Knot Physics on Entangled Vortex-Membranes: Classification, Dynamics and Effective Theory

In this paper, knot physics on entangled vortex-membranes are studied including classification, knot dynamics and effective theory. The physics objects in this paper are entangled vortex-membranes that are called composite knot-crystals. Under projection, a composite knot-crystal is reduced into coupled zero-lattices. In the continuum limit, the effective theories of coupled zero-lattices become quantum field theories. After considering the topological interplay between knots and different types of zero-lattices, gauge interactions emerge. Based on a particular composite knot-crystal with (N=4, M=3) (we call it standard knot-crystal), the derived effective model becomes the (one-flavor) Standard model. As a result, the knot physics may provide an alternative interpretation on quantum field theory.


I. INTRODUCTION
A vortex (point-vortex, vortex-line, vortex-membrane) is among the most important and most studied objects of fluid mechanics that consists of the rotating motion of fluid around a common center (a point, a line, or a membrane). In three dimensional (3D) superfluid (SF), it has been known that a vortex-line is subject to wavy distortions called Kelvin waves [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 entangled vortex-rings, there may exist the leapfrogging motion in classical fluids according to the works of Helmholtz and Kelvin [5][6][7][8][9][10]. Kelvin came to the idea that classical atoms were knots of swirling vortices in the luminiferous aether. Chemical elements correspond to knots and links. The study of knotted vortex-lines and their dynamics has attracted scientists from diverse settings, including classical fluid dynamics and superfluid dynamics [11,12].
In the paper [13], the Kelvin wave and knot dynamics on three dimensional vortexmembranes in five dimensional fluid were studied. A new theoryknot physics is developed to characterize the entanglement evolution of 3D leapfrogging vortex-membranes. Owning to the the conservation conditions of the volume of the knot in 5D space, the shape and the volume of knot is never changed and the knot can only split and knot-pieces evolutes following the equation of motion of Schrödinger equation. Three dimensional quantum Dirac model is derived to describe the entanglement evolution of the entangled vortex-membranes: The elementary excitations are knots with a projected zero; The physics quality to describe local deformation is knot density or the density of zeros between two projected vortexmembranes; The Biot-Savart equation for Kelvin waves becomes Schrödinger equation for probability waves of knots; The angular frequency for leapfrogging motion turns into the mass of knots; etc. The knot physics may give an alternative interpretation on quantum mechanics.
In this paper, we will study the Kelvin wave and knot dynamics on complex entangled vortex-membranes -a composite knot-crystal. By projecting entangled vortex-membranes into several coupled zero-lattices (T-zero-lattices and W-zero-lattices), the information of the system becomes the coupled zero-lattices with internal degrees of freedom. After considering the topological interplay between knots and zero-lattices, different kinds of gauge interactions emerge. In particular, it is the 3D quantum gauge field theories that characterize the knot dynamics of the composite knot-crystal. The knot physics may give a complete inter-

II. REVIEW ON BIOT-SAVART MECHANICS
In the paper [13], to characterize the entanglement evolution of vortex-membranes, a new theory -knot physics was developed from Biot-Savart mechanics. In this paper, firstly we review the key points of the Biot-Savart mechanics.
The 3D vortex-membrane is defined by a given singular vorticity Ω = κδ P in the 5D inviscid incompressible fluid (∇ · v ≡ 0), the singular δ-type vorticity denotes the submanifold P in 5D space, and κ is the constant circulation strength. For 5D case, we have 3D vortex-membranes with Marsden-Weinstein (MW) symplectic structure [14].
The generalized Biot-Savart equation for a 3D vortex-piece under local induction approximation (LIA) can be described by Hamiltonian formulȧ where the Hamiltonian on the vortex-membranes is just 3-volume H volume (P ) = (κα ln ǫ) · volume(P ) (2) with volume(P ) = P dV P and α = Γ( 5 2 ) 6π 5 2 . Here ǫ is defined by ǫ = ℓ a 0 where ℓ is the length of the order of the curvature radius (or inter-vortex distance when the considered infinitesimal vortex is a part of a vortex tangle) and a 0 denotes the infinitesimal vortex radius which is much smaller than any other characteristic size in the system. In complex description, z = ξ + iη, above equation can also be written into [14] i dz dt = δH volume (P ) For Kelvin waves on a 3D helical vortex-membrane, the plane-wave is described by a complex field, z( x, t) = r 0 e ±i k· x−iωt+iφ 0 where k is the winding wave vector along a direction on 3D vortex-membrane with k = π a and a is a fixed length that denotes the half pitch of the windings. The (Lamb impulse) momentum and the (Lamb impulse) angular momentum alongẽ-direction on vortex-membrane with a plane Kelvin wave are given by p Lamb = P Lamb · e = ± 1 2 ρκV P a 2 k and |J Lamb | = |J Lamb ·ẽ| = 1 2 ρκV P a 2 , respectively. V p is the total volume length of the vortex-membrane. Because the projected (Lamb impulse) angular momentum is a constant on the vortex-membrane, the effective Planck constant knot is derived as angular momentum in extra space that is proportional to the volume of the knot in 5D space, i.e., eff = J Lamb = 1 2 ρ 0 κV P r 2 0 where V p is the total volume of the vortex-membrane and ρ 0 is the superfluid mass density.
For two entangled vortex-membranes, the nonlocal interaction leads to leapfrogging motion [10]. For leapfrogging motion, the entangled vortex-membranes exchange energy in a periodic fashion. The winding radii of two vortex-membranes oscillate with a fixed leapfrogging angular frequency ω * = (ακ ln ǫ) 2 where r 0 is the distance between two vortexmembranes.
A knot is an elementary entanglement between two vortex-membranes with fixed volume.
On the one hand, a knot is π-phase changing -a sharp, time-independent, topological phase changing, on the other hand, a knot has a phase angle. Quantum mechanics describes the dynamics of smooth, slow, non-topological phase changing of knots. From point view of information, the elementary volume-changing with a zero is a knot. Under fixed-volume condition, the knot becomes fragmentized and obeys quantum mechanics rather than pseudoquantum mechanics. This is the fundamental principle of quantum mechanics. The effective Planck constant knot is derived as angular momentum in extra space (the volume of the knot in 5D space) where V p is the total volume of the vortex-membrane and ρ 0 is the superfluid mass density.
We pointed out that the function of a Kelvin wave with an extra fragmentized knot describes the distribution of the knot-pieces and z( x, t) plays the role of the wave-function in quantum mechanics as The angle ∆φ( x, t) becomes the quantum phase angle of wave-function, the knot density ρ knot = ∆K ∆V P becomes the probability density of knot-pieces n knot ( x). For a plane wave, ψ( x, t) = 1 √ V P e −iω·t+i k· x , the projected (Lamb impulse) energy of a knot is and the projected (Lamb impulse) momentum of a knot is where the effective Planck constant knot is obtained as projected (Lamb impulse) angular momentum of a knot (the elementary volume-changing of two entangled vortex-membranes) In general, the energy and momentum for a knot are described by operators The energy-momentum relationship E = H( p) becomes the equation of motion for wavefunction, We also use path-integral formulation to describe quantum processes in knot physics. For a multi-knot system, the probability amplitude is defined by where S knot = ω, p S ω, p = L knot dtd 3 x with S ω, p = ψ † p (i eff ω( p) − H knot ( p))ψ p and L knot = However, there is an unsolved problem in Ref. [13]: because free Dirac model is noninteracting, we don't know how to do a physical dynamic projection. In this paper, we develop an effective quantum gauge field theory and solve above problem.

III. COMPOSITE KNOT-CRYSTAL: DEFINITION, CLASSIFICATION AND GENERALIZED TRANSLATION SYMMETRY
In solid state physics, a basic theory is about atom-crystal and its lattices. The atomcrystal has its inherent symmetry, by which we may classify different types of symmetries (translational symmetry, rotation symmetry, mirror symmetry). For example, there are 230 distinct space groups in 3D space. In addition to simple crystal with monatomic lattices, there exist composite crystals with polyatomic lattices that have more than one type of atoms in a unit cell.
In the paper [13], a periodic entanglement-pattern between two vortex-membranes is called knot-crystal. The definition of a "knot-crystal" is based on periodic structures of knots that is similar to atom-crystal where the atoms form a periodic arrangement. However, knot-crystal is different from traditional atom-crystal. Fig.1(a) shows a 1D knot-crystal in 3D space. Due to the existence of rotation symmetry and generalized translation symmetry, the properties of knot-crystals are much different from that of atom-crystals. In this paper, under local induction approximation (LIA), we discuss the properties of composite knotcrystals, a periodic entanglement-pattern between multi-vortex-membrane. The name of composite knot-crystal comes from the similarity to the polyatomic atom-crystal with a composite lattice.

A. Definition
Firstly we consider an arbitrary d dimensional composite knot-crystal that comes from a periodic entanglement pattern by entangled vortex-membranes in d + 2 dimensional space To characterize a composite knot-crystal, we define the function by an N × M matrix where x = (x 1 , x 2 , ..., x d ). N and M denote the vortex-index and the level-index, respectively. We point out that Z( x, t) is just matrix representation to show the properties of knot-crystal and there exist N × M independent functions. z j ( x, t) denotes the function of j-th vortex-membrane. A generalized definition of a knot-crystal is given by N × M independent functions of j-th level i-th vortex-membrane z i,j ( x, t) that is an element of the matrix as where α I i,j 2 + β I i,j 2 = 1 and I = x 1 , x 2 , ..., x d . φ I i,j and (φ I i,j ) 0 denote the winding phase angle and the constant phase angle along given x I -direction j-th level i-th vortex-membrane, respectively. ω i,j is rotating velocity and r i,j is the radius of j-th level winding of i-th vortexmembrane, respectively. In general, to guarantee the stability of the composite knot-crystal, we consider LIA, In particular, j-th level i-th vortex-membrane (that is a Kelvin wave) becomes center membrane of (j-1)th level i-th vortex-membrane.
To determine a composite knot-crystal, there is a hierarchy recurrence relationship be-tween two nearest-neighbor levels along x I -direction denotes winding vector along the winding direction, and a I i,M is the length that denotes the half pitch of largest windings of i-th vortex-membrane. Each number of the vector ∆ I i,(j−1,j) is the positive winding number of (j-1)-th level windings in a j-th level winding for i-th vortex-membrane along x I -direction. Thus, according to the hierarchy recurrence relationship, we have where As a result, the function for a knot-crystal with M-level N vortex-membranes is described by the N × M winding angle φ I ij , In this paper, for simplify, we only consider the cases of As a result, the different vortex-membranes of the same level have the same winding length.
The hierarchy series of the particular type of composite knot-crystal is given by

B. Classification
To classify a knot-crystal, we introduce three indices: dimension d, number of vortexmembranes N , level M. Each d-dimensional composite knot-crystal is denoted by (N , The first index is dimension of knot-crystal, d. In this paper we focus on 3D knot-crystal Fig.1 is an illustration of two types of 1D knot-crystals in 3D space. The second index is the number of vortex-membranes, N . For a composite knot-crystal with (N = 1, M), the function is where |α i | 2 + |β i | 2 = 1 and r i ≫ r i−1 . Fig.1(b) is an example of a 1D composite knot-crystal from one vortex-membrane, of which the function is given by The third index is level of knot-crystal, M. To define the concept of level, we introduce composite knot-crystal that corresponds to polyatomic atom-crystal with a composite lattice.
In addition to the three indices, d, N , M, to characterize a composite knot-crystal, one need to define its tensor network state that denotes the entanglement pattern along different directions. In Ref. [13], we have studied a 3D simple knot-crystal with two vortex-membranes (N = 2, M = 1) -spin-orbital coupling (SOC) knot-crystal that is characterized by the following tensor network state

C. Generalized spatial translation symmetry
One of the most important properties of a composite knot-crystal is generalized spatial translation symmetry.
It is obvious that the knot-crystals break continuous translation symmetry, i.e., where Z( x, t) is function for a knot-crystal and T (δx I ) is translation operator for knotcrystal. The knot-crystals have discrete translation symmetry as Here a I is the length that denotes the half pitch of largest windings of vortex-membrane.
However, we point out that all knot-crystals have generalized spatial translation symmetry.
To define the generalized translation symmetry for a knot-crystal, we do a translation operation, under which all vortex-membranes shift a distance δx I along the x I -direction. i.e., Under the global generalized translation symmetry, we have We can define generalized spatial translation symmetry for each level of composite knotcrystal, T l (δx I ), i.e., For the case of l = j, we have δ lj = 1; For the case of l = j, we have δ lj = 0. Firstly, we consider the simplest knot-crystal -1-level winding knot-crystal with (N = 1, A 3D level winding knot-crystal with (N = 1, M = 1) is just a pure state of Kelvin wave from a vortex-membrane in five dimensional space (x, y, z, ξ, η) that is described by To distinguish the travelling Kelvin wave and standing Kelvin wave, we have introduced the spin network representation (a reduction representation of tensor network state) of Kelvin waves [13]. Different spin network states of Kelvin waves have different spin directions = n I σ , I = x, y, z where σ I is 2 × 2 Pauli matrices for helical degree of freedom. Different knot-crystal with (N = 1, M = 1) is characterized by different spin network states n σ . We then review the properties of another simple knot-crystal -three dimensional 1-level knot-crystal with (N = 2, M = 1) (two entangled vortex-membranes) in five dimensional space (x, y, z, ξ, η) that is described by Z( = n I σ ⊗ 1, I = x, y, z where σ I , τ I are 2 × 2 Pauli matrices for helical and vortex degrees of freedom, respec- tively. An interesting type of knot-crystal is spin-orbital coupling (SOC) knot-crystal that is characterized by the tensor network states There always exists leapfrogging motion for the two entangled vortex-membranes with a fixed leapfrogging angular frequency ω * and fixed distance r 0 . As a result, along x-direction, the function of the Kelvin waves becomes along y-direction, the function becomes along z-direction, the function becomes 3. 2-level composite winding knot-crystal with (N = 2, M = 2) A 2-level composite winding knot-crystal is an object of two entangled vortex-membranes A and B. To generate a 2-level winding knot-crystal, we firstly tangle two symmetric vortexmembranes A and B and get a knot-crystal. Next, we winding the knot-crystal and get a The hierarchy series is just a number . (37) b. Example In this paper, we focus on a particular type of 2-level composite knot-   becomes the function of an SOC knot-crystal with leapfrogging motion. The tensor network state of is effective σ z -type of knot-crystal with leapfrogging motion. The tensor network state is In this paper, the hierarchy series ∆ (1,2) is considered to be a large positive integer number.
c. Twist-writhe locking condition We then introduce a topological constraint condition for a 2-level composite knot-crystaltwist-writhe locking condition.
To characterize a 2-level composite knot-crystal with (N = 2, M = 2), we introduce three types of 1D translation symmetry projected topological invariable: linking-number, writhenumber, and twist-number. So, there are three types of vectors for topological invariable to describe entanglement between two vortex-membranes: a 1D linking-number densityvector, a writhe-number density vector, a twist-number density vector. We point out that there exists an important topological relationship between these topological invariable -the twist-writhe locking condition.
Firstly, we discuss the entanglement between two entangled vortex-membranes for 2-level composite knot-crystal with (N = 2, M = 2) (z A,1 ( x, t) and z B,1 ( x, t)), of which the vector of (translation symmetry projected) 1D linking numbers ζ (AB),1D = (ζ x (AB),1D , ζ y (AB),1D , ζ z (AB),1D ) [13] where Here s I A/B = r · e I is the spatial vector of vortex-membranes along a given direction e I (I = x, y, z). We decompose the linking number ζ I where a unit span-wise vector N I =n I cos θ I +b I sin θ I determines the twisting of the vortexmembranes along x I -direction.n I andb I are local normal and bi-normal (unit vectors) of the vortex-membrane in 5D fluid along x I -direction, respectively (θ I is the corresponding mixing angle).
Because the linking number is a topological invariant for two entangled vortexmembranes, we have a topological constraint condition -twist-writhe locking condition [15,16], Theoretically under continuous deformation of the vortex-membranes, the writhe number From above discussion, we point out that for two entangled vortex-membranes (2-level composite knot-crystal with (N = 2, M = 2)) when the two vortex-membranes have additional global winding, finite δW I (AB),1D leads to finite −δT I (AB),1D that is really additional entanglement between two vortex-membranes; vice versa.
It was known that the vector of linking-number density operators for two entangled vortex-membranes (2-level composite knot-crystal with (N = 2, M = 2)) are defined bŷ respectively. For a 2-level composite knot-crystal with (N = 2, M = 2), the three 1D (spatial translation symmetry protected) linking-numbers ζ I 1D (I = x, y, z) are conserved. For 3D 2-level composite knot-crystal with (N = 2, M = 2) in 5D space, we define the (spatial translation symmetry projected) 1D writhe density vector and the (spatial translation symmetry projected) 1D twist density vector Due to the twist-writhe locking condition and spatial translation symmetry, for 2-level composite knot-crystal with (N = 2, M = 2), we have the following equation between two vortex-membranes A 1 , A 2 and B 1 , B 2 , respectively. The phase angles of z 2,A/B (x I , t) are φ I A/B,2 (x I , t) and the winding radii of z A/B,2 ( x, t) are r A/B,2 ( x, t). The winding radii r A i /B i ,1 are the local winding radii of vortex- around its center-membrane A/B. Then, to characterize a knot on winding entangled vortex-membranes, we need to two types phase angles -one is φ I A/B,2 ( x, t) that describe the winding position of the knot on centre-membrane z A/B,2 ( x, t), the other is φ I A i /B i ,1 (x I , t) that are the phase angle of internal windings.
In particular, we consider a perturbative condition, The hierarchy series is also a number .
b. Example In this paper, we focus on a particular type of 2-level composite knotcrystal with (N = 4, M = 2). Z 2 ( x, t) becomes an effective SOC knot-crystal with leapfrogging motion, i.e., The tensor network state of Z 2 ( x, t) is also Z ′ A,1 ( x, t) and Z ′ B,1 ( x, t) are effective SOC knot-crystal with leapfrogging motion. The tensor network state is The hierarchy series ∆ (1,2) is considered to be a positive integer number n (for example, where ζ I (AB),1D , W I (AB),1D , and T I (AB),1D are linking number, writhe number, and twist number, respectively. In this section, we consider the case of composite knot-crystal -a composite system with four entangled vortex-membranes (Z( x, t)) and discuss the twist-writhe locking relation for it.
For the entanglement between z A 1 ,1 ( x, t) and z A 2 ,1 ( x, t) or that between z B 1 ,1 ( x, t) and z B 2 ,1 ( x, t), the twist-writhe locking relation is similar to that the entanglement between z A,2 ( x, t) and z B,2 ( x, t) in a composite winding knot-crystal with (N = 2, M = 2): There are three types of vectors for topological invariable to describe entanglement between z A 1 ,1 ( x, t) and z A 2 ,1 ( x, t): a 1D linking-number density-vector a writhe-number density vector , a twist-number density vector There are three types of vectors for topological invariable to describe entanglement between z B 1 ,1 ( x, t) and z B 2 ,1 ( x, t): a 1D linking-number density-vector a writhe-number density vector , a twist-number density vector The twist-writhe locking condition for two entangled vortex-membranes or The twist-writhe locking condition for two entangled vortex-membranes In particular, there exist the following topological locking relationships and Because W I 1D,A or W I 1D,B is time-dependent due to leapfrogging motion, W I A 1 ,1D , W I A 2 ,1D , W I B 1 ,1D , W I B 2 ,1D change with time. See the illustration in Fig.4. According to twist-writhe locking conditions, δW I and From above discussion, we point out that for uniformly entangled vortex-membranes when the two vortex-membranes have additional global winding, finite δW I (AB),1D leads to finite −δT I (AB),1D that is really additional entanglement between two vortex-membranes; vice versa. That means a global winding of A-knot-crystal and B-knot-crystal leads to additional internal twisting between vortex-membranes A 1 , A 2 and vortex-membranes B 1 , B 2 . . (71) denotes the centre-membranes of A-knot-crystal and B-knot-crystal. The phase and  also denote the centre-membranes of A-knot-crystal and Bknot-crystal, respectively. z 2,A ( x, t) denotes the global position of the entangled vortexmembranes A 1 , A 2 and z B,2 ( x, t) denotes the global position of the entangled vortex- between two vortex-membranes A 1 , A 2 and B 1 , B 2 , respectively. The phase angles of In particular, we consider a perturbative condition, The hierarchy series is . (77) b. Example In this paper, we focus on a particular type of 3-level composite knot- becomes an SOC knot-crystal with leapfrogging motion, i.e., The tensor network states of Z 2 ( x, t) are are SOC knot-crystal with leapfrogging motion. The tensor states are also

A. Projection of vortex-membranes
There are two types projections on vortex-membranes: the projection for single vortexmembrane and that for entangled vortex-membranes. We call the projection for single vortex-membrane W-type projection and that for entangled vortex-membranes T-type projection. See the illustration in Fig.6.

Projection for single vortex-membrane
Firstly, we discuss the projection for single vortex-membrane, of which the function is To locally characterize the windings of a helical vortex-membrane, we define the projec- constant. Thus, the projected helical vortex-membrane is described by the function ξ( x, t).
A crossing between a helical vortex-membrane and a straight one (z( x, t) = 0) in its center corresponds to a solution of the equation that is ξ( x, t) = 0. We call the equation zero equation and its solution zero solution.

Projection for entangled vortex-membranes
Next, we discuss the projection for entangled vortex-membranes. For two entangled vortex-membranes described by For two projected vortex-membranes described by ξ A,θ ( x, t) and

B. Zero-lattice
We then introduce two types of zero-lattices by the two types of projections.
1. W-type zero-lattice from W-type projection on a helical vortex-membrane Firstly, we consider the zero-lattice from W-type projection on a helical vortex-membrane.
The function of a 1D helical vortex-line in a 3D fluid is where r 0 is the winding radius of vortex-line that is set be constant, k 0 = π a > 0 and a is a fixed length that denotes the half pitch of the windings. φ 0 is a constant angle. ± denotes two possible chiralities: left-hand with clockwise winding, or right-hand with counterclockwise winding.
For a helical vortex-line, from the zero solution ξ(x, t) = 0, we get the zero solutions to be ±x(t) = a · X − a π ω 0 t where X is an integer along x-direction and θ = − π 2 + φ 0 . From the projection of a helical vortex-membrane, we have a crystal of crossings. Because the winding-number of helical vortex-line is half of crossing number, each crossing corresponds to a piece of helical vortex-line with half winding-number. We call the object with half winding-number a knot. As a result, the system can be regarded as a crystal of knots. It is obvious that the global rotation doesn't change the winding-number density. For a helical vortex-membrane with ω 0 = 0, we have a finite velocity of the system, a π ω 0 . For an arbitrary 1D Kelvin wave with different spin states, the zero solution doesn't change i.e., σ Z = n Z σ = (0, 0, 1) → n σ = (n x , n y , n x ) with | n σ | = 1 [13]. Therefore, in the following parts, we call the crystal with discrete lattice sites described by the integer numbers X to be "zero-lattice" [17].
For an 3D SOC Kelvin wave of single vortex-membrane, we can use similar W-type projection to obtain a 3D W-type zero-lattice.

T-type zero-lattice from T-type projection on two entangled vortex-membranes
For two entangled vortex-membranes, there exists leapfrogging motion. So we call it leapfrogging knot-crystal. A leapfrogging knot-crystal (two entangled vortex-membranes) is described by where r A = r 0 cos( ω * t 2 ) and r B = −r 0 i sin( ω * t 2 ). According to the knot-equationP wherex I F,0 = x · e I is the coordination on the axis along a given direction e I and X I is an integer number. As a result, we also have a periodic distribution of zeroes (knots) that is a T-type zero-lattice.

Generalized spatial translation symmetry for zero-lattices
For both types of zero-lattice, owing to the generalized spatial translation symmetry for the vortex-membranes there exist corresponding generalized spatial translation symmetries. For example, for a helical vortex-membrane, by doing a spatial translation operation Under the spatial transformation, the zero-lattices shift, i.e., However after changing the projection angle, θ → θ + π a ∆x I , the zero-lattice is invariant, Therefore, for the zero-lattices, the generalized spatial translation operation is also a combination of a continuum spatial translation operation and a global phase rotation operation.

C. Zeroes and knots
From the point view of "information", each zero becomes the element of a zero-lattice.
Thus, the information of vortex-membranes is characterized by the distribution of zeroes.
For the case of an extra zero, we have a knot; for the case of missing zero, we have an anti-knot. According to the existence of two types of zero-lattices, there are two types of zeroes: W-type zero and T-type zero. Obviously, a W-type zero is an element object of W-type zero-lattice and a T-type zero is an element object of T-type zero-lattice.

W-type zero and W-type knot
The element of W-type zero-lattice is W-type zero that corresponds to a crossing of a helical vortex-membrane and a straight line in its center. In the following parts, we call a knot with half winding-number corresponding to W-type zero W-type knot.

From point view of information, a knot is an information unit with fixed geometric
properties that is always anti-phase changing along arbitrary direction e. When there exists a knot, the periodic boundary condition of Kelvin waves along arbitrary direction is changed into anti-periodic boundary condition. Based on the projected vortex-membranes, we define a knot by a monotonic function where x denotes the position along the given direction e. So the sign-switching character can be labeled by winding number w 1D . The winding number w 1D for a knot along given On the other hand, from the topological character of a knot, there must exist a point, each knot corresponds to a zero between two vortex-membranes along the given direction.
The inset in Fig.7 is an illustration of a 1D unified W-type knot. Its function is given by where and where + denotes a clockwise winding and − denotes a counterclockwise winding. There is a linear relationship between φ(x) and x as φ(x) ∝ x − x 0 in the winding region of Thus, we obtain an anti-periodic boundary condition for the system, Under projection, we have the knot equation as In Fig.7(a), the relation between the phase and the coordination of a unified W-type knot is shown. It is obvious that there always exists a single knot solution for a unified W-type knot.

T-type zero and T-type knot
The element of the projected two entangled vortex-membranes is T-type zero that corresponds to a crossing of the two projected vortex-membranes. In the following parts, we call a knot with half twisting-number corresponding to T-type zero T-type knot.
Based on the projected vortex-membranes, we define a knot by a monotonic function where x denotes the position along the given direction e. From the topological character of a knot, each knot corresponds to a zero between two vortex-membranes along the given direction. See the illustration in Fig.7(d).
A knot (a zero) 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. For example, for an up-spin knot and where x = x· e is the coordination on the axis along a given direction e and φ 0 is an arbitrary constant angle, k 0 = π a .
D. Examples: The level-2 W-type zero-lattice of the composite winding knot-crystal with (N = 2, M = 2) is obtained by level-2 W-type projection of centre-membranes of vortex-membrane-A and vortex-membrane-B, i.e.,P that is ξ 2 ( x, t) = 0. The solution of zero-lattice is given byx I (t) = a 2 · X I − a 2 π ω 0 t where X I is an integer along x I -direction and θ 2 = − π 2 + φ 2,0 . a 2 is a fixed length that denotes the half pitch of the windings of z 2 ( x, t).
The level-1 T-type zero-lattice of the 2-level composite knot-crystal with (N = 2, M = 2) is obtained by level-1 T-type projection of vortex-membrane-A and vortex-membrane-B, i.e., In particular, there exists an intrinsic relationship between the number of T-type zeroes and level-2 W-type zeroes,  are described by another knot-crystal that characterizes the centre-membrane of A-knot-crystal by z A,2 ( x, t) and that of B-knot-crystal by z B,2 ( x, t), respectively. The A- and B-knot-crystal (the entangled vortex-membranes B 1 , After projection, there are five zero-lattices: level-2 W-type zero-lattice for A-knotcrystal, level-2 W-type zero-lattice for B-knot-crystal, level-2 T-type zero-lattice between A-knot-crystal and B-knot-crystal, level-1 T-type zero-lattice between two entangled vortexmembrane-A 1 , A 2 for A-knot-crystal, level-1 T-type zero-lattice between two entangled vortex-membrane-B 1 , B 2 for B-knot-crystal.
The level-2 W-type zero-lattice for A-knot-crystal is obtained by level-2 W-type projection of vortex-membrane-A, i.e.,P that is ξ A,2 ( x, t) = 0. The solution of the zero-lattice is obtained as where X I is an integer number along x I -direction. φ 2W,A,0 is a constant phase angle and a 2 is a fixed length that denotes the half pitch of the half windings of z A,2 ( x, t).
The level-2 W-type zero-lattice for B-knot-crystal is obtained by level-2 W-type projection of vortex-membrane-B, i.e.,P that is ξ B,2 ( x, t) = 0. The solution of the zero-lattice is obtained as where X I is an integer number along x I -direction. φ 2W,B,0 is a constant phase angle and a 2 is a fixed length that denotes the half pitch of the windings of z B,2 ( x, t).
The level-2 T-type zero-lattice is obtained by level-2 T-type projection between A-knotcrystal and B-knot-crystal, i.e.,P The solution of the zero-lattice is obtained as where X I is an integer number along x I -direction. φ 2T,0 is a constant phase angle and a 2 is a fixed length that denotes the half pitch of the twistings of z A/B,2 ( x, t).
The level-1 T-type zero-lattice between two entangled vortex-membrane-A 1 , A 2 for Aknot-crystal is obtained by level-1 T-type projection of A-knot-crystal, i.e.,that is The solution of the zero-lattice is obtained as where X I is an integer number along x I -direction. φ 1T,A,0 is a constant phase angle and a 1 is a fixed length that denotes the half pitch of the windings of z A 1 ,1 ( x, t).
The level-1 T-type zero-lattice between two entangled vortex-membrane-B 1 , B 2 for Bknot-crystal is obtained by level-1 T-type projection of B-knot-crystal, i.e.,that is The solution of the zero-lattice is obtained as where X I is an integer number along x I -direction. φ 1T,B,0 is a constant phase angle and a 1 is a fixed length that denotes the half pitch of the windings of z B 1 ,1 ( x, t).
In particular, there exist two intrinsic relationships between the number of T-type zeroes and level-2 W-type zeroes. One is twist-writhe locking relation for A-knot-crystal ,1D are linking number, writhe number, and twist number between vortex-membranes A 1 , A 2 , respectively. Here, the number of level-1 T-type zeroes and level-2 W-type zeroes are equal to be the writhe number W I (A 1 ,A 2 ),1D and the twist number T I (A 1 ,A 2 ),1D , respectively. The other is twist-writhe locking relation for B-knot-crystal directions. As a result, there exists an additional W-type zero-lattice by W-type projection on the center membrane of A-knot-crystal and B-knot-crystal -the level-3 W-type zerolattice. We use z 3 ( x, t) to denote the centre-membranes of A-knot-crystal and B-knot-crystal.
So the level-3 W-type zero-lattice of the 3-level composite knot-crystal with (N = 4, that is ξ 3 ( x, t) = 0. The solution of zero-lattice is given bȳ where X I is an integer along x I -direction and θ 3 = − π 2 + φ 3W,0 . φ 3W,0 is a constant phase angle and a 3 is a fixed length that denotes the half pitch of the windings of z 3 ( x, t).
Owing to the existence the additional level-3 W-type zero-lattice, there exists an additional intrinsic relationship between the number of level-2 T-type zeroes and level-3 W-type zeroes, where ζ I A W-type knot with a W-type zero is a half-winding of 1-level knot-crystal with (N = 1, M = 1). Fig.7(c) is an example of a W-type knot. We then introduce the operation for the on a constant complex field z 0 = 0 (we use [0] to denote the flat vortex-line) to generate a single W-type knot, i.e.,Û ̥ knot (r 0 ) is an expanding operator by shifting 0 to r 0 in the winding region of a knot (for example, x ∈ (x 0 , x 0 + a]). HereK = −i d dφ is knot number operator and and where + denotes a clockwise winding and − denotes a counterclockwise winding. In Fig.7(a), the relation between the phase and the coordination of a 1D unified W-type knot is shown.
It is obvious that there always exists a single knot solution for a unified W-type knot.
The knot number for 1D W-type knot can be obtained by the following equation where z * (x) is a complex conjugation of z(x). In physics, K measures the total phase changing for a knot with half winding-number that can be regarded as an anti-phase domain wall along given direction in a 1D complex field z(x), i.e., K = 1. The knot density (the density of crossings) and the density of winding-numbers are defined by ρ I knot = K ∆x and ρ wind = 2ρ knot = 2K ∆x , respectively. We call the extended object unified W-type knot. The definition can be generalized to d-dimensional W-type knot. The knot number for d-dimensional W-type knot along x I -direction can be obtained by the following equation In physics, K I measures the total phase changing for a knot with half winding-number that can be regarded as an anti-phase domain wall along given direction in a 1D complex field z(x I ), i.e., K I = 1. The knot density (the density of crossings) and the density of winding-numbers are defined by ρ I knot = KI ∆x I and ρ I wind = 2ρ I knot = 2K I ∆x I , respectively. The total density of a knot is defined by a. Fragmentized W-type knot However, because W-type knot comes from the winding of a helical vortex-membrane, it is not a rigid object. Instead, it can split and be fragmentized. We then introduce the concept of "fragmentized W-type knot" by breaking a W-type knot into N pieces (N → ∞), each of which is an identical 1 N -knot with π N phase-changing. The function of Kelvin wave with a fragmentized W-type knot (a composite object of N where N identical 1 N -knots are at (x 0 ) 1 , (x 0 ) 2 ,..., (x 0 ) N . For each 1 N -knot, the knot number K is 1 N and the corresponding phase changing is ∆φ = π N . Thus, for a fragmentized knot, there also exists only a single knot solution and the knot number is conserved. At the limit of N → ∞, we have a uniform distribution of the N identical 1 N -knots.

B. Emergent quantum mechanics
Knots can be regarded as quantum particles that obey emergent quantum mechanics and that the distribution of fragmentized knots is determined by Schrödinger equation.
The function of the Kelvin wave with a fragmentized knot describes the distribution of the N identical 1 N -knots and plays the role of the wave function in emergent quantum mechanics as and Ω( x, t) = ρ knot ( x, t) ⇐⇒ n knot ( x, t) where the function of the Kelvin wave with a fragmentized knot is equal to the knot density ρ knot and thus becomes the probability density for finding a knot n knot ( x). The geometrical Kelvin waves turn into "probability wave" for finding knots and the functions of Kelvin waves become the wave functions.
In emergent quantum mechanics, the projected energy and the projected momentum become operators for a fragmentized knot.
The projected momentum of a fragmentized knot on helical vortex-membrane with an excited Kelvin wave ψ( x, t) = 1 √ V e −i∆ωt+i∆ k· x is defined to be where the effective Planck constant eff is obtained as the projected angular momentum of a knot Given the superposition principle of Kelvin waves, a generalized wave function is ψ( ). For an arbitrary wave function ψ( x, t), we have p knot = This result indicates that the projected momentum for a fragmentized knot becomes operator p knot →p knot = −i eff Using a similar approach, one can see that the projected energy for a fragmentized knot becomes operator E knot →Ĥ knot = i eff d dt .

C. Quantum Fermionic lattice model for knots
Because the knots on 1-level knot-crystal with (N = 1, M = 1) has only one chirality (we assume right-hand chirality), the effective quantum model is that of Weyl fermions [18].
First, we discuss the statistics of a knot. In quantum mechanics, particles with wave functions antisymmetric under exchange are called Fermions. To illustrate the Fermi statistics of knots, we define the Fermionic operator for knots with right-hand chirality . It is obvious that the wave function's antisymmetry by exchanging two knots is a result of the π-phase changing nature of knots, i.e., As a result, knots obey Fermi statistics, Next, we derive the effective quantum field theory for Weyl fermions. A knot has two spin degrees of freedom ↑ or ↓ from the helicity degrees of freedom. The basis to define the microscopic structure of a knot is given by |↑ , |↓ . We define operator of knot states by the region of the phase angle of a knot: for the case of φ 0 mod(2π) ∈ (−π, 0], we have c † |0 ; For the case of φ 0 mod(2π) ∈ (0, π], we have (c † |0 ) † .
To characterize the energy cost from global winding, we use an effective Hamiltonian to describe the coupling between 2-knot states along x I -direction on 3D SOC knot-crystal Then, we get the total kinetic term as where i, j denotes the nearest-neighbor knots. Fig.8(a) is an illustration of entanglement pattern of a 2D SOC knot-crystal with (N = 1, M = 1) and Fig.8(d) is an illustration of a knot that changes entanglement along x and y directions. In Fig.8(a), each circle denotes a zero.
We then use path-integral formulation to characterize the effective Hamiltonian for a knot-crystal as where S = Ldt and L = i To describe the knot states on 3D knotcrystal, we have introduced a two-component fermion field as ↑, ↓ label two spin degrees of freedom that denote the two possible winding directions along a given direction e.
In continuum limit, we have where the dispersion of knots is where k 0 = ( π 2 , π 2 , π 2 ) and c eff = 2aJ is the velocity. In the following part we ignore k 0 .
From above equation, in the limit k → 0 we derive low energy effective Hamiltonian as We then re-write the effective Hamiltonian to be where p = knot k is the momentum operator. c eff play the role of light speed where a is a fixed length that denotes the half pitch of the windings on the knot-crystal.
The Schrödinger equation for knot becomes With help of Schrödinger equation, we can predict the spacial distribution of fragmentized 1 Nknots by varying the rotating velocity, ω 0 → ω 0 +∆ω. In the following parts, we set knot = 1 and c eff = 1. The dynamic of T-type knot on a 1-level SOC knot-crystal with (N = 2, M = 1) has been developed in Ref. [13]. In Ref. [13], the low energy effective Hamiltonian of knot has been obtained as where is the annihilation operator of four-component fermions. m knot c 2 eff = 2 knot ω * plays role of the mass of knots. In the following parts, we set knot = 1 and c eff = 1.
The low energy effective Lagrangian of 3D SOC knot-crystal is =Ψ(iγ µ∂ µ − m knot )Ψ whereΨ = Ψ † γ 0 , γ µ are the reduced Gamma matrices, and In this paper, we only consider the knot dynamics in the limit of ∆ {1,2} = a 2 a 1 ≫ 1. Now, owing to the existence of two types zeroes (the level-2 W-type zeroes and the level-1 T-type zeroes), there exist two types of knots: the level-2 W-type knots and the level-1 T-type knots. So, a knot is defined by changing half linking number, i.e., According to the substitution effect of level-1 T-type of zeroes by level-2 W-type zeroes, the property of the level-2 W-type knots is same to that of level-1 T-type knots.

B. Low energy effective model for two types of knots
Firstly, we consider the low energy effective model for level-1 T-type knots. the two circles is considered to a very larger number (here the number is 6). Fig.9(b) is an illustration of level-2 W-type knot and Fig.9(c) is an illustration of level-1 T-type knot.
Therefore, on level-2 W-type zero-lattice, the effective Hamiltonian for level-1 T-type knots turns intoĤ where p X = 1 a 2 i d d X and p ϕ = 1 a 2 i d d ϕ . Because of ϕ j ∈ (0, π], quantum number of p ϕ is angular momentum L ϕ and the energy spectra are 1 a 2 L ϕ . If we focus on the low energy physics E ≪ 1 a 2 (or L ϕ = 0), we may get the low energy effective Hamiltonian aŝ The low energy effective Hamiltonian for level-1 T-type knots indicates that the knot-pieces of level-1 T-type knots have a uniform distribution inside the unit cell of level-2 W-type zero-lattice.
Next, we consider the low energy effective model for level-2 W-type knots.
The level-2 W-type knots change half linking number between two entangled vortexmembranes by winding globally. However, the level-2 W-type knots on a 2-level knot-crystal with (N = 2, M = 2) are similar to the W-type knots on a 1-level knot-crystal with (N = 1, M = 1) and have one chirality. We may set a W-type knot to be left-hand.
As a result, in the limit of ∆ {1,2} = a 2 a 1 ≫ 1, every left-hand W-type knots can be regarded as a replacement of a left-hand T-type knot on a uniform knot-crystal. As a result, in longwave length limit the left-hand W-type knot has same properties and we cannot distinguish a left-hand W-type knot and a left-hand T-type knot. If we focus on the low energy physics E ≪ 1 a 2 (or L ϕ = 0), we may get the low energy effective Hamiltonian of left-hand W-type knot asĤ The low energy effective Hamiltonian for level-2 W-type knots indicates that the knot-pieces of level-2 W-type knots also have a uniform distribution inside the unit cell of level-2 W-type zero-lattice.
Finally, the effective Lagrangian of the two types of knots becomes There is no mass term for the left-hand level-2 W-type knots, i.e., m L 2 W = 0.
These considerations lead us to assign the left-handed components of the knots to doublets of SU weak (2) In physics, there exists mixed knot state as α( The right-handed components are assigned to singlets of SU weak (2) that has level-1 T-type knot and we have As a result, we have where U SU weak (2) ( X) = e i τ θ( X) and τ is the three Pauli matrices. field", A µ (x) = 3 a=1 A a µ (x)τ a → A a µ τ a , a = 1, 2, 3 that belong to the adjoint representation of SU weak (2) [19]. In the paper of [19], neutrons and protons had been considered to the doublets of SU weak (2). However, in this paper, a left-hand level-2 W-type knot ψ L 2 W and a left-hand level-2 T-type knot ψ L 1 T,L are the doublets of SU weak (2).
The non-Abelian gauge symmetry is represented by and The gauge strength is defined by W µν as or where g is coupling constant of SU weak (2) gauge field. The Lagrangian of Yang-Mills field can only be written as: where the weak current is We write down the effective Lagrangian of SU weak (2) gauge theory where W µ denotes the gauge fields associated to SU weak (2) respectively, of which the corresponding field strengths are W µν . Because linking number of composite knot-crystal can only be changed for left-hand knots, the charged W 's couple only to the left-handed components of the two types of knots.

D. Higgs mechanism and spontaneous symmetry breaking
In this part we focus on leapfrogging motion of the composite knot-crystal, of which the wave functions of level-1 T-type knots become time-dependent with fixed angular velocity We then consider the fluctuating angular velocity of leapfrogging motion that plays the role of Higgs field Φ( X, t) [21], i.e., ω( X, t) ←→ Φ( X, t)/2.
And the condensation of Higgs field Φ( X, t) = 0 corresponds to a finite leapfrogging angular velocity of the knot-crystal ω * = 0. It is the leapfrogging motion that gives masses to knots. Due to chirality, without leapfrogging motion left-hand level-2 W-type knot doesn't obtain mass.
We then study the properties of leapfrogging field ω( X, t) (that is really the Higgs field Φ( X, t)/2). The effect of leapfrogging motion is to change ψ L 1 T,L ( X, t) (that is ψ L 1 T,A ) to ) and there appears an extra term in Hamiltonian as On the other hand, due to ω( X, t) must be an SU weak (2) complex doublet as Next, we write down an effective Lagrangian of the leapfrogging field ω( X, t). Because the leapfrogging field ω( X, t) is an SU weak (2) complex doublet, we get the kinetic term of leapfrogging field ω( X, t) that is To obtain the finite leapfrogging angular velocity, we add a phenomenological potential term V (ω( X, t)). Finally, by adding Yukawa coupling between the leapfrogging field and fermions, the full Lagrangian of leapfrogging field ω( X, t) is given by A finite leapfrogging angular velocity is given by minimizing ω( X, t), of which the expected value is ω * . Then, the weak gauge symmetry is spontaneously broken, we get a finite leapfrogging angular velocity: The finite leapfrogging angular velocity plays the role of Higgs condensation. Under Higgs condensation, there exists Higgs mechanism. The Higgs mechanism breaks the original gauge symmetry according to SU weak (2) → Z2. As a result, the SU weak (2) gauge fields obtain masses from the following terms [22] The mass for the charged vector bosons W µ is m W = ω * g.
A finite leapfrogging angular velocity creates a mass term for the level-1 T-type knots, leaving the W-types massless, In addition, the leapfrogging field also has mass m Higgs . This is an SU weak (2) gauge theory with Higgs mechanism due to spontaneous symmetry breaking. In this part, we will derive the low energy effective model for knots on the composite knotcrystal with (N = 4, M = 2). The composite knots correspond to elementary particles in particle physics, including electron and quarks.
The knots for the composite knot-crystal are defined to correspond to the level-2 T-type zeroes by level-2 T-type projection between A-knot-crystal and B-knot-crystal. The level-2 T-type knots have four degrees of freedom, two spin degrees of freedom and two vortex degrees of freedom. For knot on a composite knot-crystal, the linking number ζ I 1D along given direction is changed by ± 1 2 , i.e., δζ I 1D = ± 1 2 . A knot (an anti-knot) removes (or adds) a projected zero of level-2 T-type zero-lattice that corresponds to removes (or adds) half of "lattice unit" on the level-2 T-type zero-lattice according to ∆x i = ±a 2 . So the size of the knot is However, by trapping different internal zeroes, there exist different types of level-2 T-type knots. We use the following number series to label different types of knots, where n L 2 is the half linking number of level-2 entangled knot-crystals that is equal to the number of level-2 T-type zeroes n L 2 T as n L 1 is the half linking number of level-1 entangled vortex-membranes that is equal to the sum of the number of level-1 T-type zeroes n 2T and the number of level-2 W-type zeroes For different types of knots, due to trapping half linking number of two entangled knotcrystals, we must have An object with level-2 half linking number is a knot and an object with level-1 half linking number is an internal-knot. For a free level-2 composite knot with changing ∆n L 2 = 1, the effective Planck constant is For a free level-1 internal knot with changing ∆n L 1 = 1, the effective Planck constant is As a result, the effective Planck constant for different knots dependents on the number of internal zeroes n k , In this paper, we focus on the case of weak coupling limit (r 1 ≪ r 2 ) and have [1,n k In general, the quantum state of a composite knot with [1, n i ] is denoted by where τ = A/B labels the vortex index, σ =↑ / ↓ labels spin-degrees of freedom. A knot with [1, n i ] is a composite object with a knot and n i internal knots. Because the knot with [1, n] corresponds to electron, its quantum state is denoted by There is n internal knots inside electron. Except for electrons, other types knots correspond to quarks with n i internal knots, of which quantum states are denoted by We then show several 2-level composite knot-crystals with (N = 4, M = 2).
For the case of n = 1, there exists one type of knot without internal additional zero (n k = 0) that is just electron.
For the case of n = 2, except for electrons, there exists one type of quark with one internal additional zero (or an extra internal knot), n k = n quark = 1. Due to the existence of two "sites" inside the knots, the internal additional zero has two degenerate internal states that are described by For the case of n = 3, except for electrons, there exists two types of quarks, u-quark and d-quark. The d-quarks are composite knots with one internal additional zero (or an extra internal knot), n k = n d−quark = 1, of which there are three degenerate states described by The three degenerate states of d-quarks are called red d-quark, blue d-quark and green dquark, respectively. The u-quarks are composite knots with 2 internal-zeroes (or two extra internal knots), n k = n u−quark = 2, of which the three degenerate states are described by The three degenerate states of u-quarks are called red u-quark, blue u-quark and green u-quark, respectively.

B. Emergent Dirac model
In this part we discuss the emergent quantum field theory for composite knots without taking into account the dynamics of internal zeroes. For different types of knots on 2-level knot-crystal with (N = 4, M = 2), there also exist two types of energy costs -the kinetic term and the mass term from leapfrogging motion. As a result, the low energy effective model is similar to that of knot on 1-level knot-crystal with (N = 2, M = 1).
For simplicity, we take the case of n = 3 as an example. By introducing operator representation, we use the traditional effective Hamiltonian to describe the dynamics of composite knot-crystal asĤ where T ij is translation operator from i-site to j-site, and The seven types of knots generated by i have four degrees of freedom, two spin degrees of freedom and two vortex (or chiral) degrees of freedom.
After considering the leapfrogging motion, we write down the low energy effective Lagrangian as where 1 is a 7-by-7 matrix and 1 is 2-by-2 unit matrix.
Finally, the Lagrangian of Dirac fermion of composite knot-crystal is derived by

Phase changing from writhe-twist locking condition
In this section, we study the knot dynamics and entanglement evolution of internal twisting after considering the twist-writhe locking condition.
There exist two intrinsic relationships between the number of level-1 T-type zeroes and level-2 W-type zeroes. One is twist-writhe locking relation for A-knot-crystal -the number of level-1 T-type zeroes and level-2 W-type zeroes are equal to be the writhe number W I and the twist number T I (A 1 ,A 2 ),1D , respectively. The other is twist-writhe locking relation for B-knot-crystal -the number of level-1 T-type zeroes and level-2 W-type zeroes are equal to be the writhe number W I (B 1 ,B 2 ),1D and the twist number T I (B 1 ,B 2 ),1D , respectively. Now, the knot with half-linking number on a composite knot-crystal cannot be a free "particle". Instead, it becomes a composite object by trapping half of writhe number (a level-2 W-type zero) and half of internal twist number (a level-1 T-type zero), i.e., That means a knot traps half of global winding of the center-membranes and half of internal twisting between two vortex-membranes A 1 , According to twist-writhe locking conditions, δW I , a global winding of A/B-knot-crystal leads to additional internal twisting between vortex-membranes A 1 , A 2 or vortex-membranes B 1 , B 2 .
The quantum phase angle of knot state is the phase angle of ψ( x, t) (or the phase angle of z A/B,2 ( x, t)) are φ( x, t) (that is φ A/B,2 ( x, t)); The phase angle of the internal twisting is 1 ( x, t). Under the phase transformation of knots, according to twist-writhe locking conditions, the phase of internal twisting changes

Emergent U(1) gauge symmetry
The U(1) gauge symmetry comes from indistinguishable phase of internal twistings inside the composite knot. Based on the hidden information of internal twistings (or internal knots), we show the physics picture of gauge symmetry. It is known that gauge symmetry appears as the redundancy to define the particles. There exists redundancy to define composite knots: the exact initial phase of an internal zero inside a composite knot is not a physical observable value. The rule to settle down the redundancy is the rule to fix the gauge for gauge field. The U(1) gauge symmetry indicates that we may locally reorganize the knots by different ways and get the same result. In Fig.11, we show the theoretical structure of emergent U(1) Abelian gauge symmetry and gauge fields.
Let us show the details. To well define a composite knot with an addition T-type zero, we must choose a given initial phase angle of internal twistings (φ A/B,1 ( x, t)) 0 on a given Owing to twist-writhe locking conditions, the initial phase angle of knot state However, the absolute phase angle of internal twistings is independent on the quantum phase angle of knots. For example, we can set the initial phase angle of internal twistings to (φ A/B,1 ( x, t)) ′ 0 = (φ A/B,1 ( x, t)) 0 . Different choices of initial phase angle of internal twistings lead to same physics result. This mechanism leads to the existence of local U(1) gauge symmetry.  According to the local U(1) gauge symmetry, the internal states of knots change as following equation, Due to the local U(1) gauge symmetry, one cannot distinguish the state |ψ( x, t) with t) . And the two knot states |ψ( x, t) and |ψ ′ ( x, t) = U U(1) ( x, t) |ψ( x, t) can be same by changing the initial phase angle of internal twistings.
As a result, the phase of the knot at site j changes as Here, the phase changing ∆φ j is relative to phase angle of internal twistings ∆φ j,A/B,1 .
After considering the local changing of phase angle induced by additional internal twisting ∆φ j,A/B,1 , the local coupling between two knot states changes, i.e., = Jψ † j e in k ∆φ j,A/B,1 · T j, j ′ · e −i∆φ j ′ ,A/B,1 n k ψ j ′ where T j, j ′ is translation operator from j-site to j ′ -site. We define a vector field to characterize the local additional internal twistings ∆φ j,1,A/B , The local coupling between two knot states becomes The total kinetic energy for knots becomeŝ It is obvious that the vector field A j, j ′ that characterizes the local position perturbation of internal zero-lattice plays the role of U(1) gauge field. To illustrate the local U(1) gauge symmetry, we do a local U(1) gauge transformation U j,U(1) = e i∆φ 0, j n k via changing the initial phase angle of internal twistings φ 0, j,A/B,1 → ∆φ ′ 0, j,A/B,1 = φ 0, j,A/B,1 + ∆φ 0, j,A/B,1 .
Under above local U(1) gauge transformation, we have and The total kinetic energy for knots turns intô According to the twist-writhe locking condition φ j,A/B,1 = −∆φ j , the Hamiltonian doesn't change,Ĥ On the other hand, the situation for tempo phase changing is similar to that for spatial phase changing. To characterize the tempo twist-writhe locking condition, we introduce a Lagrangian variable A 0, j to path-integral formulation as In continuum limit, we have U j, Abelian gauge symmetry is represented by and Finally, we derive the path-integral formulation to characterize the effective Hamiltonian in continuum limit for gauge field where S EM = L EM dtd 3 x and The gauge field strength F µν is defined by F µν = ∂ µ A ν − ∂ ν A µ . The electric charge for a internal twisting zero is e 0 . As a result, the total electric charge of a composite knot with n-internal zeroes is just e = ne 0 . It is obvious that L EM has local U(1) gauge symmetry. On the contrary, people always obtain above formula (L EM = eA µ j µ (em) − 1 4 F µν F µν ) from point view of local gauge symmetry.
From Eq.230, we can derive the Maxwell equations where j µ (em) is the electric current. In addition, to give a correct definition of knots in a composite knot-crystal, we must set down the "gauge" f (A µ ) = 0 that leads to a fixed φ 0, j,A/B,1 ( x, t) at a give position ( x, t) for internal twisting.
For the case of n = 1, we derive the U(1) gauge theory for composite knot-crystal. Now, the charge of an electron is e 0 . As a result, the effective Lagrangian for knots turns into For the case of n = 2, the charge of an electron is 2e 0 ; the charge of a quark is n quark · e 0 = e 0 . As a result, the effective Lagrangian for knots turns into For the case of n = 3, the charge of an electron is 3e 0 ; the charge of a u-quark is n u−quark · e 0 = 2e 0 , the charge of a d-quark is n d−quark = e 0 . As a result, the effective Lagrangian for knots turns into Here, we have set the charge of an electron 3e 0 to be e. Because the longitudinal fluctuation of internal twisting is "eaten" knots due to writhe-twist locking condition, the fluctuations of internal twisting have only transverse modes. This is why the gauge fields are transverse waves with spin-1.
In addition, we compare the physics picture of U(1) gauge symmetry in composite knotcrystal and that in Kaluza-Klein theory [23]. In knot physics, each internal zero corresponds to a π-flux in extra space (x d+1 -x d+2 space) and the deformation of internal zero-lattice become electromagnetic field, the fluctuations of internal zero-lattice become Maxwell waves. In Fig.13, we show the theoretical structure of emergent SU(n) non-Abelian gauge symmetry and non-Abelian gauge fields.
Thus, we have a local SU(n) symmetry that denotes the indistinguishable states of internal-knots inside the composite knot.
For the case of n = 2, there are two internal zeroes inside a knot that is labeled by 1, 2.
The corresponding basis of states of internal knots is Due to the local SU(2) gauge symmetry, the basis can be arbitrary re-defined, i.e., is the matrix of the representation of SU(2) group. Θ ( x, t) = 3 a=1 θ a ( x, t) τ a and θ a are a set of 3 constant parameters, and τ a are 2 × 2 Pauli matrices representing the 3 generators of the Lie algebra of SU(2) [19].
When there exists a knot with [1,1], the knot states are denoted by where |q 1 ( x, t) denotes a knot state with an internal knot of 1-th internal-zero and |q 2 ( x, t) denotes a knot state with an internal knot of 2-th internal-zero. q † ( x, t) is the creation operator of level-2 knot with two internal knot states. Because both the internal knot state |q 1 ( x, t) and the internal knot state |q 2 ( x, t) change level-1 half linking number, we cannot distinguish the knots with the two different internal states by detecting phase changing on level-2. As a result, the knot states  has local SU(2) gauge symmetry, i.e., Therefore, the quarks also have local SU(2) gauge symmetry. This leads to the existence of SU(2) non-Abelian gauge fields. According to the local SU(2) gauge symmetry, the internal states of quarks change as following equation, Due to the local SU(2) gauge symmetry, one cannot distinguish the state    Fig.15(b), Fig.15(c), and Fig.15(d) illustrate [1,1] knot, [1,2] knot, [1,3] knot, respectively.
We then derive the formulation of the gauge theory for the quarks with local SU(n) symmetry. For simplicity, we denote the basis of quarks with n internal knot states by where ψ i,quark describes the quantum state of the composite knots.
Due to local SU(n) symmetry of internal states for internal-knots, ψ i,quark and ψ i ′ ,quark can be changed into each other by choosing different local gauges. Thus, Ψ quark can be transformed locally according to an n-dimensional representation, where U SU(n) (x) = e iΘ(x) is the matrix of the representation of SU(n) group. To well define a composite knot with n internal zeroes, we must choose arbitrary n nearestneighbor zeroes. Gauge symmetries appear as the redundancies to choose which n internal zeroes to make up this composite knot. The redundancy protected by internal symmetry for n internal zeroes leads to a non-Abelian gauge symmetry that is really the SU(n) gauge symmetry for strong interaction. As a result, we have an SU(n) gauge symmetry that will never be broken. The rule to settle down the redundancy is the rule to fix the gauge for gauge field.
Due to the existence of the local SU(n) gauge symmetry, there exists SU(n) gauge field.
Under changing internal knots inside a composite knot, the quark state at site j changes as HereŨ SU(n) is a changing of internal knot states based on certain basis considering the local changing of basis induced by U j,SU(n) , the local coupling between two quark states changes, i.e., where T j, j ′ is translation operator from j-sie to j ′ -site. We define a vector field A j, j ′ to characterize the local changing of basis where g is coupling constant of SU(n) non-Abelian gauge field. So, for perturbation case where δŨ j,SU(n) =Ũ j,SU(n) −Ũ j ′ ,SU(n) . The local coupling between two knot states becomes The total kinetic energy for knots becomeŝ It is obvious that the vector field A j, j ′ that characterizes the local position perturbation of internal zero-lattice plays the role of SU(n) gauge field. To illustrate the local SU(n) gauge symmetry, we do a local SU(n) gauge transformation U j,SU(n) that is the transformation of basis of internal knot states, i.e., Under a local SU(n) gauge transformation, we have and gA j, j ′ → gA ′ j, j ′ = gU j,SU(n) A j, j ′ (U j ′ ,SU(n) ) −1 (253) where δU j,SU(n) = (U j,SU(n) − U j ′ ,SU(n) ). Here, we have used the following result, The total kinetic energy for knots turns intô The Hamiltonian doesn't change,Ĥ coupling =Ĥ ′ coupling .
On the other hand, the situation for tempo phase changing is similar to that for spatial phase changing. To characterize the tempo twist-writhe locking condition, we introduce a Lagrangian variable A 0, j to path-integral formulation as In continuum limit, we have, The non-Abelian gauge symmetry is represented by and The gauge strength is defined by G µν as or The Lagrangian of Yang-Mills field can only be written as: where J µ YM = iq quark γ µ q quark . The Lagrangian density L YM (SU(n)) is invariant under the gauge transformations with an x-dependent U SU(n) ( x, t). For a simple knot-crystal, the knot becomes chargeless electron and the effective model is a free Dirac model, of which the effective Lagrangian is where m e = 2ω * .
2. Physics of composite knot-crystal with n = 1 For the composite knot-crystal without internal winding (n = 0), the composite knot becomes electron with n e = 1. The effective model is U(1) gauge theory, of which the effective Lagrangian is given by where j µ (em) (x) = iē(x)γ µ e(x). m e = 2ω * are masses for electron. The phonons A µ (x) characterize the fluctuations of internal zero-lattice. This model gives QED.

Physics of composite knot-crystal with n = 2
For the composite knot-crystal with n = 2, the (composite) knots are electron with n e = 2, quarks with n quark = 1. The effective model is SU(2)⊗U(1) gauge theory, of which the effective Lagrangian is given by where e = 2e 0 . m quark = 2ω * are masses for quark and electron, respectively. The electric current is The SU(2) color current is J a,µ YM = iΨ quark γ µ T a Ψ quark . The U(1) gauge field characterizes the interaction from the phase fluctuations of internal zero-lattice. The SU(2) gauge field characterizes the interaction from the number fluctuations of internal zero-lattice.  [24] had proposed preons to be the fundamental constituent particles. Then, other types of models were developed, for example, the Rishon Model proposed simultaneously by Harari and Shupe [25,26], the helon model by S.
O. Bilson-Thompson [27], the tangles by C. Schiller [28]. In knot physics, we show a correspondence between different types of elementary particles and different types of composite knots.
By trapping different types of zeroes, there exist different types of knots. We use the following number series to label different types of knots, where n L 2 is the half linking number of level-2 entangled knot-crystals that is equal to the sum of the number of level-2 T-type zeroes n L 2 T and the number of level-3 W-type zeroes n L 3 W as n L 1 is the half linking number of level-1 entangled vortex-membranes that is equal to the sum of the number of level-1 T-type zeroes n L 1 T and the number of level-2 W-type zeroes For different types of knots, due to trapping half linking number of two entangled knotcrystals, we must have n L 2 = 1. So the classification of the knot type is based on the half internal linking number n L 1 (the half linking number of level-1 entangled vortex-membranes).
As a result, in low energy limit, the effective Lagrangian of knots on a 3-level composite knot-crystal with (N = 4, M = 3) and ∆ {1,2} = n = 3 becomes There is no mass term for the left-hand neutrinos.
where e = 3e 0 . m ν = 0, m u = 2ω * , m d = 2ω * , and m e = 2ω * are masses for neutrino, u-quark, d-quark and electron, respectively. The electric current is The SU Strong (3) color current for strong interaction is This model gives QED and QCD. In particular, neutrino doesn't couple to SU Strong (3)⊗U EM (1) gauge fields.
An important fact is that unit internal zero has supercharge Y to be − 2 3 , i.e., For a right-hand electron, there are 3 internal zeroes. The 3 internal zeroes have supercharge Y to be 3 × − 2 3 = −2. Therefore, right-hand electrons e R have −2 supercharge as Y(ψ Lepton,R ) = −2.
(287) 3. Effective Lagrangian of electro-weak SU weak (2) ⊗ U Y (1) gauge theory Finally, we write down the effective Lagrangian of electro-weak SU weak (2) ⊗ U Y (1) gauge theory L = L fermion + L Y (U Y (1)) + L weak (SU weak (2)) (296) where W µ and B µ denote the gauge fields associated to weak SU weak (2) and super-charge U Y (1) respectively, of which the corresponding field strengths are W µν and B µν . The two coupling constants g and g ′ correspond to the groups SU weak (2) and U Y (1) respectively.
Because neutrino has only left-hand degrees of freedom, the charged W 's couple only to the left-handed components of the lepton fields.

Higgs mechanism and spontaneous symmetry breaking for standard knot-crystal
We then study the properties of angular velocity of leapfrogging motion ω( X, t) (that is really the Higgs field Φ( X, t)/2 in Standard model), that is ψ ′ ( X, t) = e 2iτxω( X,t)·t · ψ( X, t).
As a result, the fluctuating leapfrogging angular velocity of a standard knot-crystal ω 0 → ω( X, t) plays the role of Higgs field Φ( X, t) in Standard model [21].
Next, we write down the Lagrangian of the leapfrogging field ω( X, t). Because the leapfrogging field ω( X, t) is an SU weak (2) complex doublet and has supercharge Y = 1, we get the kinetic term of leapfrogging field ω( X, t) as To obtain the finite leapfrogging velocity, we also add a phenomenological term V (ω( X, t)).
A finite leapfrogging angular velocity is given by minimizing ω( X, t), of which the expected value is ω * . Then, the weak gauge symmetry is spontaneously broken, we get a finite angular velocity of leapfrogging motion as ω( X, t) = 1 √ 2   0 ω *   + δω( X, t).
1 √ 2 G e ω * .For the system with finite angular velocity of leapfrogging motion, we produce masses for the quarks given by m u = G u ω * , m d = G d ω * . Because there is no right-hand neutrino, the mass of neutrino is zero, m ν ≡ 0. In addition, the Higgs field also has mass that is m Higgs = 0.
The finite angular velocity of leapfrogging motion plays the role of Higgs condensation and the Higgs mechanism of 3-level composite knot-crystal with (N = 4, M = 3) breaks the original gauge symmetry according to SU weak (2) ⊗ U Y (1) → U EM (1).
As a result, the SU weak (2) gauge fields obtain masses from the following terms [22] The mass for the charged vector bosons After diagonalization, the gauge fields B µ and W 3 µ are transformed into gauge fields Z µ and A µ from the following relations A µ = cos θ W B µ + sin θ W W 3 µ , with tan θ W = g ′ /g, of which the masses are m A = 0.
The "Weinberg angle" θ W becomes the angle between the original U(1) and the one left unbroken.
The neutral gauge bosons A µ are massless and will be identified with the photons. Now, the gauge symmetry U EM (1) accompanying A µ is to change the position of the internal zeroes of the composite knots that will never be broken. That is ψ( X) = L weak (SU weak (2)) = −Tr − V (ω) +ψ quark,L G quark ωψ quark,R +ψ Lepton,L G Lepton ωψ Lepton,R + h.c..
where the electric current is j µ (em) = iēγ µ e + i 2 3ū γ µ u (318) the weak current is j µ w + = iνγ µ e + id T γ µ u, and the color current is This is exact one-flavor Standard model, an SU Strong (3) ⊗ (SU(2)) weak ⊗U Y (1) gauge theory with Higgs mechanism due to spontaneous symmetry breaking.

X. SUMMARY AND DISCUSSION
In the end, we give a summary. In this paper, knot dynamics on composite knot-crystal is studied. From knot physics, the knot-crystal becomes fundamental physical object, of which  Grand Unified Theory (GUT) is a dream of physicists to unify all non-gravitational interactions. There are several approaches towards GUT. String theory is a possible theory of GUT [32]. According to string theory, matter consists of vibrating strings (or strands) and different oscillatory patterns of strings become different particles with different masses.
In condensed matter physics, the idea of our universe as an "emergent" phenomenon has become increasingly popular. In emergence approach, a deeper and unified understanding of the universe is developed based on a complicated many-body system. Different quantum fields correspond to different many-body systems: the vacuum corresponds to the ground state and the elementary particles correspond to the excitations of the systems. According to string-net picture proposed by Wen, our universe (such as gauge interaction, Fermi statistics, ...) emerges from a frustrated quantum spin model [33]. In addition, there exist many other proposals of GUT from different points of view, such as loop quantum gravity theory [34,35], G. Lisi's E8 theory [36], C. Schiller's Strand Model [28], ...
Based on knot physics, we may guess that our universe becomes a composite knot-crystal -standard knot-crystal, of which the hierarchy series is {3, N} where N is a very large number, N ≫ 1 (for example, N ∼ 10 15 ).
In this paper there are two important issues that we don't discuss: 1) the flavor physics, including the origin of three-flavor, the value of each elements of Cabibbo-Kobayashi-Maskawa mass matrix [37] and the mechanism of weak charge-parity (CP) violence; 2) quark confinement that may be relevant to the dynamics of internal knots inside the quarks. In the future, we will study these issues and develops a complete knot theory for particle physics and quantum field theory.