1. Introduction

A vortex (point-vortex, vortex-line, vortex-membrane) consists of the rotating motion of fluid around a common centerline. It is defined by the vorticity in the fluid, which measures the rate of local fluid rotation. In three dimensional (3D) superfluid (SF), the quantization of the vorticity manifests itself in the quantized circulation <sup>∮</sup> <sup>v</sup> � <sup>d</sup><sup>l</sup> <sup>¼</sup> <sup>h</sup> <sup>m</sup> where h is Planck constant and m is atom mass of SF. Vortex-lines can twist around its equilibrium position (common centerline) forming a transverse and circularly polarized wave (Kelvin wave) [1, 2]. Because Kelvin waves are relevant to Kolmogorov-like turbulence [3, 4], a variety of approaches have been used to study this phenomenon. For two vortex-lines, owing to the interaction, the leapfrogging motion has been predicted in classical fluids from the works of Helmholtz and Kelvin [5–10]. Another interesting issue is entanglement between two vortex-lines. In mathematics, vortex-line-entanglement can be characterized by knots with different linking

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

numbers. The study of knotted vortex-lines and their dynamics has attracted scientists from diverse settings, including classical fluid dynamics and superfluid dynamics [11, 12].

Because a knot-crystal is a plane Kelvin wave with fixed wave vector k0, we can use the tensor

! ττþ 1 ! τ0

, <sup>τ</sup><sup>I</sup> are 2 � 2 Pauli matrices for helical and vortex degrees of freedom,

Topological Interplay between Knots and Entangled Vortex-Membranes

<sup>σ</sup> <sup>¼</sup> ð Þ <sup>0</sup>; <sup>1</sup>; <sup>0</sup> , <sup>σ</sup><sup>Z</sup> <sup>⊗</sup> <sup>1</sup>

.

; t � �

; t � �:

knot�crystal

� � (2)

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

! i ¼ n !<sup>Z</sup>

2

!� � [13],

(3)

35

2 <sup>p</sup> <sup>r</sup><sup>0</sup>

(4)

(5)


<sup>σ</sup> ¼ ð Þ 0; 0; 1 :

<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

!I <sup>σ</sup>σ<sup>I</sup> � � <sup>⊗</sup> <sup>n</sup>

respectively. For example, a particular knot-crystal is called SOC knot-crystal Zknot–crystal x

For the SOC knot-crystal, along <sup>x</sup>-direction, the plane Kelvin wave becomes zð Þ¼ <sup>x</sup> ffiffiffi

For a knot-crystal, another important property is generalized spatial translation symmetry that

; <sup>t</sup> � � ! <sup>T</sup> <sup>Δ</sup>xI � �<sup>Z</sup> xi

<sup>i</sup> b<sup>k</sup> I �Δx<sup>I</sup> � �

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

a result, we may use the winding angles along different directions to denote a given point

For a 1D leapfrogging knot-crystal that describes two entangled vortex-lines with leapfrogging

!� � <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> � � wind-

<sup>i</sup>� b<sup>k</sup><sup>I</sup> 0�ΔxI � ��Γ~<sup>I</sup>

knot�crystalZ xi

dx<sup>I</sup> ð Þ I ¼ x; y; z . For example, for the knot states on 3D SOC knot-crystal, the

�

� <sup>σ</sup><sup>I</sup> <sup>⊗</sup> <sup>1</sup> ! Þ:

! i ¼ n !<sup>Y</sup>

representation to characterize knot-crystals [13],

where 1!

Here bk I

Φ ! x is �<sup>i</sup> <sup>d</sup>

translation operation along xI

2.2. Winding space and geometric space

ing space. See the illustration in Figure 1(d).

motion, the function is given by

<sup>¼</sup> 1 0

σ<sup>X</sup> ⊗ 1 ! i ¼ n !<sup>X</sup>

0 1 � � and <sup>σ</sup><sup>I</sup>

of which the tensor state is given by

Γ~I

<sup>σ</sup> <sup>¼</sup> ð Þ <sup>1</sup>; <sup>0</sup>; <sup>0</sup> , <sup>σ</sup><sup>Y</sup> <sup>⊗</sup> <sup>1</sup>

<sup>z</sup>-direction, the plane Kelvin wave becomes zð Þ¼ <sup>z</sup> <sup>r</sup>0eik�<sup>z</sup>

is defined by the translation operation <sup>T</sup> <sup>Δ</sup>xI � � <sup>¼</sup> <sup>e</sup>

D D D

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

Z xI


<sup>T</sup> <sup>Δ</sup>xI � � <sup>¼</sup> <sup>e</sup>

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>

¼ e <sup>i</sup>� b<sup>k</sup><sup>I</sup> 0�Δxi � ��Γ~<sup>I</sup>

knot–crystal ¼ n

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 interpretation on quantum mechanics.

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

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.
