, so that

$$\mathbf{e}^{\rm i\theta} = \mathbf{e}^{2i\pi s} = (-1)^{2s} \boldsymbol{\mu}\_2 \boldsymbol{\mu}\_1. \tag{49}$$

The loop braid group, *LBBIO*

<sup>2</sup>*<sup>i</sup>* <sup>¼</sup> <sup>1</sup>*;* <sup>⋯</sup>*; <sup>n</sup>* � <sup>1</sup> and *<sup>τ</sup><sup>i</sup>* <sup>∈</sup> *Bc*

*τ*2

*group of affine transform of* 3 *behavior fields in genetic code σ*

*, σim*� *, σ<sup>m</sup>*� ∈*B<sup>c</sup>*

*ρiτ<sup>j</sup>* ¼ *τiρ<sup>j</sup>*

*<sup>τ</sup><sup>i</sup>*þ<sup>1</sup>*σ<sup>i</sup>* <sup>¼</sup> *<sup>ρ</sup>iσ*�<sup>1</sup>

*σi; ρ<sup>i</sup>* ∈ *B*<sup>3</sup>

**Figure 4.**

*element in σ*

*B*3 2⋊*B<sup>c</sup>* 3*.*

½ � *<sup>A</sup><sup>μ</sup>* <sup>1</sup> *; σ* ½ � *<sup>A</sup><sup>μ</sup>* <sup>1</sup> *; σ* ½ � *<sup>A</sup><sup>μ</sup>* 1 <sup>∈</sup>*B*<sup>3</sup>

*field in the genetic code σ<sup>ω</sup>*�

*LBBIO*

**109**

anyon *Type* � *I* for biology,

*ρType*�*<sup>I</sup>*

ð Þ *<sup>σ</sup><sup>i</sup> <sup>ρ</sup>Type*�*<sup>I</sup>*

<sup>¼</sup> *<sup>ρ</sup>Type*�*<sup>I</sup>*

*<sup>n</sup>* , for a genetic code has three types of generators,

<sup>3</sup>*. We have affine group as loop braid group in genetic code by*

<sup>2</sup>*. The red color line represents the curvature from the physiology of biological*

(50)

<sup>3</sup>*, the structure*

<sup>2</sup> *action on affine fibre bundle of behavior*

*<sup>n</sup>* and in

,

<sup>3</sup>*<sup>i</sup>* <sup>¼</sup> <sup>1</sup>*;* <sup>⋯</sup>*; <sup>n</sup>* fulfill the relations

½ � *<sup>A</sup><sup>μ</sup> <sup>i</sup>* ∈*B*<sup>3</sup>

*<sup>σ</sup>* <sup>¼</sup> ð Þ *<sup>σ</sup>D; <sup>σ</sup>R; <sup>σ</sup><sup>P</sup>* , *<sup>ρ</sup>* <sup>¼</sup> *<sup>σ</sup>*½ � *<sup>A</sup> ; ; <sup>σ</sup>*½ � *<sup>U</sup> ; ; <sup>σ</sup>*½ � *<sup>C</sup> ; ; <sup>σ</sup>*½ � *<sup>G</sup>* , and *<sup>τ</sup>* <sup>¼</sup> *<sup>σ</sup>ω; ; <sup>σ</sup>im; ; <sup>σ</sup><sup>m</sup>* . The generators

*The left picture shows biological Artin braid element σ*<sup>1</sup> ∈*B*<sup>2</sup> *in complex plane* C*. There are eight equivalent classes span by eight orders of σ*1*. It is a representation of eight states in three genetic codes in codon as braid*

*time series data, and the blue color represents the active and passive behavior field layers. The right picture*

*shows a member of loop braid group. Three circles S*<sup>1</sup> *represent the sources of closed* 3*-balls B<sup>c</sup>*

*Application of Spin-Orbit Coupling in Exotic Graphene Structures and Biology*

*DOI: http://dx.doi.org/10.5772/intechopen.88486*

*τiτ<sup>j</sup>* ¼ *τjτi, i* 6¼ *j*

lation as an anyon, we have *<sup>ρ</sup>* : *<sup>G</sup>* ! *<sup>U</sup>*ð Þ¼ <sup>1</sup> *<sup>S</sup>*<sup>2</sup> as a representation in <sup>Ω</sup>*BIO*

three types of loop braid group operations in biology. All loop elements of the representation of amino acids arose from group operations over the superspace of time series data. These tree types are a translation, reflection, and rotation, *ρType*�*<sup>I</sup>*

*ρType*�*II*, and *ρType*�*III*. For the translation as a string of amino acids, we have the

ð Þ *<sup>σ</sup><sup>i</sup>*þ<sup>1</sup> *<sup>ρ</sup>Type*�*<sup>I</sup>*

ð Þ *<sup>σ</sup><sup>i</sup>*þ<sup>1</sup> *<sup>ρ</sup>Type*�*<sup>I</sup>*

ð Þ *σ<sup>i</sup>*

ð Þ *<sup>σ</sup><sup>i</sup> <sup>ρ</sup>Type*�*<sup>I</sup>*

ð Þ *σ<sup>i</sup>*þ<sup>1</sup> *,*

*<sup>i</sup>* ¼ 1*, i* ¼ 1*,* ⋯*, n σiτ<sup>j</sup>* ¼ *τjσi,* ∣*i* � *j*∣>1

*,* ∣*i* � *j*∣>1 *τiρ<sup>i</sup>* ¼ *ρiτ<sup>i</sup>*þ<sup>1</sup>*, i* ¼ 1*,* ⋯*, n* � 1 *τiσ<sup>i</sup>* ¼ *σiτ<sup>i</sup>*þ<sup>1</sup>*, i* ¼ 1*,* ⋯*, n* � 1

For a group operation of the genotype *G* and a representation *ρ* for gene trans-

*<sup>n</sup>* . In order to visualize 3D folding structure of the protein structure, we define

*<sup>i</sup> ρiτi, i* ¼ 1*,* ⋯*, n* � 1*:*

In a 3-dimensional position space, the geneon and retrotransposon statistics operators are �1 and þ1, respectively. By the same way, in two-dimensional position space, the abelian anyonic statistics operators ei*<sup>θ</sup>* are 1-dimensional representations of eight loop braid elements *σ*1*, σ*<sup>2</sup> <sup>1</sup>*, σ*<sup>3</sup> <sup>1</sup>*,* ⋯*, σ*<sup>8</sup> <sup>1</sup> in circular Artin braid group *B*<sup>3</sup> 2 acting on the space of wave functions (**Figure 4**).

#### **3.1 Classification of loop braid group in genetic code**

We classify three types of loop braid group operations; it is a representation of an anyon for protein folding. For two-dimensional representation of D-brane in loop braid group for the genetic code, we define abelian anyon for biology in (2 + 1) dimensions, the extra dimensions used to represent the homotopy path of protein folding.

*Application of Spin-Orbit Coupling in Exotic Graphene Structures and Biology DOI: http://dx.doi.org/10.5772/intechopen.88486*

#### **Figure 4.**

1000 ≤*dt* <sup>∗</sup> ≤2000. The size of the duplicate sequence at the end of this open chromosome is amplified by six braids caused by six superspaces in time series data of organisms. The G alphabet might be suitable to be chosen as hidden time scale in

*Solid State Physics - Metastable, Spintronics Materials and Mechanics of Deformable…*

ment is Grothendieck topology over an adjoint cofunctor; it is a self-

*, <sup>σ</sup><sup>R</sup>* ! � <sup>1</sup>

*, <sup>σ</sup>R*<sup>∗</sup> ! <sup>1</sup>

*<sup>R</sup>* , and *<sup>σ</sup>P*<sup>∗</sup> <sup>¼</sup> *<sup>σ</sup>*�<sup>1</sup>

2 *;* � 1 2 *;* � 1 2

*<sup>R</sup> <sup>σ</sup><sup>D</sup>* <sup>¼</sup> *<sup>σ</sup>*�<sup>1</sup>

<sup>3</sup> ! *Bc*

*<sup>D</sup>* , *<sup>σ</sup>R*<sup>∗</sup> <sup>¼</sup> *<sup>σ</sup>*�<sup>1</sup>

*<sup>σ</sup>*½ � *<sup>G</sup>* <sup>≔</sup> ½ � <sup>0</sup>*;* <sup>0</sup>*;* <sup>0</sup> *<sup>σ</sup>*�<sup>1</sup>

*<sup>σ</sup>*½ � *<sup>A</sup>* <sup>≔</sup> ½ � <sup>0</sup>*;* <sup>0</sup>*;* <sup>1</sup> *<sup>σ</sup>*�<sup>1</sup>

*<sup>σ</sup>*½ � *<sup>U</sup>* <sup>≔</sup> ½ � <sup>0</sup>*;* <sup>1</sup>*;* <sup>0</sup> *<sup>σ</sup>*�<sup>1</sup>

*<sup>σ</sup>*½ � *<sup>C</sup>* <sup>≔</sup> ½ � <sup>1</sup>*;* <sup>0</sup>*;* <sup>0</sup> *<sup>σ</sup>*�<sup>1</sup>

*<sup>σ</sup>*½ � *NA* <sup>≔</sup> ½ � <sup>0</sup>*;* <sup>1</sup>*;* <sup>1</sup> *<sup>σ</sup>*�<sup>1</sup>

*<sup>σ</sup>*½ � *NU* <sup>≔</sup> ½ � <sup>1</sup>*;* <sup>1</sup>*;* <sup>0</sup> *<sup>σ</sup>*�<sup>1</sup>

*<sup>σ</sup>*½ � *NC* <sup>≔</sup> ½ � <sup>1</sup>*;* <sup>0</sup>*;* <sup>1</sup> *<sup>σ</sup>*�<sup>1</sup>

*<sup>σ</sup>*½ � *NG* <sup>≔</sup> ½ � <sup>1</sup>*;* <sup>1</sup>*;* <sup>1</sup> *<sup>σ</sup>*�<sup>1</sup>

retrotransposon and half-integer for geneon, so that

acting on the space of wave functions (**Figure 4**).

**3.1 Classification of loop braid group in genetic code**

tions of eight loop braid elements *σ*1*, σ*<sup>2</sup>

symmetric group by a chosen basis in Clifford algebra as

Here we assume that all genetic code cannot be completely separated and biological clock in telomere length is parametrized by a hidden state of the number of *dt* <sup>∗</sup> <sup>≔</sup> ½ � *<sup>G</sup>* alphabet in ð Þ *TTAGGG <sup>n</sup>*¼*dt* <sup>∗</sup> repeated pattern of the telomere. The ele-

3. The loop braid generator for *B<sup>c</sup>*

*, <sup>σ</sup><sup>P</sup>* ! <sup>1</sup>

*<sup>P</sup>* . We have

*<sup>R</sup> σ<sup>D</sup> ,*

*<sup>R</sup> σ<sup>D</sup> ,*

*<sup>R</sup> σ<sup>D</sup> ,* 2 *;* � 1 2 *;* 1 2

*, <sup>σ</sup>P*<sup>∗</sup> ! � <sup>1</sup>

*<sup>R</sup> σ<sup>D</sup> ,*

*<sup>R</sup> σD,*

*<sup>R</sup> σ<sup>D</sup> ,*

> *<sup>R</sup> σ<sup>D</sup> :*

*ψ*2*ψ*1*:* (49)

<sup>1</sup> in circular Artin braid group *B*<sup>3</sup>

2 *;* 1 2 *;* � 1 2

in genetic code. We define their explicit forms and their permutations over the

2 *;* 1 2 *;* 1 2

therefore one can write eight bases for spinor field in the genetic code in braid

*<sup>R</sup> σD,*

*<sup>R</sup> <sup>σ</sup><sup>D</sup>* <sup>¼</sup> ð Þ *<sup>σ</sup>Rσ<sup>P</sup>* ð Þ *<sup>σ</sup>Pσ<sup>D</sup> <sup>σ</sup>*�<sup>1</sup>

*<sup>R</sup> <sup>σ</sup><sup>D</sup>* <sup>¼</sup> ð Þ *<sup>σ</sup>Dσ<sup>R</sup>* ð Þ *<sup>σ</sup>Rσ<sup>P</sup> <sup>σ</sup>*�<sup>1</sup>

*<sup>R</sup> <sup>σ</sup><sup>D</sup>* <sup>¼</sup> ð Þ *<sup>σ</sup>Pσ<sup>D</sup>* ð Þ *<sup>σ</sup>Dσ<sup>R</sup> <sup>σ</sup>*�<sup>1</sup>

*<sup>R</sup> <sup>σ</sup><sup>D</sup>* <sup>¼</sup> ð Þ *<sup>σ</sup>Rσ<sup>P</sup>* ð Þ *<sup>σ</sup>Pσ<sup>D</sup>* ð Þ *<sup>σ</sup>Dσ<sup>R</sup> <sup>σ</sup>*�<sup>1</sup>

*<sup>R</sup> <sup>σ</sup><sup>D</sup>* <sup>¼</sup> ð Þ *<sup>σ</sup>Dσ<sup>R</sup> <sup>σ</sup>*�<sup>1</sup>

*<sup>R</sup> <sup>σ</sup><sup>D</sup>* <sup>¼</sup> ð Þ *<sup>σ</sup>Pσ<sup>D</sup> <sup>σ</sup>*�<sup>1</sup>

*<sup>R</sup> <sup>σ</sup><sup>D</sup>* <sup>¼</sup> ð Þ *<sup>σ</sup>Rσ<sup>P</sup> <sup>σ</sup>*�<sup>1</sup>

We may also use *θ* ¼ 2*πs* with spin quantum number *s* being an integer for

In a 3-dimensional position space, the geneon and retrotransposon statistics operators are �1 and þ1, respectively. By the same way, in two-dimensional position space, the abelian anyonic statistics operators ei*<sup>θ</sup>* are 1-dimensional representa-

<sup>1</sup>*, σ*<sup>3</sup>

We classify three types of loop braid group operations; it is a representation of an anyon for protein folding. For two-dimensional representation of D-brane in loop braid group for the genetic code, we define abelian anyon for biology in (2 + 1) dimensions, the extra dimensions used to represent the homotopy path of protein

<sup>1</sup>*,* ⋯*, σ*<sup>8</sup>

ei*<sup>θ</sup>* <sup>¼</sup> <sup>e</sup><sup>2</sup>*iπ<sup>s</sup>* ¼ �ð Þ<sup>1</sup> <sup>2</sup>*<sup>s</sup>*

<sup>3</sup> is a quaternionic field

*,* (46)

*,* (47)

(48)

2

the biological clock.

diffeomorphism *ξ* : *Bc*

and *<sup>σ</sup>D*<sup>∗</sup> <sup>¼</sup> *<sup>σ</sup>*�<sup>1</sup>

*<sup>σ</sup>D*<sup>∗</sup> ! � <sup>1</sup>

form as follows

folding.

**108**

*<sup>σ</sup><sup>D</sup>* ! <sup>1</sup> 2 *;* 1 2 *;* � 1 2

> 2 *;* � 1 2 *;* 1 2

*The left picture shows biological Artin braid element σ*<sup>1</sup> ∈*B*<sup>2</sup> *in complex plane* C*. There are eight equivalent classes span by eight orders of σ*1*. It is a representation of eight states in three genetic codes in codon as braid element in σ* ½ � *<sup>A</sup><sup>μ</sup>* <sup>1</sup> *; σ* ½ � *<sup>A</sup><sup>μ</sup>* <sup>1</sup> *; σ* ½ � *<sup>A</sup><sup>μ</sup>* 1 <sup>∈</sup>*B*<sup>3</sup> <sup>2</sup>*. The red color line represents the curvature from the physiology of biological time series data, and the blue color represents the active and passive behavior field layers. The right picture shows a member of loop braid group. Three circles S*<sup>1</sup> *represent the sources of closed* 3*-balls B<sup>c</sup>* <sup>3</sup>*, the structure group of affine transform of* 3 *behavior fields in genetic code σ* ½ � *<sup>A</sup><sup>μ</sup> <sup>i</sup>* ∈*B*<sup>3</sup> <sup>2</sup> *action on affine fibre bundle of behavior field in the genetic code σ<sup>ω</sup>*� *, σim*� *, σ<sup>m</sup>*� ∈*B<sup>c</sup>* <sup>3</sup>*. We have affine group as loop braid group in genetic code by B*3 2⋊*B<sup>c</sup>* 3*.*

The loop braid group, *LBBIO <sup>n</sup>* , for a genetic code has three types of generators, *<sup>σ</sup>* <sup>¼</sup> ð Þ *<sup>σ</sup>D; <sup>σ</sup>R; <sup>σ</sup><sup>P</sup>* , *<sup>ρ</sup>* <sup>¼</sup> *<sup>σ</sup>*½ � *<sup>A</sup> ; ; <sup>σ</sup>*½ � *<sup>U</sup> ; ; <sup>σ</sup>*½ � *<sup>C</sup> ; ; <sup>σ</sup>*½ � *<sup>G</sup>* , and *<sup>τ</sup>* <sup>¼</sup> *<sup>σ</sup>ω; ; <sup>σ</sup>im; ; <sup>σ</sup><sup>m</sup>* . The generators *σi; ρ<sup>i</sup>* ∈ *B*<sup>3</sup> <sup>2</sup>*<sup>i</sup>* <sup>¼</sup> <sup>1</sup>*;* <sup>⋯</sup>*; <sup>n</sup>* � <sup>1</sup> and *<sup>τ</sup><sup>i</sup>* <sup>∈</sup> *Bc* <sup>3</sup>*<sup>i</sup>* <sup>¼</sup> <sup>1</sup>*;* <sup>⋯</sup>*; <sup>n</sup>* fulfill the relations

$$\begin{aligned} \tau\_i \tau\_j &= \tau\_j \tau\_i, & i \neq j \\ \tau\_i^2 &= 1, & i = 1, \cdots, n \\ \sigma\_i \tau\_j &= \tau\_j \sigma\_i, & |i - j| > 1 \\ \rho\_i \tau\_j &= \tau\_i \rho\_j, & |i - j| > 1 \\ \tau\_i \rho\_i &= \rho\_i \tau\_{i+1}, & i = 1, \cdots, n - 1 \\ \tau\_i \tau\_i &= \sigma\_i \tau\_{i+1}, & i = 1, \cdots, n - 1 \\ \tau\_{i+1} \sigma\_i &= \rho\_i \sigma\_i^{-1} \rho\_i \tau\_i, & i = 1, \cdots, n - 1. \end{aligned} \tag{50}$$

For a group operation of the genotype *G* and a representation *ρ* for gene translation as an anyon, we have *<sup>ρ</sup>* : *<sup>G</sup>* ! *<sup>U</sup>*ð Þ¼ <sup>1</sup> *<sup>S</sup>*<sup>2</sup> as a representation in <sup>Ω</sup>*BIO <sup>n</sup>* and in *LBBIO <sup>n</sup>* . In order to visualize 3D folding structure of the protein structure, we define three types of loop braid group operations in biology. All loop elements of the representation of amino acids arose from group operations over the superspace of time series data. These tree types are a translation, reflection, and rotation, *ρType*�*<sup>I</sup>* , *ρType*�*II*, and *ρType*�*III*. For the translation as a string of amino acids, we have the anyon *Type* � *I* for biology,

$$\begin{aligned} &\rho^{\mathsf{T}\mathsf{type}-I}(\sigma\_i)\rho^{\mathsf{T}\mathsf{type}-I}(\sigma\_{i+1})\rho^{\mathsf{T}\mathsf{type}-I}(\sigma\_i) \\ & \qquad = \rho^{\mathsf{T}\mathsf{type}-I}(\sigma\_{i+1})\rho^{\mathsf{T}\mathsf{type}-I}(\sigma\_i)\rho^{\mathsf{T}\mathsf{type}-I}(\sigma\_{i+1}), \end{aligned}$$

*Solid State Physics - Metastable, Spintronics Materials and Mechanics of Deformable…*

$$\begin{split} \rho^{\mathsf{T}\mathsf{type}-I}(\sigma\_{i+1})\rho^{\mathsf{T}\mathsf{type}-I}(\sigma\_{i})\rho^{\mathsf{T}\mathsf{type}-I}(\sigma\_{i+1})\\ = \rho^{\mathsf{T}\mathsf{type}-I}(\sigma\_{i+1})\rho^{\mathsf{T}\mathsf{type}-I}(\sigma\_{i+1})\rho^{\mathsf{T}\mathsf{type}-I}(\sigma\_{i}),\\ \rho^{\mathsf{T}\mathsf{type}-I}(\sigma\_{i}) = \rho^{\mathsf{T}\mathsf{type}-I}(\sigma\_{i+1}). \end{split} \tag{51}$$

Thus the elements *di* and *di*þ<sup>1</sup> exchange places in DNA strand by an analogy with genetic variation. If *di* is twisted by the inner automorphism corresponding to *di*þ<sup>1</sup> at the position *<sup>i</sup>* <sup>¼</sup> *dt* <sup>∗</sup> , the product of the *<sup>d</sup>* components remains the identity element. It may be checked that the braid group relations are satisfied and this

*Application of Spin-Orbit Coupling in Exotic Graphene Structures and Biology*

The spin-orbit coupling is an important quantity, which is measurable in the carbon nanostructures, including the graphitic wormhole (or its nanotubular part); it can also help to identify the wormhole structure in details. SOC in a graphene could be induced by the nonzero curvature; in the particular case of the wormhole with negative curvature, the chiral fermions penetrating through the connecting nanotube in the wormhole structure could be created. 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; the effect is stronger if the radius of the wormhole bridge is smaller. Moreover, one can detect permanently oriented flow when the chiral fermions prefer only one direction of the massive or massless fermionic current from the upper graphene sheet to the

We also 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. The Chern-Simons current, coming from ghost and anti-ghost fields of supersymmetry theory, can be used to define a spectrum of gene expression in new time series data where a spinor field, as alternative representation of a gene, is adopted instead of using the alphabet sequence of standard bases *A*, *T*, *C*, *U*, and *G*. Effort is also directed toward the explanation of the adaptive behavior of immunosystem and to find the source of cancer from the physiology of telomere malfunction in DNA repairing state. Similar examination of Holo-Hilbert spectral analysis of the Chern-Simons current in V3 loop genotypes was performed recently in [33]. A genetic variation in V3 loop genotypes was forecast by using the imaging generated from tensor correlation network with an autoregressive integrated moving average model, support spinor model, and convolutional neural network algo-

The reported results of the work have promissory perspective for their extension to interdisciplinary areas as machine learning, econophysics, or biological sciences.

The work is partly supported by Scientific Grant Agency VEGA Grant No. 2/ 0009/19 and No. 2/0153/17. R. Pinčák would like to thank the TH division in CERN

*<sup>n</sup>* on *Xt*.

formula indeed defines a group action of *LBBIO*

*DOI: http://dx.doi.org/10.5772/intechopen.88486*

lower one, depending on the wormhole curvature.

**4. Conclusions**

rithms.

**Acknowledgements**

for hospitality.

**111**

It is isomorphic to the mapping class group of the infinitely punctured disk, a discrete set of punctures limiting to the boundary of the disk.

By an analogy with the action of the symmetric group by permutations in various mathematical settings, there exists a natural action of the braid group on *n*tuples of objects or on the *n*-folded tensor product that involves some twistors. Let us consider an arbitrary group *G*, and let *X* be the set of all *n*-tuples of elements of *G* whose product is the identity element of *G*. The kernel of the homomorphism *LBBIO <sup>n</sup>* ! <sup>Ω</sup>*BIO <sup>n</sup>* is a subgroup of *LBBIO <sup>n</sup>* called pure loop braid group for biology on *n* strands and denoted as *LPBIO <sup>n</sup>* . In the pure braid, the beginning and end of each strand are in the same position. Pure braid groups fit into a short exact sequence

$$\mathbf{1} \to L F\_{n-1} \to L P\_n^{\mathrm{RO}} \to L P\_{n-1}^{\mathrm{RO}} \to \mathbf{1}.\tag{52}$$

This sequence splits, and therefore pure braid groups are realized as iterated semi-direct products of free groups.

The braid group *B*<sup>3</sup> is the universal central extension of the modular group *PSL*ð Þ 2*;* ℤ , with these sitting as lattices inside the universal covering group. If we define O*D*, O*R*, and O*<sup>P</sup>* as active layers over the superspace of DNA, RNA, and protein, O*D*<sup>∗</sup> , O*R*<sup>∗</sup> , and O*P*<sup>∗</sup> , as passive layers, we can define braid group in genetic code by a curvature inside DNA, RNA, and protein folding structure. It is a source of an acceleration of biological clock in the epigenetic code. We let *σD*, *σR*, *σP*, *σD*<sup>∗</sup> , *σR*<sup>∗</sup> , *σP*<sup>∗</sup> , and *σ*½ � *<sup>A</sup><sup>μ</sup>* and *A<sup>μ</sup>* <sup>¼</sup> ½ � *<sup>A</sup> , U*½ �*, C*½ �*, G*½ �*, NA* ½ �*, NU* ½ �*, NC* ½ �*, NG* ½ � be a loop braid group elements in the genetic code. They are the circular Artin braid groups for the genetic code. One can then define

$$
\Psi^{\mathbb{R}} = \sigma\_{\mathcal{D}} \sigma\_{\mathcal{R}} \sigma\_{\mathcal{D}}, \quad \Psi^{\mathbb{P}} = \sigma\_{\mathcal{D}} \sigma\_{\mathcal{R}}, \quad \Psi^{\mathbb{P}^\*} = \sigma\_{\mathcal{R}} \sigma\_{\mathcal{D}} \sigma\_{\mathcal{R}}.\tag{53}
$$

The braid group operation gives Ψ*<sup>P</sup>*

$$
\sigma\_{\rm D} \Psi\_{\rm P} \sigma\_{\rm D}{}^{-1} = \sigma\_{\rm R} \Psi^{\rm P} \sigma\_{\rm R}{}^{-1} = \Psi^{\rm P} \tag{54}
$$

implying that Ψ*<sup>P</sup>* is in the center of *B*3. It is a wave function of protein transition anyon state. If *<sup>G</sup>* <sup>¼</sup> *LBBIO <sup>n</sup>* acts on *Xt*, we get

$$\begin{aligned} \sigma\_i^{D^\ast} \Psi\_i(d\_1, \dots, d\_{i-1}, d\_i, d\_{i+1}, \dots, d\_n) \\ = \Psi\_i(d\_1, \dots, d\_{i-1}, d\_{i+1}, d\_{i+1}^{-1} d\_i d\_{i+1}, d\_{i+2}, \dots, d\_n). \end{aligned} \tag{55}$$

If *<sup>G</sup>* <sup>¼</sup> *LBBIO <sup>n</sup>* acts on *Yt*, we get

$$\begin{aligned} \sigma\_i^{\mathbb{R}^\*} \Psi\_i(r\_1, \dots, r\_{i-1}, r\_i, r\_{i+1}, \dots, r\_n) \\ = \Psi\_i(r\_1, \dots, r\_{i-1}, r\_{i+1}, r\_{i+1}^{-1} r\_i r\_{i+1}, r\_{i+2}, \dots, r\_n) . \end{aligned} \tag{56}$$

If *<sup>G</sup>* <sup>¼</sup> *LBBIO <sup>n</sup>* acts on *Pt* ¼ *Xt=Yt*, we get

$$\begin{aligned} \sigma\_i^{p^\*} \Psi\_i(p\_1, \dots, p\_{i-1}, p\_i, p\_{i+1}, \dots, p\_n) \\ = \Psi\_i(p\_1, \dots, p\_{i-1}, p\_{i+1}, p\_{i+1}^{-1} p\_i p\_{i+1}, p\_{i+2}, \dots, p\_n). \end{aligned} \tag{57}$$

*Application of Spin-Orbit Coupling in Exotic Graphene Structures and Biology DOI: http://dx.doi.org/10.5772/intechopen.88486*

Thus the elements *di* and *di*þ<sup>1</sup> exchange places in DNA strand by an analogy with genetic variation. If *di* is twisted by the inner automorphism corresponding to *di*þ<sup>1</sup> at the position *<sup>i</sup>* <sup>¼</sup> *dt* <sup>∗</sup> , the product of the *<sup>d</sup>* components remains the identity element. It may be checked that the braid group relations are satisfied and this formula indeed defines a group action of *LBBIO <sup>n</sup>* on *Xt*.

## **4. Conclusions**

*ρType*�*<sup>I</sup>*

*<sup>n</sup>* is a subgroup of *LBBIO*

*LBBIO*

*<sup>n</sup>* ! <sup>Ω</sup>*BIO*

strands and denoted as *LPBIO*

*σR*<sup>∗</sup> , *σP*<sup>∗</sup> , and *σ*½ � *<sup>A</sup><sup>μ</sup>* and *A<sup>μ</sup>*

anyon state. If *<sup>G</sup>* <sup>¼</sup> *LBBIO*

If *<sup>G</sup>* <sup>¼</sup> *LBBIO*

If *<sup>G</sup>* <sup>¼</sup> *LBBIO*

**110**

*σ<sup>P</sup>*<sup>∗</sup>

*σ<sup>D</sup>*<sup>∗</sup>

*σ<sup>R</sup>*<sup>∗</sup>

*<sup>i</sup>* <sup>Ψ</sup>*<sup>i</sup> <sup>p</sup>*1*;* …*; pi*�<sup>1</sup>*; pi*

semi-direct products of free groups.

for the genetic code. One can then define

The braid group operation gives Ψ*<sup>P</sup>*

ð Þ *<sup>σ</sup>i*þ<sup>1</sup> *<sup>ρ</sup>Type*�*<sup>I</sup>*

<sup>¼</sup> *<sup>ρ</sup>Type*�*<sup>I</sup>*

*ρType*�*<sup>I</sup>*

discrete set of punctures limiting to the boundary of the disk.

ð Þ *<sup>σ</sup><sup>i</sup> <sup>ρ</sup>Type*�*<sup>I</sup>*

*Solid State Physics - Metastable, Spintronics Materials and Mechanics of Deformable…*

ð Þ *<sup>σ</sup>i*þ<sup>1</sup> *<sup>ρ</sup>Type*�*<sup>I</sup>*

ð Þ¼ *<sup>σ</sup><sup>i</sup> <sup>ρ</sup>Type*�*<sup>I</sup>*

It is isomorphic to the mapping class group of the infinitely punctured disk, a

By an analogy with the action of the symmetric group by permutations in various mathematical settings, there exists a natural action of the braid group on *n*tuples of objects or on the *n*-folded tensor product that involves some twistors. Let us consider an arbitrary group *G*, and let *X* be the set of all *n*-tuples of elements of *G* whose product is the identity element of *G*. The kernel of the homomorphism

strand are in the same position. Pure braid groups fit into a short exact sequence

This sequence splits, and therefore pure braid groups are realized as iterated

The braid group *B*<sup>3</sup> is the universal central extension of the modular group *PSL*ð Þ 2*;* ℤ , with these sitting as lattices inside the universal covering group. If we define O*D*, O*R*, and O*<sup>P</sup>* as active layers over the superspace of DNA, RNA, and protein, O*D*<sup>∗</sup> , O*R*<sup>∗</sup> , and O*P*<sup>∗</sup> , as passive layers, we can define braid group in genetic code by a curvature inside DNA, RNA, and protein folding structure. It is a source of an acceleration of biological clock in the epigenetic code. We let *σD*, *σR*, *σP*, *σD*<sup>∗</sup> ,

braid group elements in the genetic code. They are the circular Artin braid groups

<sup>Ψ</sup>*<sup>R</sup>* <sup>¼</sup> *<sup>σ</sup>DσRσD,* <sup>Ψ</sup>*<sup>P</sup>* <sup>¼</sup> *<sup>σ</sup>DσR,* <sup>Ψ</sup>*<sup>P</sup>*<sup>∗</sup>

*<sup>σ</sup>D*Ψ*PσD*�<sup>1</sup> <sup>¼</sup> *<sup>σ</sup>R*Ψ*<sup>P</sup>σ<sup>R</sup>*

<sup>¼</sup> <sup>Ψ</sup>*<sup>i</sup> <sup>d</sup>*1*;* …*; di*�<sup>1</sup>*; di*þ<sup>1</sup>*; <sup>d</sup>*�<sup>1</sup>

<sup>¼</sup> <sup>Ψ</sup>*<sup>i</sup> <sup>r</sup>*1*;* …*;ri*�<sup>1</sup>*;ri*þ<sup>1</sup>*;r*�<sup>1</sup>

*; pi*þ<sup>1</sup>*;* …*; pn*

<sup>¼</sup> <sup>Ψ</sup>*<sup>i</sup> <sup>p</sup>*1*;* …*; pi*�<sup>1</sup>*; pi*þ<sup>1</sup>*; <sup>p</sup>*�<sup>1</sup>

*<sup>n</sup>* acts on *Xt*, we get

*<sup>i</sup>* Ψ*<sup>i</sup> d*1*;* …*; di*�<sup>1</sup>*; di* ð Þ *; di*þ<sup>1</sup>*;* …*; dn*

*<sup>i</sup>* Ψ*<sup>i</sup> r*1*;* …*;ri*�<sup>1</sup>*;ri* ð Þ *;ri*þ<sup>1</sup>*;* …*;rn*

*<sup>n</sup>* acts on *Pt* ¼ *Xt=Yt*, we get

*<sup>n</sup>* acts on *Yt*, we get

implying that Ψ*<sup>P</sup>* is in the center of *B*3. It is a wave function of protein transition

<sup>1</sup> ! *LFn*�<sup>1</sup> ! *LPBIO*

ð Þ *σi*þ<sup>1</sup>

ð Þ *σi*þ<sup>1</sup> *:*

ð Þ *<sup>σ</sup>i*þ<sup>1</sup> *<sup>ρ</sup>Type*�*<sup>I</sup>*

*<sup>n</sup>* called pure loop braid group for biology on *n*

*<sup>n</sup>*�<sup>1</sup> ! 1*:* (52)

¼ *σRσDσR:* (53)

�<sup>1</sup> <sup>¼</sup> <sup>Ψ</sup>*<sup>P</sup>* (54)

*<sup>i</sup>*þ<sup>1</sup>*didi*þ<sup>1</sup>*; di*þ<sup>2</sup>*;* …*; dn :* (55)

*<sup>i</sup>*þ<sup>1</sup>*riri*þ<sup>1</sup>*;ri*þ<sup>2</sup>*;* …*;rn :* (56)

*pi*þ<sup>1</sup>*; pi*þ<sup>2</sup>*;* …*; pn :* (57)

*<sup>i</sup>*þ<sup>1</sup>*pi*

*<sup>n</sup>* . In the pure braid, the beginning and end of each

<sup>¼</sup> ½ � *<sup>A</sup> , U*½ �*, C*½ �*, G*½ �*, NA* ½ �*, NU* ½ �*, NC* ½ �*, NG* ½ � be a loop

*<sup>n</sup>* ! *LPBIO*

ð Þ *<sup>σ</sup><sup>i</sup> ,* (51)

The spin-orbit coupling is an important quantity, which is measurable in the carbon nanostructures, including the graphitic wormhole (or its nanotubular part); it can also help to identify the wormhole structure in details. SOC in a graphene could be induced by the nonzero curvature; in the particular case of the wormhole with negative curvature, the chiral fermions penetrating through the connecting nanotube in the wormhole structure could be created. 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; the effect is stronger if the radius of the wormhole bridge is smaller. Moreover, one can detect permanently oriented flow when the chiral fermions prefer only one direction of the massive or massless fermionic current from the upper graphene sheet to the lower one, depending on the wormhole curvature.

We also 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. The Chern-Simons current, coming from ghost and anti-ghost fields of supersymmetry theory, can be used to define a spectrum of gene expression in new time series data where a spinor field, as alternative representation of a gene, is adopted instead of using the alphabet sequence of standard bases *A*, *T*, *C*, *U*, and *G*. Effort is also directed toward the explanation of the adaptive behavior of immunosystem and to find the source of cancer from the physiology of telomere malfunction in DNA repairing state. Similar examination of Holo-Hilbert spectral analysis of the Chern-Simons current in V3 loop genotypes was performed recently in [33]. A genetic variation in V3 loop genotypes was forecast by using the imaging generated from tensor correlation network with an autoregressive integrated moving average model, support spinor model, and convolutional neural network algorithms.

The reported results of the work have promissory perspective for their extension to interdisciplinary areas as machine learning, econophysics, or biological sciences.

## **Acknowledgements**

The work is partly supported by Scientific Grant Agency VEGA Grant No. 2/ 0009/19 and No. 2/0153/17. R. Pinčák would like to thank the TH division in CERN for hospitality.

*Solid State Physics - Metastable, Spintronics Materials and Mechanics of Deformable…*

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