2. Knot-crystal and the corresponding zero-lattice

#### 2.1. Knot-crystal

Knot-crystal is a system of two periodically entangled vortex-membranes that is described by a special pure state of Kelvin waves with fixed wave length Zknot–crystal x !; t � � [13, 14]. In emergent quantum mechanics, we consider knot-crystal as a ground state for excited knot states, i.e.,

$$\mathbf{Z\_{knot-crystal}}(\overrightarrow{\mathbf{x}},t) = \begin{pmatrix} \mathbf{z\_A}(\overrightarrow{\mathbf{x}},t) \\\\ \mathbf{z\_B}(\overrightarrow{\mathbf{x}},t) \end{pmatrix} \to |\text{vacuum}\rangle. \tag{1}$$

On the one hand, a knot is a piece of knot-crystal and becomes a topological excitation on it; on the other hand, a knot-crystal can be regarded as a composite system with multi-knot, each of which is described by same tensor state.

Because a knot-crystal is a plane Kelvin wave with fixed wave vector k0, we can use the tensor representation to characterize knot-crystals [13],

$$\tilde{\Gamma}^{l}\_{\text{knot-crystal}} = \left(\overrightarrow{n}^{l}\_{o}\sigma^{l}\right) \otimes \left(\overrightarrow{n}\_{\text{\textpi}}\tau + \overrightarrow{1}\,\tau\_{0}\right) \tag{2}$$

where 1! <sup>¼</sup> 1 0 0 1 � � and <sup>σ</sup><sup>I</sup> , <sup>τ</sup><sup>I</sup> are 2 � 2 Pauli matrices for helical and vortex degrees of freedom, respectively. For example, a particular knot-crystal is called SOC knot-crystal Zknot–crystal x !� � [13], of which the tensor state is given by

$$\left\langle \sigma^{\underline{\boldsymbol{Y}}} \otimes \ \overline{\boldsymbol{1}} \right\rangle = \overline{\boldsymbol{n}}\_{\sigma}^{\underline{\boldsymbol{X}}} = (1,0,0) \left\langle \sigma^{\underline{\boldsymbol{Y}}} \otimes \ \overline{\boldsymbol{1}} \right\rangle = \overline{\boldsymbol{n}}\_{\sigma}^{\underline{\boldsymbol{Y}}} = (0,1,0) \left\langle \sigma^{\underline{\boldsymbol{Z}}} \otimes \ \overline{\boldsymbol{1}} \right\rangle = \overline{\boldsymbol{n}}\_{\sigma}^{\underline{\boldsymbol{Z}}} = (0,0,1). \tag{3}$$

For the SOC knot-crystal, along <sup>x</sup>-direction, the plane Kelvin wave becomes zð Þ¼ <sup>x</sup> ffiffiffi 2 <sup>p</sup> <sup>r</sup><sup>0</sup> cos ð Þ <sup>k</sup><sup>0</sup> � <sup>x</sup> ; along <sup>y</sup>-direction, the plane Kelvin wave becomes zð Þ¼ <sup>y</sup> <sup>1</sup>ffiffi 2 <sup>p</sup> <sup>r</sup><sup>0</sup> <sup>e</sup>ik�<sup>y</sup> <sup>þ</sup> ie�ik�<sup>y</sup> � �; along <sup>z</sup>-direction, the plane Kelvin wave becomes zð Þ¼ <sup>z</sup> <sup>r</sup>0eik�<sup>z</sup> .

For a knot-crystal, another important property is generalized spatial translation symmetry that is defined by the translation operation <sup>T</sup> <sup>Δ</sup>xI � � <sup>¼</sup> <sup>e</sup> <sup>i</sup>� b<sup>k</sup><sup>I</sup> 0�ΔxI � ��Γ~<sup>I</sup> knot�crystal

$$\begin{split} \mathbf{Z}(\mathbf{x}^{l},t) &\to \mathcal{T}(\Delta \mathbf{x}^{l}) \mathbf{Z}(\mathbf{x}^{i},t) \\ = \mathbf{c}^{i} \widehat{\left(k\_{0}^{l} \cdot \Delta \mathbf{x}^{l}\right)} \dot{\mathbf{r}}\_{\text{knot-crystal}}^{l} \mathbf{Z}(\mathbf{x}^{i},t) . \end{split} \tag{4}$$

Here bk I is �<sup>i</sup> <sup>d</sup> dx<sup>I</sup> ð Þ I ¼ x; y; z . For example, for the knot states on 3D SOC knot-crystal, the translation operation along xI -direction becomes

$$\mathcal{T}\left(\Delta \mathbf{x}^{l}\right) = \boldsymbol{\varepsilon}^{\hat{i}\left(\stackrel{\scriptstyle l}{k} \cdot \Delta \mathbf{x}^{l}\right) \cdot \left(\boldsymbol{\sigma}^{l} \otimes \stackrel{\scriptstyle \mathcal{I}}{1}\right)} \tag{5}$$

#### 2.2. Winding space and geometric space

numbers. The study of knotted vortex-lines and their dynamics has attracted scientists from

In the paper [13], the Kelvin wave and knot dynamics in high dimensional vortex-membranes were studied, including the leapfrogging motion and the entanglement between two vortexmembranes. A new theory—knot physics is developed to characterize the entanglement evolution of 3D leapfrogging vortex-membranes in five-dimensional (5D) inviscid incompressible fluid [13, 14]. According to knot physics, it is the 3D quantum Dirac model that describes the knot dynamics of leapfrogging vortex-membranes (we have called it knot-crystal, that is really plane Kelvin-waves with fixed wave-length). The knot physics may give a complete interpre-

In this paper, we will study the Kelvin wave and knot dynamics on 3D deformed knot-crystal, particularly the topological interplay between knots and the lattice of projected zeroes (we call it zero-lattice). Owing to the existence of local Lorentz invariance and diffeomorphism invariance, the gravitational interaction emerges: on the one hand, the deformed zero-lattice can be denoted by curved space-time; on the other hand, the knots deform the zero-lattice that

The paper is organized as below. In Section 2, we introduce the concept of "zero-lattice" from projecting a knot-crystal. In addition, to characterize the entangled vortex-membranes, we introduce geometric space and winding space. In Section 3, we derive the massive Dirac model in the vortex-representation of knot states on geometric space and that on winding space. In Section 4, we consider the deformed knot-crystal as a background and map the problem onto Dirac fermions on a curved space-time. In Section 5, the gravity in knot physics emerges as a topological interplay between zero-lattice and knots and the knot dynamics on deformed knot-crystal is

described by Einstein's general relativity. Finally, the conclusions are drawn in Section 6.

Knot-crystal is a system of two periodically entangled vortex-membranes that is described by a

zA x !; t � � 1

zB x !; t � �

On the one hand, a knot is a piece of knot-crystal and becomes a topological excitation on it; on the other hand, a knot-crystal can be regarded as a composite system with multi-knot, each of

quantum mechanics, we consider knot-crystal as a ground state for excited knot states, i.e.,

0 B@

¼

!; t � �

CA ! j i vacuum : (1)

[13, 14]. In emergent

indicates matter may curve space-time (see below discussion).

2. Knot-crystal and the corresponding zero-lattice

Zknot–crystal x

which is described by same tensor state.

special pure state of Kelvin waves with fixed wave length Zknot–crystal x

!; t � �

diverse settings, including classical fluid dynamics and superfluid dynamics [11, 12].

tation on quantum mechanics.

34 Superfluids and Superconductors

2.1. Knot-crystal

For a knot-crystal, we can study it properties on a 3D space (x, y, z). In the following part, we call the space of (x, y, z) geometric space. According to the generalized spatial translation symmetry, each spatial point (x, y, z) in geometric space corresponds to a point denoted by three winding angles <sup>Φ</sup>xð Þ<sup>x</sup> ; <sup>Φ</sup>yð Þ<sup>y</sup> ; <sup>Φ</sup>zð Þ<sup>z</sup> � � where <sup>Φ</sup>xI xI � � is the winding angle along <sup>x</sup><sup>I</sup> -direction. As a result, we may use the winding angles along different directions to denote a given point Φ ! x !� � <sup>¼</sup> <sup>Φ</sup>xð Þ<sup>x</sup> ; <sup>Φ</sup>yð Þ<sup>y</sup> ; <sup>Φ</sup>zð Þ<sup>z</sup> � �. We call the space of winding angles <sup>Φ</sup>xð Þ<sup>x</sup> ; <sup>Φ</sup>yð Þ<sup>y</sup> ; <sup>Φ</sup>zð Þ<sup>z</sup> � � winding space. See the illustration in Figure 1(d).

For a 1D leapfrogging knot-crystal that describes two entangled vortex-lines with leapfrogging motion, the function is given by

Figure 1. (a) An illustration of a 1D knot-crystal; (b) the relationship between winding angle Φ and coordinate position x. The red dots consist of a 1D zero-lattice in geometric space and the blue dots consist of a zero-lattice in winding space; (c) an illustration of a 3D uniform zero-lattice in geometric space; and (d) an illustration of a 3D uniform zero-lattice in winding space.

$$\mathbf{Z}\left(\overrightarrow{\mathbf{x}},t\right) = r\_0 \begin{pmatrix} \cos\left(\frac{\omega^\*t}{2}\right) \\\\ -i\sin\left(\frac{\omega^\*t}{2}\right) \end{pmatrix} e^{i\frac{2\pi}{\omega}} e^{-i\omega\_0t + i\omega^\*t/2},\tag{6}$$

2.3. Zero-lattice

membranes zA=<sup>B</sup> x

where ξ<sup>A</sup>=B,<sup>θ</sup> x

� � sin <sup>θ</sup> � <sup>η</sup><sup>A</sup>=<sup>B</sup> <sup>x</sup>

the function ξ<sup>A</sup>=B,<sup>θ</sup> x

crystal is described by

where <sup>θ</sup> ¼ � <sup>π</sup>

zeroes (knots).

σ ⊗ 1 ! i ¼ n !

x !; t

ξB,<sup>θ</sup> x !; t !; t

!; t

� � <sup>¼</sup> <sup>ξ</sup><sup>A</sup>=<sup>B</sup> <sup>x</sup>

!; t

!; t

� �, a zero is solution of the equation

Pb <sup>θ</sup> zA x

!; t h i � � � <sup>ξ</sup>A,<sup>θ</sup> <sup>x</sup>

ZKC x !; t � � <sup>¼</sup> <sup>r</sup><sup>0</sup>

� � <sup>¼</sup> <sup>ξ</sup><sup>A</sup>=<sup>B</sup> <sup>x</sup>

Pbθ

!; t

0 B@ !; t

ξ<sup>A</sup>=<sup>B</sup> x !; t � �

η<sup>A</sup>=<sup>B</sup> x !; t � �

� � cos <sup>θ</sup> <sup>þ</sup> <sup>η</sup><sup>A</sup>=<sup>B</sup> <sup>x</sup>

� � <sup>þ</sup> <sup>i</sup>η<sup>A</sup>=<sup>B</sup> <sup>x</sup>

Before introduce zero-lattice, we firstly review the projection between two entangled vortex-

!; t

0 B@

!; t

� � <sup>¼</sup> <sup>P</sup>b<sup>θ</sup> zB <sup>x</sup>

After projection, the knot-crystal becomes a zero lattice. For example, a 1D leapfrogging knot-

ω<sup>∗</sup>t 2 � �

�<sup>i</sup> sin <sup>ω</sup><sup>∗</sup><sup>t</sup> 2 � �

x<sup>0</sup> ¼ a � X þ

similar situation—the solution of zeroes does not change when the tensor order changes, i.e.,

tion of zeroes to be zero-lattice. See the illustration of a 1D zero-lattice in Figure 1(b) and 3D

! σ � � � � �

ξ<sup>A</sup>=B,<sup>θ</sup> x !; t � �

η<sup>A</sup>=B,<sup>θ</sup> x !; t h i � �

� � cos <sup>θ</sup> is constant. So the projected vortex-membrane is described by

!; t h i � � � <sup>ξ</sup>B,<sup>θ</sup> <sup>x</sup>

1

CCCA e i π ax e

a π

<sup>2</sup> and x<sup>0</sup> is the position of zero. As a result, we have a periodic distribution of

�iω0tþiω∗t=<sup>2</sup>

!; t � � <sup>¼</sup>

!, after shifting the distance a, the phase angle of vortex-membranes in

� �. For two projected vortex-membranes described by <sup>ξ</sup>A,<sup>θ</sup> <sup>x</sup>

1

!; t

cos

0

BBB@

According to the knot-equation Pbθ½ �¼ zKC,Að Þx Pbθ½ � zKC,Bð Þx , we have

For a 3D leapfrogging SOC knot-crystal described by ZKC x

<sup>σ</sup> ¼ nx; ny; nx

!

<sup>D</sup> � � with <sup>n</sup>

<sup>σ</sup> ¼ ð Þ! 0; 0; 1 n

zero-lattice in Figure 1(c).

Along a given direction e

knot-crystal changes π, i.e.,

CA <sup>¼</sup>

� � along a given direction <sup>θ</sup> in 5D space by

Topological Interplay between Knots and Entangled Vortex-Membranes

1

!; t

ω0t (12)

zKC,<sup>A</sup> x !; t � �

0 B@

zKC,<sup>B</sup> x !; t � �

� <sup>¼</sup> 1 [13]. We call the periodic distribu-

CA (9)

http://dx.doi.org/10.5772/intechopen.72809

!; t h i � �

� �: (10)

: (11)

1

CA, we have

<sup>0</sup> <sup>¼</sup> <sup>ξ</sup><sup>A</sup>=<sup>B</sup>

37

!; t � � and

0

� � sin <sup>θ</sup> is variable and <sup>η</sup><sup>A</sup>=B,<sup>θ</sup> <sup>x</sup>

where ω<sup>∗</sup> is angular frequency of leapfrogging motion. For the 1D σz-knot-crystal, the coordinate on winding space is <sup>Φ</sup>ð Þ¼ <sup>x</sup> <sup>π</sup> <sup>a</sup> x. Another example is 3D SOC knot-crystal [10], of which the function is given by

$$\mathbf{Z\_{KC}}\left(\overrightarrow{\mathbf{x}},t\right) = \begin{pmatrix} \mathbf{Z\_{KC,A}}\left(\overrightarrow{\mathbf{x}},t\right) \\\\ \mathbf{Z\_{KC,B}}\left(\overrightarrow{\mathbf{x}},t\right) \end{pmatrix} = r\_0 \begin{pmatrix} \cos\left(\frac{\omega^\*t}{2}\right) \\\\ -i\sin\left(\frac{\omega^\*t}{2}\right) \end{pmatrix} e^{-i\omega\_0t + i\omega^\*t/2} \tag{7}$$
 
$$\cdot \sqrt{2}r\_0 \cos\left(\Phi\_\mathbf{x}(\mathbf{x})\right) \cdot \left(\frac{1}{\sqrt{2}}r\_0\left(e^{i\Phi\_\mathbf{y}(\mathbf{y})} + ie^{-i\Phi\_\mathbf{y}(\mathbf{y})}\right)\right) e^{i\Phi\_\mathbf{z}(\mathbf{z})},$$

where the coordinates on winding space are <sup>Φ</sup>xð Þ¼ <sup>x</sup> <sup>π</sup> <sup>a</sup> <sup>x</sup>, <sup>Φ</sup>yð Þ¼ <sup>y</sup> <sup>π</sup> <sup>a</sup> <sup>y</sup>, <sup>Φ</sup>zð Þ¼ <sup>z</sup> <sup>π</sup> <sup>a</sup> z, respectively. In addition, there exists generalized spatial translation symmetry on winding space. On winding space, the translation operation T ΔΦ<sup>I</sup> � � becomes

$$\mathcal{T}\left(\Delta\Phi^{l}\right) = \varepsilon^{i\sum\_{l}\Delta\Phi^{l}\cdot\mathbf{f}\_{\text{knot-crystal}}^{l}}\tag{8}$$

where ΔΦ<sup>I</sup> denotes the distance on winding space.

#### 2.3. Zero-lattice

Z x !; t � �

¼

� ffiffiffi 2

where the coordinates on winding space are <sup>Φ</sup>xð Þ¼ <sup>x</sup> <sup>π</sup>

ing space, the translation operation T ΔΦ<sup>I</sup> � � becomes

where ΔΦ<sup>I</sup> denotes the distance on winding space.

0

BB@

nate on winding space is <sup>Φ</sup>ð Þ¼ <sup>x</sup> <sup>π</sup>

ZKC x !; t � �

the function is given by

winding space.

36 Superfluids and Superconductors

¼ r<sup>0</sup>

zKC,<sup>A</sup> x !; t � �

zKC,<sup>B</sup> x !; t � �

<sup>p</sup> <sup>r</sup><sup>0</sup> cos ð Þ� <sup>Φ</sup>xð Þ<sup>x</sup>

<sup>T</sup> ΔΦ<sup>I</sup> � � <sup>¼</sup> <sup>e</sup>

cos

0

BBB@

ω<sup>∗</sup>t 2 � �

Figure 1. (a) An illustration of a 1D knot-crystal; (b) the relationship between winding angle Φ and coordinate position x. The red dots consist of a 1D zero-lattice in geometric space and the blue dots consist of a zero-lattice in winding space; (c) an illustration of a 3D uniform zero-lattice in geometric space; and (d) an illustration of a 3D uniform zero-lattice in

where ω<sup>∗</sup> is angular frequency of leapfrogging motion. For the 1D σz-knot-crystal, the coordi-

1

CCCA e i π ax e

cos

0

BBBB@

�<sup>i</sup> sin <sup>ω</sup><sup>∗</sup><sup>t</sup> 2 � �

�iω0tþiω∗t=<sup>2</sup>

<sup>a</sup> x. Another example is 3D SOC knot-crystal [10], of which

1

CCCCA e

�iω0tþiω∗t=<sup>2</sup>

e <sup>i</sup>Φ<sup>z</sup> ð Þ<sup>z</sup> ,

<sup>a</sup> <sup>y</sup>, <sup>Φ</sup>zð Þ¼ <sup>z</sup> <sup>π</sup>

knot�crystal (8)

ω<sup>∗</sup>t 2 � �

<sup>i</sup>Φyð Þ<sup>y</sup> <sup>þ</sup> ie�iΦyð Þ<sup>y</sup> � � � �

<sup>a</sup> <sup>x</sup>, <sup>Φ</sup>yð Þ¼ <sup>y</sup> <sup>π</sup>

, (6)

(7)

<sup>a</sup> z, respectively.

�<sup>i</sup> sin <sup>ω</sup><sup>∗</sup><sup>t</sup> 2 � �

1

CCA <sup>¼</sup> <sup>r</sup><sup>0</sup>

1 ffiffiffi 2 p r<sup>0</sup> e

In addition, there exists generalized spatial translation symmetry on winding space. On wind-

i� P i ΔΦ<sup>I</sup> �Γ~I Before introduce zero-lattice, we firstly review the projection between two entangled vortexmembranes zA=<sup>B</sup> x !; t � � <sup>¼</sup> <sup>ξ</sup><sup>A</sup>=<sup>B</sup> <sup>x</sup> !; t � � <sup>þ</sup> <sup>i</sup>η<sup>A</sup>=<sup>B</sup> <sup>x</sup> !; t � � along a given direction <sup>θ</sup> in 5D space by

$$
\hat{P}\_{\theta} \begin{pmatrix} \xi\_{\mathsf{A}/\mathsf{B}} \left( \overrightarrow{\mathbf{x}}, t \right) \\\\ \eta\_{\mathsf{A}/\mathsf{B}} \left( \overrightarrow{\mathbf{x}}, t \right) \end{pmatrix} = \begin{pmatrix} \xi\_{\mathsf{A}/\mathsf{B},\theta} \left( \overrightarrow{\mathbf{x}}, t \right) \\\\ \left[ \eta\_{\mathsf{A}/\mathsf{B},\theta} \left( \overrightarrow{\mathbf{x}}, t \right) \right]\_{0} \end{pmatrix} \tag{9}
$$

where ξ<sup>A</sup>=B,<sup>θ</sup> x !; t � � <sup>¼</sup> <sup>ξ</sup><sup>A</sup>=<sup>B</sup> <sup>x</sup> !; t � � cos <sup>θ</sup> <sup>þ</sup> <sup>η</sup><sup>A</sup>=<sup>B</sup> <sup>x</sup> !; t � � sin <sup>θ</sup> is variable and <sup>η</sup><sup>A</sup>=B,<sup>θ</sup> <sup>x</sup> !; t h i � � <sup>0</sup> <sup>¼</sup> <sup>ξ</sup><sup>A</sup>=<sup>B</sup> x !; t � � sin <sup>θ</sup> � <sup>η</sup><sup>A</sup>=<sup>B</sup> <sup>x</sup> !; t � � cos <sup>θ</sup> is constant. So the projected vortex-membrane is described by the function ξ<sup>A</sup>=B,<sup>θ</sup> x !; t � �. For two projected vortex-membranes described by <sup>ξ</sup>A,<sup>θ</sup> <sup>x</sup> !; t � � and ξB,<sup>θ</sup> x !; t � �, a zero is solution of the equation

$$
\widehat{P}\_{\theta}\left[\mathbf{z}\_{\mathsf{A}}\left(\overrightarrow{\mathbf{x}},t\right)\right] \equiv \xi\_{\mathsf{A},\theta}\left(\overrightarrow{\mathbf{x}},t\right) = \widehat{P}\_{\theta}\left[\mathbf{z}\_{\mathsf{B}}\left(\overrightarrow{\mathbf{x}},t\right)\right] \equiv \xi\_{\mathsf{B},\theta}\left(\overrightarrow{\mathbf{x}},t\right).\tag{10}
$$

After projection, the knot-crystal becomes a zero lattice. For example, a 1D leapfrogging knotcrystal is described by

$$\mathbf{Z\_{KC}}\left(\vec{\mathbf{x}},t\right) = r\_0 \begin{pmatrix} \cos\left(\frac{\omega^\* t}{2}\right) \\\\ -i\sin\left(\frac{\omega^\* t}{2}\right) \end{pmatrix} e^{\frac{i\omega}{\omega}t} e^{-i\omega\_0 t + i\omega^\* t/2}.\tag{11}$$

According to the knot-equation Pbθ½ �¼ zKC,Að Þx Pbθ½ � zKC,Bð Þx , we have

$$\overline{X}\_0 = a \cdot X + \frac{a}{\pi} a \wp t \tag{12}$$

where <sup>θ</sup> ¼ � <sup>π</sup> <sup>2</sup> and x<sup>0</sup> is the position of zero. As a result, we have a periodic distribution of zeroes (knots).

For a 3D leapfrogging SOC knot-crystal described by ZKC x !; t � � <sup>¼</sup> zKC,<sup>A</sup> x !; t � � zKC,<sup>B</sup> x !; t � � 0 B@ 1 CA, we have

similar situation—the solution of zeroes does not change when the tensor order changes, i.e., σ ⊗ 1 ! i ¼ n ! <sup>σ</sup> ¼ ð Þ! 0; 0; 1 n ! <sup>σ</sup> ¼ nx; ny; nx <sup>D</sup> � � with <sup>n</sup> ! σ � � � � � � <sup>¼</sup> 1 [13]. We call the periodic distribution of zeroes to be zero-lattice. See the illustration of a 1D zero-lattice in Figure 1(b) and 3D zero-lattice in Figure 1(c).

Along a given direction e !, after shifting the distance a, the phase angle of vortex-membranes in knot-crystal changes π, i.e.,

$$
\overrightarrow{\Phi}\left(\overrightarrow{\mathbf{x}},t\right) \rightarrow \overrightarrow{\Phi}\left(\overrightarrow{\mathbf{x}} + \mathbf{a} \cdot \overrightarrow{\mathbf{e}},t\right) = \overrightarrow{\Phi}\left(\overrightarrow{\mathbf{x}},t\right) + \pi. \tag{13}
$$

p !

as

and

knot ¼ ℏknot k

and <sup>c</sup>eff <sup>¼</sup> <sup>a</sup>�<sup>J</sup>

where <sup>ψ</sup> <sup>¼</sup> <sup>ψ</sup>†

j i <sup>i</sup> <sup>þ</sup> <sup>3</sup>; <sup>A</sup>; <sup>↑</sup> <sup>∗</sup>

….

metry, the transfer matrices T<sup>I</sup>

!

pitch of the windings on the knot-crystal.

is the momentum operator. mknotc<sup>2</sup>

γ0, γμ are the reduced Gamma matrices,

<sup>γ</sup><sup>1</sup> <sup>¼</sup> <sup>γ</sup><sup>0</sup>

3.1.2. Dirac model in vortex-representation on geometric space

of vortex degrees of freedom. We call it vortex-representation.

describe the coupling between two-knot states along x<sup>I</sup>

Γ1

<sup>γ</sup><sup>0</sup> <sup>¼</sup> <sup>Γ</sup><sup>5</sup>

the microscopic structure of a knot is given by j i ↑; A , j i ↑; B , j i ↓; A , j i ↓; B .

shown in Figure 2, we label the knots by Wannier state j i <sup>i</sup>; <sup>A</sup>; <sup>↑</sup> , j i <sup>i</sup> <sup>þ</sup> <sup>1</sup>; <sup>A</sup>; <sup>↑</sup> <sup>∗</sup>

Jc† A=Bi TI

with the annihilation operator of knots at the site <sup>i</sup>, <sup>c</sup><sup>A</sup>=B,i <sup>¼</sup> <sup>c</sup><sup>A</sup>=B, <sup>↑</sup>,i

<sup>A</sup>=B,A=<sup>B</sup> along xI

TI <sup>A</sup>,<sup>A</sup> <sup>¼</sup> <sup>T</sup><sup>I</sup>

, <sup>γ</sup><sup>2</sup> <sup>¼</sup> <sup>γ</sup><sup>0</sup>

, <sup>γ</sup><sup>5</sup> <sup>¼</sup> <sup>i</sup>γ<sup>0</sup>

In this paper, we derive the effective Dirac model for a knot-crystal based on a representation

In Ref. [13], it was known that a knot has four degrees of freedom, two spin degrees of freedom ↑ or ↓ from the helicity degrees of freedom, the other two vortex degrees of freedom from the vortex degrees of freedom that characterize the vortex-membranes, A or B. The basis to define

We define operator of knot states by the region of the phase angle of a knot: for the case of <sup>ϕ</sup><sup>0</sup> mod 2ð Þ <sup>π</sup> <sup>∈</sup>ð � �π; <sup>0</sup> , we have <sup>c</sup>†j i<sup>0</sup> ; for the case of <sup>ϕ</sup><sup>0</sup> mod 2ð Þ <sup>π</sup> <sup>∈</sup> ð � <sup>0</sup>; <sup>π</sup> , we have <sup>c</sup>†j i<sup>0</sup> � �†

To characterize the energy cost from global winding, we use an effective Hamiltonian to

constant between two nearest-neighbor knots. According to the generalized translation sym-

<sup>B</sup>,<sup>B</sup> ¼ e

ia b<sup>k</sup> I �σI � �


Γ2

γ1 γ2

, <sup>γ</sup><sup>3</sup> <sup>¼</sup> <sup>γ</sup><sup>0</sup>

Γ3

<sup>ℏ</sup>knot ¼ 2aω<sup>0</sup> play the role of light speed where a is a fixed length that denotes the half

In addition, the low energy effective Lagrangian of knots on 3D SOC knot-crystal is obtained

eff <sup>¼</sup> <sup>2</sup>ℏknotω<sup>∗</sup> plays role of the mass of knots

http://dx.doi.org/10.5772/intechopen.72809

39

, (19)

. As

(22)

, j i i þ 2; A; ↑ ,

. J is the coupling

γ<sup>3</sup>: (20)


<sup>A</sup>=B,A=<sup>B</sup>c<sup>A</sup>=B,iþeI (21)

c<sup>A</sup>=B, <sup>↓</sup>,i !

<sup>L</sup>3D <sup>¼</sup> <sup>ψ</sup> <sup>i</sup>γμb∂<sup>μ</sup> � <sup>m</sup>knot � �<sup>ψ</sup> (18)

Topological Interplay between Knots and Entangled Vortex-Membranes

Thus, on the winding space, we have a corresponding "zero-lattice" of discrete lattice sites described by the three integer numbers

$$\overrightarrow{X} = (X, Y, Z) = \frac{1}{\pi} \overrightarrow{\Phi} - \frac{1}{\pi} \overrightarrow{\Phi} \text{ mod } \pi. \tag{14}$$

See the illustration of a 1D zero-lattice in Figure 1(b) and 3D zero-lattice in Figure 1(d).
