# Knot Universes in Bianchi Type I Cosmology

Authors: Ratbay Myrzakulov
Journal: Advances in High Energy Physics
 (2012)

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## Abstract

We investigate the trefoil and figure eight knot universes from Bianchi type I cosmology. In particular, we construct several concrete models describing the knot universes related to
the cyclic universe and examine those cosmological features and properties in detail. Finally some examples of unknotted closed curves solutions (spiky and Mobius strip universes) are presented.


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## Body

## 1. Introduction

Inflation is one of the most important phenomena in modern cosmology and has been confirmed by recent observations on cosmic microwave background (CMB) radiation [1–4]. Furthermore, it is suggested by the cosmological and astronomical observations of Type Ia supernovae [5, 6], CMB radiation [1–4], large scale structure (LSS) [7, 8], baryon acoustic oscillations (BAO) [9], and weak lensing [10] that the expansion of the current universe is accelerating. In order to explain the late time cosmic acceleration, we need to introduce so-called dark energy in the framework of general relativity or modify the gravitational theory, which can be regarded as a kind of geometrical dark energy (for reviews on dark energy, see, e.g., [11–16], and for reviews on modified gravity, see, e.g., [17–23]).It is considered that there happened a Big Bang singularity in the early universe. In addition, at the dark energy dominated stage, the finite-time future singularities will occur [24–70]. There also exists the possibility that a Big Crunch singularity will happen. To avoid such cosmological singularities, there are various proposals such as the cyclic universe [71–80] (in other approach of the cyclic universe, see [81]), the ekpyrotic scenario [82–85], and the bouncing universe [86–97].On the other hand, as a related theory to the cyclic universe, the trefoil and figure-eight knot universes have been explored in [98–103]. In the homogeneous and isotropic Friedmann-Lemaître-Robertson-Walker (FLRW) and the homogeneous and anisotropic Bianchi-type I cosmologies, the geometrical description of these knot theories corresponds to oscillating solutions of the gravitational field equations. Note that the terms “the trefoil knot universe” and “the figure-eight knot universe” were introduced for the first time in [98–103]. Moreover, the Weierstrass ℘(t), ζ(t), and σ(t) functions and the Jacobian elliptic functions have been applied to solve several issues on astrophysics and cosmology [104–106]. In particular, very recently, by combining the reconstruction method in [17, 18, 66, 107, 108] with the Weierstrass and Jacobian elliptic functions, the equation of state (EoS) for the cyclic universes [109] and periodic generalizations of Chaplygin gas type models [110–112] for dark energy [113] have been examined. This procedure can be considered to be a novel approach to cosmological models in order to investigate the properties of dark energy.In this paper, we explore the cosmological features and properties of the trefoil and figure-eight knot universes from Bianchi-type I cosmology in detail. In particular, we construct several concrete models describing the trefoil and figure-eight knot universes based on Bianchi-type I spacetime. In our previous work [98–103], the models of the knot universes from the homogeneous and isotropic FLRW spacetime were studied. By using the equivalent procedure, as continuous investigations, in this work we explicitly demonstrate that the knot universes can be constructed by Bianchi-type I spacetime. In other words, our purpose is to establish the formalism which can describe the knot universes.It is significant to emphasize that according to the recent cosmological data analysis [1–4], it is implied that the universe is homogeneous and isotropic. In fact, however, recently the feature of anisotropy of cosmological phenomena such as anisotropic inflation [114, 115] has also been studied in the literature. In such a cosmological sense, it can be regarded as reasonable to consider the anisotropic universe including Bianchi-type I spacetime. The units of the gravitational constant 8πG=c=1 with G and c being the gravitational constant and the seed of light are used.The organization of the paper is as follows. In Section2, we explain the model and derive the basic equations. In Section 3, we investigate the trefoil knot universe. Next, we study the figure-eight knot universe in Section 4. In Section 5 we present some unknotted closed curve solutions of the model. Finally, we give conclusions in Section 6.


## 2. The Model

In this section we briefly review some basic facts about Einstein's field equation. We start from the standard gravitational action (chosen units arec=8πG=1)
(2.1)S=14∫‍d4x-g(R-2Λ+Lm),
where R is the Ricci scalar, Λ is the cosmological constant, and Lm is the matter Lagrangian. For a general metric gμν, the line element is
(2.2)ds2=gμνdxμdxν,(μ,ν=0,1,2,3).
The corresponding Einstein field equations are given by
(2.3)Rμν+(Λ-12R)gμν=-κ2Tμν,
where Rμν is the Ricci tensor. This equation forms the mathematical basis of the theory of general relativity. In (2.3), Tμν is the energy-momentum tensor of the matter field defined as
(2.4)Tμν=2-gδLmδgμν,
and satisfies the conservation equation
(2.5)∇μTμν=0,
where ∇μ is the covariant derivative which is the relevant operator to smooth a tensor on a differentiable manifold. Equation (2.5) yields the conservations of energy and momentums, corresponding to the independent variables involved. The general Einstein Equation (2.3) is a set of nonlinear partial differential equations. We consider the Bianchi-I metric
(2.6)ds2=-dτ2+A2dx12+B2dx22+C2dx32,
where we assume that τ=t/t0, xi=xi′/xi0, A,B,C are dimensionless (usually we put t0=xi0=1). Here the metric potentials A,B, and C are functions of τ=t alone. This insures that the model is spatially homogeneous. The statistical volume for the anisotropic Bianchi type-I model can be written as
(2.7)V=ABC.
The Ricci scalar is
(2.8)R=gijRij=2(A¨A+B¨B+C¨C+A˙B˙AB+A˙C˙AC+B˙C˙BC),
where A˙=dA/dτ and so on. The  nonvanishing components of Einstein tensor
(2.9)Gij=Rij-0.5gijR
are
(2.10)G00=A˙B˙AB+A˙C˙AC+B˙C˙BC,GAA=-A2(B¨B+C¨C+B˙C˙BC),GBB=-B2(A¨A+C¨C+A˙C˙AC),GCC=-C2(B¨B+A¨A+B˙A˙BA).
We define a=(ABC)1/3 as the average scale factor so that the average Hubble parameter may be defined as
(2.11)H=a˙a=13(A˙A+B˙B+C˙C).
We write this average Hubble parameter H sometimes as
(2.12)H=13(H1+H2+H3),
where
(2.13)H1=A˙A,H2=B˙B,H3=C˙C
are the directional Hubble parameters in the directions of x1, x2, and x3, respectively. Hence we get the important relations
(2.14)A=A0e∫H1dt,B=B0e∫H2dt,C=C0e∫H3dt,
where A0,B0,C0 are integration constants. The other important cosmological quantity is the deceleration parameter q, which for our model reads as
(2.15)q=-aa¨a˙2.Next, we assume that the energy-momentum tensor of fluid has the form(2.16)Tij=diag[T00,T11,T22,T33]=diag[ρ,-p1,-p2,-p3].
Here pi are the pressures along the xi axes, recpectively, ρ is the proper density of energy. Then the Einstein equations (with gravitational units, 8πG=1 and c=1) read as
(2.17)Rij-12Rgij=-Tij,
where we assumed Λ=0. For the metric (2.6) these equations take the form
(2.18)A˙B˙AB+B˙C˙BC+C˙A˙CA-ρ=0,B¨B+C¨C+B˙C˙BC+p1=0,C¨C+A¨A+C˙A˙CA+p2=0,A¨A+B¨B+A˙B˙AB+p3=0.
In terms of the Hubble parameters this system takes the form
(2.19)H1H2+H2H3+H1H3-ρ=0,H˙2+H˙3+H22+H32+H2H3+p1=0,H˙3+H˙1+H32+H12+H3H1+p2=0,H˙1+H˙2+H12+H22+H1H2+p3=0.
Also we can introduce the three EoS parameters as
(2.20)ω1=p1ρ,ω2=p2ρ,ω3=p3ρ
and the deceleration parameters
(2.21)q1=-A¨AA˙2,q2=-B¨BB˙2,q3=-C¨CC˙2.
Finally we want to present
(2.22)2H˙+6H2=ρ-p,
where
(2.23)p=p1+p2+p33
is the average pressure. Hence we can calculate the average parameter of the EoS as
(2.24)ω=pρ=ω1+ω2+ω33.
Let us also present the expression of R in terms of Hi. From (2.8) and (2.13) follows
(2.25)R=2(H˙1+H˙2+H˙3+H12+H22+H32+H1H2+H1H3+H2H3).
Now we want to present the knot and unknotted universe solutions of the system (2.18) or its equivalent (2.19). Consider some examples.


## 3. The Trefoil Knot Universe

Our aim in this section is to construct the simplest examples of the knot universes, namely, the trefoil knot universes. Consider the following examples.


### 3.1. Example 1

Let us assume that our universe is filled by the fluid with the following parametric EoS:(3.1)p1=-D1E1,p2=-D2E2,p3=-D3E3,ρ=D0E0,
where
(3.2)D1=(-12sin2(3τ)+36cos(3τ)+18cos2(3τ))cos(2τ)-49sin(2τ)(2649+cos(3τ))sin(3τ),E1=sin(3τ)(2+cos(3τ))sin(2τ),D2=-18sin(2τ)cos2(3τ)+(-49sin(3τ)cos(2τ)-36sin(2τ))cos(3τ)-26sin(3τ)cos(2τ)+12sin2(3τ)sin(2τ),E2=sin(3τ)(2+cos(3τ))cos(2τ),D3=-30sin(3τ)(2+cos(3τ))cos2(2τ)-38sin(2τ)(cos2(3τ)-(2738)sin2(3τ)+(5819)cos(3τ)+4019)cos(2τ)+30sin(3τ)sin2(2τ)(2+cos(3τ)),E3=(2+cos(3τ))2cos(2τ)sin(2τ),D0=(6cos2(2τ)-6sin2(2τ))cos3(3τ)+(24cos2(2τ)-22sin(3τ)sin(2τ)cos(2τ)-24sin2(2τ))cos2(3τ)+((-6sin2(3τ)+24)cos2(2τ)-52sin(3τ)sin(2τ)cos(2τ)+(6sin2(3τ)-24)sin2(2τ))cos(3τ)-(12(cos(2τ)-(34)sin(3τ)sin(2τ)))×(sin(3τ)cos(2τ)+(43)sin(2τ))sin(3τ),E0=(2+cos(3τ))2cos(2τ)sin(2τ)sin(3τ).
Substituting these expressions for the pressures and density of energy into the system (2.18), we obtain the following  solution:
(3.3)A=A0+[2+cos(3τ)]cos(2τ),B=B0+[2+cos(3τ)]sin(2τ),C=C0+sin(3τ),
where A0,B0,C0 are some real constants. We see that this solution describes the trefoil knot. In fact the solution (3.3) is the parametric equation of the trefoil knot. In Figure 1 we plot the trefoil knot for (3.3), where we assume
(3.4)A0=B0=C0=0
and the initial conditions are A(0)=3,B(0)=C(0)=0. The Hubble parameters for the solution (3.3) with (3.4) read as
(3.5)H1=-2tan(2τ)-2sin(3τ)2+cos(3τ),H2=-2cot(2τ)-2sin(3τ)2+cos(3τ),H3=3cot(3τ).
In Figure 2 we plot the evolution of Hi for thr solution (3.5) with (3.4). It is interesting to study the evolution of the volume of the trefoil knot universe. For our case it is given by
(3.6)V=[2+cos(3τ)]2cos(2τ)sin(2τ)sin(3τ).
In Figure 3 we plot the evolution of the volume of the trefoil knot universe with respect to the cosmic time τ for (3.6) with (3.4). To get V≥0, we must consider A0,B0,C0>0, if exactly for example, as A0>3,B0>3,C0>1. But below for simplicity we take the case (3.4). The other interesting quantity is the scalar curvature. For the trefoil knot solution (3.3), it has the form
(3.7)R=(6(12sin2(3τ)sin2(2τ)-28sin(3τ)cos(2τ)sin(2τ)+3sin3(3τ)cos(2τ)sin(2τ)-12sin2(3τ)cos2(2τ)-8cos(3τ)sin2(2τ)-8cos2(3τ)sin2(2τ)-2cos3(3τ)sin2(2τ)+8cos(3τ)cos2(2τ)+8cos2(3τ)cos2(2τ)+2cos3(3τ)cos2(2τ)-52sin(3τ)cos(2τ)cos(3τ)sin(2τ)-19cos(2τ)sin(2τ)sin(3τ)cos2(3τ)+6sin2(2τ)sin2(3τ)cos(3τ)-6cos2(2τ)sin2(3τ)cos(3τ)))/(sin(2τ)cos(2τ)(2+cos(3τ))2sin(3τ)).Figure 1
The trefoil knot for (3.3), where A0=B0=C0=0.Figure 2
The evolution of the Hubble parameters for (3.5).Figure 3
The evolution of the volume of the trefoil knot universe with respect to the cosmic timeτ for (3.6).In Figure4 we plot the evolution of the R with respect of the cosmic time τ.Figure 4
The evolution of theR with respect of the cosmic time τ for (3.7).So we have shown that the universe can live in the trefoil knot orbit according to the solution (3.3). It is interesting to note that this trefoil knot solution admits infinite number accelerated and decelerated expansion phases of the universe. To show this, as an example let us consider the solution for C from (3.3) that is C=C0+sin(3τ). In this case we have C¨=-9sin(3τ) so that C¨>0 (accelerating phase) as τ∈((π/3)+(2nπ/3),(2π/3)+(2nπ/3)) and C¨<0 (decelerating phase) as τ∈((2nπ/3),(π/3)+(2nπ/3)) with the transion points C˙=3cos(3τi)=0 as τi=(0.5π+nπ)/3, where n is integer that is n=0,±1,±2,±3,⋯.


### 3.2. Example 2

Now we consider the following parametric EoS:(3.8)p1=-D1E1,p2=-D2E2,p3=-D3E3,ρ=D0E0,
where
(3.9)D1=-sin2(2τ)cos2(3τ)+(-2cos(2τ)-4sin2(2τ)-3-sin(3τ)sin(2τ))cos(3τ)+sin(3τ)sin(2τ)-4cos(2τ)-sin2(3τ)-4sin2(2τ),E1=1,D2=-(2+cos2(3τ))cos2(2τ)-sin(3τ)(-1+cos(3τ))cos(2τ)+(-3+2sin(2τ))cos(3τ)+4sin(2τ)-sin2(3τ),E2=1,D3=-(2+cos2(3τ))cos2(2τ)+(-sin(2τ)cos2(3τ)+(-4sin(2τ)-2)cos(3τ)+3sin(3τ)-4-4sin(2τ)cos2(3τ))cos(2τ)+2sin(2τ)cos(3τ)+(4+3sin(3τ))sin(2τ)-sin2(3τ),E3=1,D0=(2+cos(3τ))×(((2+cos(3τ))sin(2τ)+sin(3τ))cos(2τ)+sin(3τ)sin(2τ)),E0=1.
Substituting these expressions for the pressures and density of energy into the system (3.8), we obtain the following  solution:
(3.10)H1=[2+cos(3τ)]cos(2τ)=2cos(2τ)+0.5[cos(5τ)+cos(τ)],H2=[2+cos(3τ)]sin(2τ)=2sin(2τ)+0.5[sin(5τ)-sin(τ)],H3=sin(3τ).
We see that this solution again describes the trefoil knot but for the "coordinates" Hi. Note that the scale factors we can recovered from (2.14). We get
(3.11)A=A0esin(2τ)+0.1sin(5τ)+0.5sin(τ),B=B0e-[cos(2τ)+0.1cos(5τ)-0.5cos(τ)],C=C0e-(1/3)cos(3τ),
where A0,B0,C0 are some real constants. In Figure 5 we plot the evolution of A,B,C accordingly to (3.11) and for the initial conditions A(0)=1,B(0)=e-0.6,C(0)=e-1/3, where we assume that A0=B0=C0=1. For this example, the volume of the universe is given by
(3.12)V=V0e{sin(2τ)+0.1sin(5τ)+0.5sin(τ)-[cos(2τ)+0.1cos(5τ)-0.5cos(τ)]-(1/3)cos(3τ)}.
The evolution of the volume for (3.12) is presented in Figure 6 for A0=B0=C0=V0=1 and for the intial condition V(0)=e-14/15.Figure 5
The evolution ofA,B,C accordingly to (3.11),  t∈[0,2π].Figure 6
The evolution of the volume for (3.12) with A0=B0=C0=V0=1.The scalar curvature has the form(3.13)R=(2cos2(2τ)+2sin2(2τ)+2cos(2τ)sin(2τ))cos2(3τ)+(8cos2(2τ)+(2sin(3τ)+4+8sin(2τ))cos(2τ)+6+8sin2(2τ)+(-4+2sin(3τ))sin(2τ))cos(3τ)+8cos2(2τ)+(-2sin(3τ)+8+8sin(2τ))cos(2τ)+8sin2(2τ)+(-8-2sin(3τ))sin(2τ)+2sin2(3τ).
In Figure 7 we plot the evolution of the R with respect of the cosmic time τ. Finally we conclude that the Einstein equations for the Bianchi I type metric admit the trefoil knot solution of the form (3.10) or (3.11). These solutions describe the accelerated and decelerated phases of the expansion of the universe.Figure 7
The evolution of theR with respect of the cosmic time τ for (3.13).


### 3.3. Example 3

Now we present a new kind of the trefoil knot universes. Let the system (2.18) has the solution
(3.14)A=A0+[2+cn(3τ)]cn(2τ),B=B0+[2+cn(3τ)]sn(2τ),C=C0+sn(3τ),
where cn(t)≡cn(t,k) and sn(t)≡sn(t,k) are the Jacobian elliptic functions which are doubly periodic functions, and k is the elliptic modulus. Figure 8 shows the knotted closed curve corresponding to the solution (3.14) with (3.4). Substituting (3.14) into the system (2.18) we get the corresponding expressions for ρ and pi that gives us the parametric EoS. This parametric EoS reads as
(3.15)p1=-D1E1,p2=-D2E2,p3=-D3E3,ρ=D0E0,
where
(3.16)D1=9k2cn(3τ,k)sn3(3τ,k)sn(2τ,k)-12∂∂τam(3τ,k)sn2(3τ,k)cn(2τ,k)∂∂τam(2τ,k)-4sn(2τ,k)((94)cn3(3τ,k)k2+(92)cn2(3τ,k)k2+((454)∂∂τam2(3τ,k)+cn2(2τ,k)k2+∂∂τam2(2τ,k))cn(3τ,k)+2∂∂τam2(2τ,k)+(92)∂∂τam2(3τ,k)+2cn2(2τ,k)k2)sn(3τ,k)+18cn(3τ,k)∂∂τam(3τ,k)(2+cn(3τ,k))cn(2τ,k)∂∂τam(2τ,k),E1=(2+cn(3τ,k))sn(2τ,k)sn(3τ,k),D2=9k2cn(3τ,k)sn3(3τ,k)cn(2τ,k)+12∂∂τam(3τ,k)sn2(3τ,k)∂∂τam(2τ,k)sn(2τ,k)-9cn(2τ,k)cn3(3τ,k)k2+2cn2(3τ,k)k2+((49)∂∂τam2(2τ,k)+5∂∂τam2(3τ,k)-(49)k2sn2(2τ,k))cn(3τ,k)+(89)∂∂τam2(2τ,k)+2∂∂τam2(3τ,k)-(89)k2sn2(2τ,k)sn(3τ,k)-18cn(3τ,k)∂∂τam(3τ,k)(2+cn(3τ,k))∂∂τam(2τ,k)sn(2τ,k),E2=sn(3τ,k)(2+cn(3τ,k))cn(2τ,k),D3=-4k2sn(2τ,k)(2+cn(3τ,k))2cn3(2τ,k)-30∂∂τam(2τ,k)sn(3τ,k)∂∂τam(3τ,k)(2+cn(3τ,k))cn2(2τ,k)+4sn(2τ,k)×((274)(274)(274)(274)sn2(3τ,k)∂∂τam(3τ,k)2-20∂∂τam2(2τ,k))k2(2+cn2(3τ,k))sn2(2τ,k)+(-(92)∂∂τam2(3τ,k)+(92)k2sn2(3τ,k)-5∂∂τam2(2τ,k))cn2(3τ,k)+(-20∂∂τam2(2τ,k)+9k2sn2(3τ,k)-9∂∂τam2(3τ,k))cn(3τ,k)+(274)sn2(3τ,k)∂∂τam(3τ,k)2-20∂∂τam2(2τ,k))cn(2τ,k)+30∂∂τam(3τ,k)sn(3τ,k)∂∂τam(2τ,k)sn2(2τ,k)(2+cn(3τ,k)),E3=(2+cn(3τ,k))2cn(2τ,k)sn(2τ,k),D0=-4sn(3τ,k)sn(2τ,k)cn(2τ,k)(2+cn(3τ,k))2∂∂τam2(2τ,k)+6∂∂τam(τ,k)(2+cn(3τ,k))×(cn2(3τ,k)+2cn(3τ,k)-sn2(3τ,k))(cn(2τ,k)-sn(2τ,k))×(cn(2τ,k)+sn(2τ,k))∂∂τam(2τ,k)-18sn(3τ,k)sn(2τ,k)×(-(12)sn2(3τ,k)+cn2(3τ,k)+2cn(3τ,k))cn(2τ,k)∂∂τam2(3τ,k),E0=(2+cn2(3τ,k))cn(2τ,k)sn(2τ,k)sn(3τ,k).
The volume of the universe for the solution (3.14) with (3.4) looks like
(3.17)V=[2+cn(3τ)]2cn(2τ)sn(2τ)sn(3τ).
The evolution of the volume for (3.17) is presented in Figure 9 The scalar curvature has the form
(3.18)R=((83)dn2(2τ,k)+2dn2(3τ,k)))-8sn(2τ,k)sn(3τ,k)k2(2+cn(3τ,k))2cn3(2τ,k)+12dn(2τ,k)dn(3τ,k)(2+cn(3τ,k))×(cn2(3τ,k)+2cn(3τ,k)-3sn2(3τ,k))cn2(2τ,k)-18sn(3τ,k)×(-(49)k2(2+cn(3τ,k))2sn2(2τ,k)+(-4k2cn(3τ,k)-2cn2(3τ,k)k2-dn2(3τ,k))sn2(3τ,k)+(2+cn(3τ,k))×((43)cn3(3τ,k)k2+2cn2(3τ,k)k2+(5dn2(3τ,k)+(43)dn2(2τ,k))cn(3τ,k)+(83)dn2(2τ,k)+2dn2(3τ,k)))sn(2τ,k)cn(2τ,k)-12dn(2τ,k)sn2(2τ,k)dn(3τ,k)(2+cn(3τ,k))×(cn2(3τ,k)+2cn(3τ,k)-3sn2(3τ,k))(83)dn2(2τ,k)+2dn2(3τ,k))))/(cn(2τ,k)sn(2τ,k)(2+cn(3τ,k))2sn(3τ,k)).
In Figure 10 we plot the evolution of the R with respect of the cosmic time τ.Figure 8
The knotted closed curve corresponding to the solution (3.14) with (3.4), t∈[0,4π], k=1/3.Figure 9
The evolution of the volume of the trefoil knot universe with respect to the cosmic timeτ for (3.17).Figure 10
The evolution of theR with respect of the cosmic time τ for (3.18).


### 3.4. Example 4

Our fourth example is given by(3.19)H1=[2+cn(3τ)]cn(2τ),H2=[2+cn(3τ)]sn(2τ),H3=sn(3τ),
Which again the knotted closed curve in Figure 8 but for the "coordinates" Hi. Note that the corresponding parametric EoS looks like
(3.20)p1=-D1E1,p2=-D2E2,p3=-D3E3,ρ=D0E0,
where
(3.21)D1=-(2+cn(3τ,k))2sn2(2τ,k)-sn(3τ,k)(-3∂∂τam(3τ,k)+2+cn(3τ,k))sn(2τ,k)+(-2cn(2τ,k)∂∂τam(2τ,k)-3∂∂τam(3τ,k))cn(3τ,k)-4cn(2τ,k)∂∂τam(2τ,k)-sn2(3τ,k),E1=1,D2=-(2+cn(3τ,k))2cn2(2τ,k)-sn(3τ,k)(-3∂∂τam(3τ,k)+2+cn(3τ,k))cn(2τ,k)+(-3∂∂τam(3τ,k)+2∂∂τam(2τ,k)sn(2τ,k))cn(3τ,k)+4∂∂τam(2τ,k)sn(2τ,k)-sn2(3τ,k),E2=1,D3=-(2+cn(3τ,k))2cn2(2τ,k)+(-sn(2τ,k)cn2(3τ,k)+(-4sn(2τ,k)-2∂∂τam(2τ,k))cn(3τ,k)+3sn(3τ,k)∂∂τam(3τ,k)-4∂∂τam(2τ,k)-4sn(2τ,k))×cn(2τ,k)+2∂∂τam(2τ,k)sn(2τ,k)cn(3τ,k)+(4∂∂τam(2τ,k)+3sn(3τ,k)∂∂τam(3τ,k))sn(2τ,k)-sn2(3τ,k),E3=1,D0=(((2+cn(3τ,k))sn(2τ,k)+sn(3τ,k))cn(2τ,k)+sn(2τ,k)sn(3τ,k))×(2+cn(3τ,k)),E0=1.
In Figure 11 we plot the evolution of pi,ρ for (3.20). The scalar curvature has the form
(3.22)R=2(2+cn(3τ,k))2cn2(2τ,k)+(2sn(2τ,k)cn2(3τ,k)+(8sn(2τ,k)+4dn(2τ,k)+2sn(3τ,k))cn(3τ,k)+8sn(2τ,k)+(4-6dn(3τ,k))sn(3τ,k)+8dn(2τ,k)cn2)cn(2τ,k)+2sn2(2τ,k)cn2(3τ,k)+((-4dn(2τ,k)+2sn(3τ,k))8sn2(2τ,k)+(-4dn(2τ,k)+2sn(3τ,k))sn(2τ,k)+6dn(3τ,k)sn2(2τ,k))×cn(3τ,k)+8sn2(2τ,k)+((4-6dn(3τ,k))sn(3τ,k)-8dn(2τ,k))×sn(2τ,k)+2sn(3τ,k)2,
In Figure 12 we plot the evolution of the R with respect of the cosmic time τ.Figure 11
The evolution ofpi,ρ for (3.20), t∈[0,2π], k=1/3, ρ(red), p1(blue), p2(green), p3(black).Figure 12
The evolution of theR with respect of the cosmic time τ for (3.22).


## 3.1. Example 1

Let us assume that our universe is filled by the fluid with the following parametric EoS:(3.1)p1=-D1E1,p2=-D2E2,p3=-D3E3,ρ=D0E0,
where
(3.2)D1=(-12sin2(3τ)+36cos(3τ)+18cos2(3τ))cos(2τ)-49sin(2τ)(2649+cos(3τ))sin(3τ),E1=sin(3τ)(2+cos(3τ))sin(2τ),D2=-18sin(2τ)cos2(3τ)+(-49sin(3τ)cos(2τ)-36sin(2τ))cos(3τ)-26sin(3τ)cos(2τ)+12sin2(3τ)sin(2τ),E2=sin(3τ)(2+cos(3τ))cos(2τ),D3=-30sin(3τ)(2+cos(3τ))cos2(2τ)-38sin(2τ)(cos2(3τ)-(2738)sin2(3τ)+(5819)cos(3τ)+4019)cos(2τ)+30sin(3τ)sin2(2τ)(2+cos(3τ)),E3=(2+cos(3τ))2cos(2τ)sin(2τ),D0=(6cos2(2τ)-6sin2(2τ))cos3(3τ)+(24cos2(2τ)-22sin(3τ)sin(2τ)cos(2τ)-24sin2(2τ))cos2(3τ)+((-6sin2(3τ)+24)cos2(2τ)-52sin(3τ)sin(2τ)cos(2τ)+(6sin2(3τ)-24)sin2(2τ))cos(3τ)-(12(cos(2τ)-(34)sin(3τ)sin(2τ)))×(sin(3τ)cos(2τ)+(43)sin(2τ))sin(3τ),E0=(2+cos(3τ))2cos(2τ)sin(2τ)sin(3τ).
Substituting these expressions for the pressures and density of energy into the system (2.18), we obtain the following  solution:
(3.3)A=A0+[2+cos(3τ)]cos(2τ),B=B0+[2+cos(3τ)]sin(2τ),C=C0+sin(3τ),
where A0,B0,C0 are some real constants. We see that this solution describes the trefoil knot. In fact the solution (3.3) is the parametric equation of the trefoil knot. In Figure 1 we plot the trefoil knot for (3.3), where we assume
(3.4)A0=B0=C0=0
and the initial conditions are A(0)=3,B(0)=C(0)=0. The Hubble parameters for the solution (3.3) with (3.4) read as
(3.5)H1=-2tan(2τ)-2sin(3τ)2+cos(3τ),H2=-2cot(2τ)-2sin(3τ)2+cos(3τ),H3=3cot(3τ).
In Figure 2 we plot the evolution of Hi for thr solution (3.5) with (3.4). It is interesting to study the evolution of the volume of the trefoil knot universe. For our case it is given by
(3.6)V=[2+cos(3τ)]2cos(2τ)sin(2τ)sin(3τ).
In Figure 3 we plot the evolution of the volume of the trefoil knot universe with respect to the cosmic time τ for (3.6) with (3.4). To get V≥0, we must consider A0,B0,C0>0, if exactly for example, as A0>3,B0>3,C0>1. But below for simplicity we take the case (3.4). The other interesting quantity is the scalar curvature. For the trefoil knot solution (3.3), it has the form
(3.7)R=(6(12sin2(3τ)sin2(2τ)-28sin(3τ)cos(2τ)sin(2τ)+3sin3(3τ)cos(2τ)sin(2τ)-12sin2(3τ)cos2(2τ)-8cos(3τ)sin2(2τ)-8cos2(3τ)sin2(2τ)-2cos3(3τ)sin2(2τ)+8cos(3τ)cos2(2τ)+8cos2(3τ)cos2(2τ)+2cos3(3τ)cos2(2τ)-52sin(3τ)cos(2τ)cos(3τ)sin(2τ)-19cos(2τ)sin(2τ)sin(3τ)cos2(3τ)+6sin2(2τ)sin2(3τ)cos(3τ)-6cos2(2τ)sin2(3τ)cos(3τ)))/(sin(2τ)cos(2τ)(2+cos(3τ))2sin(3τ)).Figure 1
The trefoil knot for (3.3), where A0=B0=C0=0.Figure 2
The evolution of the Hubble parameters for (3.5).Figure 3
The evolution of the volume of the trefoil knot universe with respect to the cosmic timeτ for (3.6).In Figure4 we plot the evolution of the R with respect of the cosmic time τ.Figure 4
The evolution of theR with respect of the cosmic time τ for (3.7).So we have shown that the universe can live in the trefoil knot orbit according to the solution (3.3). It is interesting to note that this trefoil knot solution admits infinite number accelerated and decelerated expansion phases of the universe. To show this, as an example let us consider the solution for C from (3.3) that is C=C0+sin(3τ). In this case we have C¨=-9sin(3τ) so that C¨>0 (accelerating phase) as τ∈((π/3)+(2nπ/3),(2π/3)+(2nπ/3)) and C¨<0 (decelerating phase) as τ∈((2nπ/3),(π/3)+(2nπ/3)) with the transion points C˙=3cos(3τi)=0 as τi=(0.5π+nπ)/3, where n is integer that is n=0,±1,±2,±3,⋯.


## 3.2. Example 2

Now we consider the following parametric EoS:(3.8)p1=-D1E1,p2=-D2E2,p3=-D3E3,ρ=D0E0,
where
(3.9)D1=-sin2(2τ)cos2(3τ)+(-2cos(2τ)-4sin2(2τ)-3-sin(3τ)sin(2τ))cos(3τ)+sin(3τ)sin(2τ)-4cos(2τ)-sin2(3τ)-4sin2(2τ),E1=1,D2=-(2+cos2(3τ))cos2(2τ)-sin(3τ)(-1+cos(3τ))cos(2τ)+(-3+2sin(2τ))cos(3τ)+4sin(2τ)-sin2(3τ),E2=1,D3=-(2+cos2(3τ))cos2(2τ)+(-sin(2τ)cos2(3τ)+(-4sin(2τ)-2)cos(3τ)+3sin(3τ)-4-4sin(2τ)cos2(3τ))cos(2τ)+2sin(2τ)cos(3τ)+(4+3sin(3τ))sin(2τ)-sin2(3τ),E3=1,D0=(2+cos(3τ))×(((2+cos(3τ))sin(2τ)+sin(3τ))cos(2τ)+sin(3τ)sin(2τ)),E0=1.
Substituting these expressions for the pressures and density of energy into the system (3.8), we obtain the following  solution:
(3.10)H1=[2+cos(3τ)]cos(2τ)=2cos(2τ)+0.5[cos(5τ)+cos(τ)],H2=[2+cos(3τ)]sin(2τ)=2sin(2τ)+0.5[sin(5τ)-sin(τ)],H3=sin(3τ).
We see that this solution again describes the trefoil knot but for the "coordinates" Hi. Note that the scale factors we can recovered from (2.14). We get
(3.11)A=A0esin(2τ)+0.1sin(5τ)+0.5sin(τ),B=B0e-[cos(2τ)+0.1cos(5τ)-0.5cos(τ)],C=C0e-(1/3)cos(3τ),
where A0,B0,C0 are some real constants. In Figure 5 we plot the evolution of A,B,C accordingly to (3.11) and for the initial conditions A(0)=1,B(0)=e-0.6,C(0)=e-1/3, where we assume that A0=B0=C0=1. For this example, the volume of the universe is given by
(3.12)V=V0e{sin(2τ)+0.1sin(5τ)+0.5sin(τ)-[cos(2τ)+0.1cos(5τ)-0.5cos(τ)]-(1/3)cos(3τ)}.
The evolution of the volume for (3.12) is presented in Figure 6 for A0=B0=C0=V0=1 and for the intial condition V(0)=e-14/15.Figure 5
The evolution ofA,B,C accordingly to (3.11),  t∈[0,2π].Figure 6
The evolution of the volume for (3.12) with A0=B0=C0=V0=1.The scalar curvature has the form(3.13)R=(2cos2(2τ)+2sin2(2τ)+2cos(2τ)sin(2τ))cos2(3τ)+(8cos2(2τ)+(2sin(3τ)+4+8sin(2τ))cos(2τ)+6+8sin2(2τ)+(-4+2sin(3τ))sin(2τ))cos(3τ)+8cos2(2τ)+(-2sin(3τ)+8+8sin(2τ))cos(2τ)+8sin2(2τ)+(-8-2sin(3τ))sin(2τ)+2sin2(3τ).
In Figure 7 we plot the evolution of the R with respect of the cosmic time τ. Finally we conclude that the Einstein equations for the Bianchi I type metric admit the trefoil knot solution of the form (3.10) or (3.11). These solutions describe the accelerated and decelerated phases of the expansion of the universe.Figure 7
The evolution of theR with respect of the cosmic time τ for (3.13).


## 3.3. Example 3

Now we present a new kind of the trefoil knot universes. Let the system (2.18) has the solution
(3.14)A=A0+[2+cn(3τ)]cn(2τ),B=B0+[2+cn(3τ)]sn(2τ),C=C0+sn(3τ),
where cn(t)≡cn(t,k) and sn(t)≡sn(t,k) are the Jacobian elliptic functions which are doubly periodic functions, and k is the elliptic modulus. Figure 8 shows the knotted closed curve corresponding to the solution (3.14) with (3.4). Substituting (3.14) into the system (2.18) we get the corresponding expressions for ρ and pi that gives us the parametric EoS. This parametric EoS reads as
(3.15)p1=-D1E1,p2=-D2E2,p3=-D3E3,ρ=D0E0,
where
(3.16)D1=9k2cn(3τ,k)sn3(3τ,k)sn(2τ,k)-12∂∂τam(3τ,k)sn2(3τ,k)cn(2τ,k)∂∂τam(2τ,k)-4sn(2τ,k)((94)cn3(3τ,k)k2+(92)cn2(3τ,k)k2+((454)∂∂τam2(3τ,k)+cn2(2τ,k)k2+∂∂τam2(2τ,k))cn(3τ,k)+2∂∂τam2(2τ,k)+(92)∂∂τam2(3τ,k)+2cn2(2τ,k)k2)sn(3τ,k)+18cn(3τ,k)∂∂τam(3τ,k)(2+cn(3τ,k))cn(2τ,k)∂∂τam(2τ,k),E1=(2+cn(3τ,k))sn(2τ,k)sn(3τ,k),D2=9k2cn(3τ,k)sn3(3τ,k)cn(2τ,k)+12∂∂τam(3τ,k)sn2(3τ,k)∂∂τam(2τ,k)sn(2τ,k)-9cn(2τ,k)cn3(3τ,k)k2+2cn2(3τ,k)k2+((49)∂∂τam2(2τ,k)+5∂∂τam2(3τ,k)-(49)k2sn2(2τ,k))cn(3τ,k)+(89)∂∂τam2(2τ,k)+2∂∂τam2(3τ,k)-(89)k2sn2(2τ,k)sn(3τ,k)-18cn(3τ,k)∂∂τam(3τ,k)(2+cn(3τ,k))∂∂τam(2τ,k)sn(2τ,k),E2=sn(3τ,k)(2+cn(3τ,k))cn(2τ,k),D3=-4k2sn(2τ,k)(2+cn(3τ,k))2cn3(2τ,k)-30∂∂τam(2τ,k)sn(3τ,k)∂∂τam(3τ,k)(2+cn(3τ,k))cn2(2τ,k)+4sn(2τ,k)×((274)(274)(274)(274)sn2(3τ,k)∂∂τam(3τ,k)2-20∂∂τam2(2τ,k))k2(2+cn2(3τ,k))sn2(2τ,k)+(-(92)∂∂τam2(3τ,k)+(92)k2sn2(3τ,k)-5∂∂τam2(2τ,k))cn2(3τ,k)+(-20∂∂τam2(2τ,k)+9k2sn2(3τ,k)-9∂∂τam2(3τ,k))cn(3τ,k)+(274)sn2(3τ,k)∂∂τam(3τ,k)2-20∂∂τam2(2τ,k))cn(2τ,k)+30∂∂τam(3τ,k)sn(3τ,k)∂∂τam(2τ,k)sn2(2τ,k)(2+cn(3τ,k)),E3=(2+cn(3τ,k))2cn(2τ,k)sn(2τ,k),D0=-4sn(3τ,k)sn(2τ,k)cn(2τ,k)(2+cn(3τ,k))2∂∂τam2(2τ,k)+6∂∂τam(τ,k)(2+cn(3τ,k))×(cn2(3τ,k)+2cn(3τ,k)-sn2(3τ,k))(cn(2τ,k)-sn(2τ,k))×(cn(2τ,k)+sn(2τ,k))∂∂τam(2τ,k)-18sn(3τ,k)sn(2τ,k)×(-(12)sn2(3τ,k)+cn2(3τ,k)+2cn(3τ,k))cn(2τ,k)∂∂τam2(3τ,k),E0=(2+cn2(3τ,k))cn(2τ,k)sn(2τ,k)sn(3τ,k).
The volume of the universe for the solution (3.14) with (3.4) looks like
(3.17)V=[2+cn(3τ)]2cn(2τ)sn(2τ)sn(3τ).
The evolution of the volume for (3.17) is presented in Figure 9 The scalar curvature has the form
(3.18)R=((83)dn2(2τ,k)+2dn2(3τ,k)))-8sn(2τ,k)sn(3τ,k)k2(2+cn(3τ,k))2cn3(2τ,k)+12dn(2τ,k)dn(3τ,k)(2+cn(3τ,k))×(cn2(3τ,k)+2cn(3τ,k)-3sn2(3τ,k))cn2(2τ,k)-18sn(3τ,k)×(-(49)k2(2+cn(3τ,k))2sn2(2τ,k)+(-4k2cn(3τ,k)-2cn2(3τ,k)k2-dn2(3τ,k))sn2(3τ,k)+(2+cn(3τ,k))×((43)cn3(3τ,k)k2+2cn2(3τ,k)k2+(5dn2(3τ,k)+(43)dn2(2τ,k))cn(3τ,k)+(83)dn2(2τ,k)+2dn2(3τ,k)))sn(2τ,k)cn(2τ,k)-12dn(2τ,k)sn2(2τ,k)dn(3τ,k)(2+cn(3τ,k))×(cn2(3τ,k)+2cn(3τ,k)-3sn2(3τ,k))(83)dn2(2τ,k)+2dn2(3τ,k))))/(cn(2τ,k)sn(2τ,k)(2+cn(3τ,k))2sn(3τ,k)).
In Figure 10 we plot the evolution of the R with respect of the cosmic time τ.Figure 8
The knotted closed curve corresponding to the solution (3.14) with (3.4), t∈[0,4π], k=1/3.Figure 9
The evolution of the volume of the trefoil knot universe with respect to the cosmic timeτ for (3.17).Figure 10
The evolution of theR with respect of the cosmic time τ for (3.18).


## 3.4. Example 4

Our fourth example is given by(3.19)H1=[2+cn(3τ)]cn(2τ),H2=[2+cn(3τ)]sn(2τ),H3=sn(3τ),
Which again the knotted closed curve in Figure 8 but for the "coordinates" Hi. Note that the corresponding parametric EoS looks like
(3.20)p1=-D1E1,p2=-D2E2,p3=-D3E3,ρ=D0E0,
where
(3.21)D1=-(2+cn(3τ,k))2sn2(2τ,k)-sn(3τ,k)(-3∂∂τam(3τ,k)+2+cn(3τ,k))sn(2τ,k)+(-2cn(2τ,k)∂∂τam(2τ,k)-3∂∂τam(3τ,k))cn(3τ,k)-4cn(2τ,k)∂∂τam(2τ,k)-sn2(3τ,k),E1=1,D2=-(2+cn(3τ,k))2cn2(2τ,k)-sn(3τ,k)(-3∂∂τam(3τ,k)+2+cn(3τ,k))cn(2τ,k)+(-3∂∂τam(3τ,k)+2∂∂τam(2τ,k)sn(2τ,k))cn(3τ,k)+4∂∂τam(2τ,k)sn(2τ,k)-sn2(3τ,k),E2=1,D3=-(2+cn(3τ,k))2cn2(2τ,k)+(-sn(2τ,k)cn2(3τ,k)+(-4sn(2τ,k)-2∂∂τam(2τ,k))cn(3τ,k)+3sn(3τ,k)∂∂τam(3τ,k)-4∂∂τam(2τ,k)-4sn(2τ,k))×cn(2τ,k)+2∂∂τam(2τ,k)sn(2τ,k)cn(3τ,k)+(4∂∂τam(2τ,k)+3sn(3τ,k)∂∂τam(3τ,k))sn(2τ,k)-sn2(3τ,k),E3=1,D0=(((2+cn(3τ,k))sn(2τ,k)+sn(3τ,k))cn(2τ,k)+sn(2τ,k)sn(3τ,k))×(2+cn(3τ,k)),E0=1.
In Figure 11 we plot the evolution of pi,ρ for (3.20). The scalar curvature has the form
(3.22)R=2(2+cn(3τ,k))2cn2(2τ,k)+(2sn(2τ,k)cn2(3τ,k)+(8sn(2τ,k)+4dn(2τ,k)+2sn(3τ,k))cn(3τ,k)+8sn(2τ,k)+(4-6dn(3τ,k))sn(3τ,k)+8dn(2τ,k)cn2)cn(2τ,k)+2sn2(2τ,k)cn2(3τ,k)+((-4dn(2τ,k)+2sn(3τ,k))8sn2(2τ,k)+(-4dn(2τ,k)+2sn(3τ,k))sn(2τ,k)+6dn(3τ,k)sn2(2τ,k))×cn(3τ,k)+8sn2(2τ,k)+((4-6dn(3τ,k))sn(3τ,k)-8dn(2τ,k))×sn(2τ,k)+2sn(3τ,k)2,
In Figure 12 we plot the evolution of the R with respect of the cosmic time τ.Figure 11
The evolution ofpi,ρ for (3.20), t∈[0,2π], k=1/3, ρ(red), p1(blue), p2(green), p3(black).Figure 12
The evolution of theR with respect of the cosmic time τ for (3.22).


## 4. The Figure-Eight Knot Universe

Our aim in this section is to demonstrate some examples of the figure-eight knot universes for the Bianchi type I metric (2.6). We give some particular figure-eight knot universe models.


### 4.1. Example 1

Again, let us assume that our universe is filled by the fluid with the following parametric EoS:(4.1)ρ=D8E8,p1=-D9E9,p2=-D10E10,p3=-D11E11,
where
(4.2)D8=(-2sin(2τ)cos(3τ)-(3(2+cos(2τ)))sin(3τ))×(-2sin(2τ)sin(3τ)+(3(2+cos(2τ)))cos(3τ))×sin(4τ)+12cos(3τ)×((2+cos(2τ))cos(3τ)-(23)sin(2τ)sin(3τ))×(2+cos(2τ))cos(4τ)-(12(2+cos(2τ)))cos(4τ)*((2+cos(2τ))sin(3τ)+(23)sin(2τ)cos(3τ))sin(3τ),E8=(2+cos(2τ))2cos(3τ)sin(3τ)sin(4τ),D9=((72+36cos(2τ))cos(4τ)-12sin(4τ)sin(2τ))cos(3τ)-(29((2429)*cos(4τ)sin(2τ)+(cos(2τ)+5029)sin(4τ)))sin(3τ),E9=sin(4τ)(2+cos(2τ))sin(3τ),D10=(-24cos(4τ)sin(2τ)+sin(4τ)(-29cos(2τ)-50))cos(3τ)-(36((2+cos(2τ))cos(4τ)-(13)sin(4τ)sin(2τ)))sin(3τ),E10=sin(4τ)(2+cos(2τ))cos(3τ),D11=-(30(2+cos(2τ)))sin(2τ)cos(3τ)2+sin(3τ)×(12sin(2τ)2-196cos(2τ)-180-53cos(2τ)2)cos(3τ)+30sin(2τ)sin(3τ)2(2+cos(2τ)),E11=(2+cos(2τ))2cos(3τ)sin(3τ).Substituting these expressions for the pressuries and the density of energy into the system (2.18), we obtain the following its solution [98–103]:
(4.3)A=A0+[2+cos(2τ)]cos(3τ),B=B0+[2+cos(2τ)]sin(3τ),C=C0+sin(4τ).
This solution is nothing but the parametric equation of the figure-eight knot as we can see from Figure 13, where we assume that A0=B0=C0=0 and the initial conditions have the form A(0)=3,B(0)=0,C(0)=0. And for that reason in [98–103] we called such models as the figure-eight knot universes. Note that the “coordinates” A,B,C with (3.4) satisfy the equation
(4.4)4(h-2)4-4(h-2)2+z2=0,
where h=2+cos(2τ). Let us calculate the volume of the universe. For our case it is given by
(4.5)V=[2+cos(2τ)]2cos(3τ)sin(3τ)sin(4τ),
where we used (3.4). In Figure 14 we present the evolution of the volume for the solution (4.3) with (3.4). The scalar curvature has the form
(4.6)R=((24(cos(4τ)cos(2τ)-(32)sin(4τ)sin(2τ)+2cos(4τ)))×(2+cos(2τ))cos2(3τ)-102sin(3τ)×(sin(4τ)cos2(2τ)+((18851)sin(4τ)+(1651)cos(4τ)sin(2τ))cos(2τ)+(3251)cos(4τ)sin(2τ)+(17251)sin(4τ)-(451)sin2(2τ)sin(4τ))cos(3τ)-24sin2(3τ)×(cos(4τ)cos(2τ)-(32)sin(4τ)sin(2τ)+2cos(4τ))(2+cos(2τ)))/(sin(3τ)cos(3τ)*(2+cos(2τ))2sin(4τ)).
In Figure 15 we plot the evolution of the R with respect of the cosmic time τ. So we found the figure-eight knot solution of the Einstein equations which again describe the accelerated and decelerated expansion phases of the universe.Figure 13
The figure-eight knot for (4.3) with (3.4).Figure 14
The evolution of the volume for the solution (4.3) with (3.4), t∈[0,2π].Figure 15
The evolution of theR with respect of the cosmic time τ for (4.6).


### 4.2. Example 2

Now we consider the system (2.19). Its solution is given by
(4.7)H1=[2+cos(2τ)]cos(3τ)=2cos(3τ)+cos(5τ)+cos(τ),H2=[2+cos(2τ)]sin(3τ)=2sin(3τ)+sin(τ)+sin(5τ),H3=sin(4τ).
Then the coorresponding scale factors read as
(4.8)A=A0e(2/3)sin(3τ)+0.2sin(5τ)+sin(τ),B=B0e-[(2/3)ຉcos(3τ)+0.2cos(5τ)+cos(τ)],C=C0e-0.25cos(4τ).
For this solution the parametric EoS looks like
(4.9)ρ=D0E0,p1=-D1E1,p2=-D2E2,p3=-D3E3,
where
(4.10)D0=(((2+cos(2τ))sin(3τ)+sin(4τ))cos(3τ)+sin(3τ)sin(4τ))(2+cos(2τ)),E0=1,D1=-(2+cos(2τ))2sin2(3τ)+(2sin(2τ)-2sin(4τ)-sin(4τ)cos(2τ))sin(3τ)-6cos(3τ)-3cos(3τ)cos(2τ)-4cos(4τ)-sin2(4τ),E1=1,D2=-(2+cos(2τ))2cos2(3τ)+(2sin(2τ)-2sin(4τ)-sin(4τ)cos(2τ))cos(3τ)-4cos(4τ)+6sin(3τ)+3sin(3τ)cos(2τ)-sin2(4τ),E2=1,D3=-3sin(τ)-64sin(τ)cos9(τ)+36sin(τ)cos5(τ)+40sin(τ)cos4(τ)+4sin(τ)cos3(τ)-6sin(τ)cos2(τ)-3sin(τ)cos(τ)-25cos2(τ)+5cos(τ)-40cos5(τ)-64cos10(τ)+96cos8(τ)-84cos6(τ)+68cos4(τ)+26cos3(τ),E3=1.
In Figure 16 we plot the EoS (4.9). For this example, the evolution of the volume of the universe is given by
(4.11)V=V0e(2/3)sin(3τ)+0.2sin(5τ)+sin(τ)-(2/3)cos(3τ)-0.2cos(5τ)-cos(τ)-0.25cos(4τ).
The evolution of the volume is presented in Figure 17 for A0=B0=C0=V0=1 and for the intial condition V(0)=e-127/60. The scalar curvature has the form
(4.12)R=2(2+cos(2τ))2cos2(3τ)+(2(2+cos(2τ))2sin(3τ)+(6+2sin(4τ))cos(2τ)+12-4sin(2τ)+4sin(4τ)(2+cos(2τ))2)cos(3τ)+2(2+cos(2τ))2sin2(3τ)+((-6+2sin(4τ))cos(2τ)+4sin(4τ)-4sin(2τ)-12)*sin(3τ)+2sin2(4τ)+8cos(4τ).
In Figure 18 we plot the evolution of the R with respect of the cosmic time τ. Again we have shown that the Einstein equations admit the figure-eight knot solution and it again describe the accelerated and decelerated expansion phases of the universe.Figure 16
The plot of the EoS (4.9), t∈[0,2π], ρ(red), p1(blue), p2(green), p3(black).Figure 17
The evolution of the volume for the expression (4.11) with V0=1, t∈[0,2π].Figure 18
The evolution of theR with respect of the cosmic time τ for (4.12).


### 4.3. Example 3

Now we present the figure-eight knot universe induced by the Jacobian elliptic functions. Let the system (2.18) have the solution
(4.13)A=A0+[2+cn(2τ)]cn(3τ),B=B0+[2+cn(2τ)]sn(3τ),C=C0+sn(4τ).
Note that cn(t) and sn(t) are the doubly periodic Jacobian elliptic functions. Figure 19 shows the knotted closed curve corresponding to the solution (4.13) with (3.4). Substituting the formulas (4.13) into the system (2.18) we get the corresponding expressions for ρ and pi that gives us the parametric EoS. The evolution of the volume of the universe for (3.4) reads as
(4.14)V=[2+cn(2τ)]2cn(3τ)sn(3τ)sn(4τ).Figure 19
The knotted closed curve corresponding to the solution (4.13) with (3.4), t∈[0,4π], k=1/3.The scalar curvature has the form(4.15)R=(274-18sn(3τ,k)sn(4τ,k)k2(2+cn(2τ,k))2cn3(3τ,k)+(24(-(32)sn(2τ,k)dn(2τ,k)sn(4τ,k)+cn(4τ,k)dn(4τ,k)(2+cn(2τ,k))(32))(32))×(2+cn(2τ,k))dn(3τ,k)cn2(3τ,k)-(32(-(916)sn(4τ,k)k2(2+cn(2τ,k))2sn2(3τ,k)+((cn2(4τ,k)k2+(2716)dn2(3τ,k)+dn2(4τ,k)-(12)k2sn2(2τ,k)+(12)dn2(2τ,k))cn2(2τ,k)+(cn2(4τ,k)k2(274)-k2sn2(2τ,k)(274)dn2(3τ,k)+dn2(2τ,k)+4dn2(4τ,k)+4cn2(4τ,k)k2(274))×cn(2τ,k)+4dn2(4τ,k)+(274)dn2(3τ,k)-(14)dn2(2τ,k)sn2(2τ,k)+4cn2(4τ,k)k2)×sn(4τ,k)+cn(4τ,k)dn×(4τ,k)dn(2τ,k)sn(2τ,k)(2+cn(2τ,k))(274))274)sn(3τ,k)cn(3τ,k)-(24(-(32)sn(2τ,k)dn(2τ,k)sn(4τ,k)+cn(4τ,k)dn(4τ,k)(2+cn(2τ,k))))×(2+cn(2τ,k)(274))sn2(3τ,k)dn(3τ,k))/(cn(3τ,k)sn(3τ,k)(2+cn(2τ,k))2sn(4τ,k)(274-18sn(3τ,k)sn(4τ,k)k2(2+cn(2τ,k))2cn3(3τ,k)).
In Figure 20 we plot the evolution of the R with respect of the cosmic time τ.Figure 20
The evolution of theR with respect of the cosmic time τ for (4.15).


### 4.4. Example 4

We now consider the following solution of the system (2.19):
(4.16)H1=[2+cn(2τ)]cn(3τ),H2=[2+cn(2τ)]sn(3τ),H3=sn(4τ),
which again the trefoil knot universe as shown in Figure 19 but for the “coordinates” Hi. The corresponding parametric EoS reads as
(4.17)ρ=D0E0,p1=-D1E1,p2=-D2E2,p3=-D3E3,
where
(4.18)D0=(((2+cn(2τ,k))sn(3τ,k)+sn(4τ,k))cn(3τ,k)+sn(3τ,k)sn(4τ,k))×(2+cn(2τ,k)),E0=1,D1=2∂∂τam(2τ,k)sn(2τ,k)sn(3τ,k)-(3(2+cn(2τ,k)))cn(3τ,k)∂∂τam(3τ,k)-4cn(4τ,k)∂∂τam(4τ,k)-(2+cn(2τ,k))2sn(3τ,k)2-sn(4τ,k)2-(2+cn(2τ,k))sn(3τ,k)sn(4τ,k),E1=1,D2=-4cn(4τ,k)∂∂τam(4τ,k)+2∂∂τam(2τ,k)sn(2τ,k)cn(3τ,k)+(3(2+cn(2τ,k)))∂∂τam(3τ,k)sn(3τ,k)-sn2(4τ,k)-(2+cn(2τ,k))2cn2(3τ,k)-(2+cn(2τ,k))cn(3τ,k)sn(4τ,k),E2=1,D3=-(2+cn(2τ,k))2cn2(3τ,k)+({6∂∂τ}-sn(3τ,k)cn2(2τ,k)+(-4sn(3τ,k)-3∂∂τam(3τ,k))cn(2τ,k)+2∂∂τam(2τ,k)sn(2τ,k)-6∂∂τam(3τ,k)-4sn(3τ,k))×cn(3τ,k)+3∂∂τam(3τ,k)sn(3τ,k)cn(2τ,k)+(6∂∂τam(3τ,k)+2∂∂τam(2τ,k)sn(2τ,k))×sn(3τ,k)-sn2(4τ,k),E3=1.
Its plot we give in Figure 21.Figure 21
The plot of the EoS (4.17), t∈[0,2π], k=1/3, ρ(red), p1(blue), p2(green), p3(black).The scalar curvature has the form(4.19)R=2(2+cn(2τ,k))2cn(3τ,k)2+(2(2+cn(2τ,k))2sn(3τ,k)+(6dn(3τ,k)+2sn(4τ,k))cn(2τ,k)+12dn(3τ,k)-4dn(2τ,k)sn(2τ,k)+4sn(4τ,k)(2+cn(2τ,k))2)×cn(3τ,k)+2(2+cn(2τ,k))2sn(3τ,k)2+((-6dn(3τ,k)+2sn(4τ,k))cn(2τ,k)+4sn(4τ,k)-4dn(2τ,k)sn(2τ,k)-12dn(3τ,k))*sn(3τ,k)+2sn(4τ,k)2+8cn(4τ,k)dn(4τ,k).
In Figure 22 we plot the evolution of the R with respect of the cosmic time τ.Figure 22
The evolution of theR with respect of the cosmic time τ for (4.19).


## 4.1. Example 1

Again, let us assume that our universe is filled by the fluid with the following parametric EoS:(4.1)ρ=D8E8,p1=-D9E9,p2=-D10E10,p3=-D11E11,
where
(4.2)D8=(-2sin(2τ)cos(3τ)-(3(2+cos(2τ)))sin(3τ))×(-2sin(2τ)sin(3τ)+(3(2+cos(2τ)))cos(3τ))×sin(4τ)+12cos(3τ)×((2+cos(2τ))cos(3τ)-(23)sin(2τ)sin(3τ))×(2+cos(2τ))cos(4τ)-(12(2+cos(2τ)))cos(4τ)*((2+cos(2τ))sin(3τ)+(23)sin(2τ)cos(3τ))sin(3τ),E8=(2+cos(2τ))2cos(3τ)sin(3τ)sin(4τ),D9=((72+36cos(2τ))cos(4τ)-12sin(4τ)sin(2τ))cos(3τ)-(29((2429)*cos(4τ)sin(2τ)+(cos(2τ)+5029)sin(4τ)))sin(3τ),E9=sin(4τ)(2+cos(2τ))sin(3τ),D10=(-24cos(4τ)sin(2τ)+sin(4τ)(-29cos(2τ)-50))cos(3τ)-(36((2+cos(2τ))cos(4τ)-(13)sin(4τ)sin(2τ)))sin(3τ),E10=sin(4τ)(2+cos(2τ))cos(3τ),D11=-(30(2+cos(2τ)))sin(2τ)cos(3τ)2+sin(3τ)×(12sin(2τ)2-196cos(2τ)-180-53cos(2τ)2)cos(3τ)+30sin(2τ)sin(3τ)2(2+cos(2τ)),E11=(2+cos(2τ))2cos(3τ)sin(3τ).Substituting these expressions for the pressuries and the density of energy into the system (2.18), we obtain the following its solution [98–103]:
(4.3)A=A0+[2+cos(2τ)]cos(3τ),B=B0+[2+cos(2τ)]sin(3τ),C=C0+sin(4τ).
This solution is nothing but the parametric equation of the figure-eight knot as we can see from Figure 13, where we assume that A0=B0=C0=0 and the initial conditions have the form A(0)=3,B(0)=0,C(0)=0. And for that reason in [98–103] we called such models as the figure-eight knot universes. Note that the “coordinates” A,B,C with (3.4) satisfy the equation
(4.4)4(h-2)4-4(h-2)2+z2=0,
where h=2+cos(2τ). Let us calculate the volume of the universe. For our case it is given by
(4.5)V=[2+cos(2τ)]2cos(3τ)sin(3τ)sin(4τ),
where we used (3.4). In Figure 14 we present the evolution of the volume for the solution (4.3) with (3.4). The scalar curvature has the form
(4.6)R=((24(cos(4τ)cos(2τ)-(32)sin(4τ)sin(2τ)+2cos(4τ)))×(2+cos(2τ))cos2(3τ)-102sin(3τ)×(sin(4τ)cos2(2τ)+((18851)sin(4τ)+(1651)cos(4τ)sin(2τ))cos(2τ)+(3251)cos(4τ)sin(2τ)+(17251)sin(4τ)-(451)sin2(2τ)sin(4τ))cos(3τ)-24sin2(3τ)×(cos(4τ)cos(2τ)-(32)sin(4τ)sin(2τ)+2cos(4τ))(2+cos(2τ)))/(sin(3τ)cos(3τ)*(2+cos(2τ))2sin(4τ)).
In Figure 15 we plot the evolution of the R with respect of the cosmic time τ. So we found the figure-eight knot solution of the Einstein equations which again describe the accelerated and decelerated expansion phases of the universe.Figure 13
The figure-eight knot for (4.3) with (3.4).Figure 14
The evolution of the volume for the solution (4.3) with (3.4), t∈[0,2π].Figure 15
The evolution of theR with respect of the cosmic time τ for (4.6).


## 4.2. Example 2

Now we consider the system (2.19). Its solution is given by
(4.7)H1=[2+cos(2τ)]cos(3τ)=2cos(3τ)+cos(5τ)+cos(τ),H2=[2+cos(2τ)]sin(3τ)=2sin(3τ)+sin(τ)+sin(5τ),H3=sin(4τ).
Then the coorresponding scale factors read as
(4.8)A=A0e(2/3)sin(3τ)+0.2sin(5τ)+sin(τ),B=B0e-[(2/3)ຉcos(3τ)+0.2cos(5τ)+cos(τ)],C=C0e-0.25cos(4τ).
For this solution the parametric EoS looks like
(4.9)ρ=D0E0,p1=-D1E1,p2=-D2E2,p3=-D3E3,
where
(4.10)D0=(((2+cos(2τ))sin(3τ)+sin(4τ))cos(3τ)+sin(3τ)sin(4τ))(2+cos(2τ)),E0=1,D1=-(2+cos(2τ))2sin2(3τ)+(2sin(2τ)-2sin(4τ)-sin(4τ)cos(2τ))sin(3τ)-6cos(3τ)-3cos(3τ)cos(2τ)-4cos(4τ)-sin2(4τ),E1=1,D2=-(2+cos(2τ))2cos2(3τ)+(2sin(2τ)-2sin(4τ)-sin(4τ)cos(2τ))cos(3τ)-4cos(4τ)+6sin(3τ)+3sin(3τ)cos(2τ)-sin2(4τ),E2=1,D3=-3sin(τ)-64sin(τ)cos9(τ)+36sin(τ)cos5(τ)+40sin(τ)cos4(τ)+4sin(τ)cos3(τ)-6sin(τ)cos2(τ)-3sin(τ)cos(τ)-25cos2(τ)+5cos(τ)-40cos5(τ)-64cos10(τ)+96cos8(τ)-84cos6(τ)+68cos4(τ)+26cos3(τ),E3=1.
In Figure 16 we plot the EoS (4.9). For this example, the evolution of the volume of the universe is given by
(4.11)V=V0e(2/3)sin(3τ)+0.2sin(5τ)+sin(τ)-(2/3)cos(3τ)-0.2cos(5τ)-cos(τ)-0.25cos(4τ).
The evolution of the volume is presented in Figure 17 for A0=B0=C0=V0=1 and for the intial condition V(0)=e-127/60. The scalar curvature has the form
(4.12)R=2(2+cos(2τ))2cos2(3τ)+(2(2+cos(2τ))2sin(3τ)+(6+2sin(4τ))cos(2τ)+12-4sin(2τ)+4sin(4τ)(2+cos(2τ))2)cos(3τ)+2(2+cos(2τ))2sin2(3τ)+((-6+2sin(4τ))cos(2τ)+4sin(4τ)-4sin(2τ)-12)*sin(3τ)+2sin2(4τ)+8cos(4τ).
In Figure 18 we plot the evolution of the R with respect of the cosmic time τ. Again we have shown that the Einstein equations admit the figure-eight knot solution and it again describe the accelerated and decelerated expansion phases of the universe.Figure 16
The plot of the EoS (4.9), t∈[0,2π], ρ(red), p1(blue), p2(green), p3(black).Figure 17
The evolution of the volume for the expression (4.11) with V0=1, t∈[0,2π].Figure 18
The evolution of theR with respect of the cosmic time τ for (4.12).


## 4.3. Example 3

Now we present the figure-eight knot universe induced by the Jacobian elliptic functions. Let the system (2.18) have the solution
(4.13)A=A0+[2+cn(2τ)]cn(3τ),B=B0+[2+cn(2τ)]sn(3τ),C=C0+sn(4τ).
Note that cn(t) and sn(t) are the doubly periodic Jacobian elliptic functions. Figure 19 shows the knotted closed curve corresponding to the solution (4.13) with (3.4). Substituting the formulas (4.13) into the system (2.18) we get the corresponding expressions for ρ and pi that gives us the parametric EoS. The evolution of the volume of the universe for (3.4) reads as
(4.14)V=[2+cn(2τ)]2cn(3τ)sn(3τ)sn(4τ).Figure 19
The knotted closed curve corresponding to the solution (4.13) with (3.4), t∈[0,4π], k=1/3.The scalar curvature has the form(4.15)R=(274-18sn(3τ,k)sn(4τ,k)k2(2+cn(2τ,k))2cn3(3τ,k)+(24(-(32)sn(2τ,k)dn(2τ,k)sn(4τ,k)+cn(4τ,k)dn(4τ,k)(2+cn(2τ,k))(32))(32))×(2+cn(2τ,k))dn(3τ,k)cn2(3τ,k)-(32(-(916)sn(4τ,k)k2(2+cn(2τ,k))2sn2(3τ,k)+((cn2(4τ,k)k2+(2716)dn2(3τ,k)+dn2(4τ,k)-(12)k2sn2(2τ,k)+(12)dn2(2τ,k))cn2(2τ,k)+(cn2(4τ,k)k2(274)-k2sn2(2τ,k)(274)dn2(3τ,k)+dn2(2τ,k)+4dn2(4τ,k)+4cn2(4τ,k)k2(274))×cn(2τ,k)+4dn2(4τ,k)+(274)dn2(3τ,k)-(14)dn2(2τ,k)sn2(2τ,k)+4cn2(4τ,k)k2)×sn(4τ,k)+cn(4τ,k)dn×(4τ,k)dn(2τ,k)sn(2τ,k)(2+cn(2τ,k))(274))274)sn(3τ,k)cn(3τ,k)-(24(-(32)sn(2τ,k)dn(2τ,k)sn(4τ,k)+cn(4τ,k)dn(4τ,k)(2+cn(2τ,k))))×(2+cn(2τ,k)(274))sn2(3τ,k)dn(3τ,k))/(cn(3τ,k)sn(3τ,k)(2+cn(2τ,k))2sn(4τ,k)(274-18sn(3τ,k)sn(4τ,k)k2(2+cn(2τ,k))2cn3(3τ,k)).
In Figure 20 we plot the evolution of the R with respect of the cosmic time τ.Figure 20
The evolution of theR with respect of the cosmic time τ for (4.15).


## 4.4. Example 4

We now consider the following solution of the system (2.19):
(4.16)H1=[2+cn(2τ)]cn(3τ),H2=[2+cn(2τ)]sn(3τ),H3=sn(4τ),
which again the trefoil knot universe as shown in Figure 19 but for the “coordinates” Hi. The corresponding parametric EoS reads as
(4.17)ρ=D0E0,p1=-D1E1,p2=-D2E2,p3=-D3E3,
where
(4.18)D0=(((2+cn(2τ,k))sn(3τ,k)+sn(4τ,k))cn(3τ,k)+sn(3τ,k)sn(4τ,k))×(2+cn(2τ,k)),E0=1,D1=2∂∂τam(2τ,k)sn(2τ,k)sn(3τ,k)-(3(2+cn(2τ,k)))cn(3τ,k)∂∂τam(3τ,k)-4cn(4τ,k)∂∂τam(4τ,k)-(2+cn(2τ,k))2sn(3τ,k)2-sn(4τ,k)2-(2+cn(2τ,k))sn(3τ,k)sn(4τ,k),E1=1,D2=-4cn(4τ,k)∂∂τam(4τ,k)+2∂∂τam(2τ,k)sn(2τ,k)cn(3τ,k)+(3(2+cn(2τ,k)))∂∂τam(3τ,k)sn(3τ,k)-sn2(4τ,k)-(2+cn(2τ,k))2cn2(3τ,k)-(2+cn(2τ,k))cn(3τ,k)sn(4τ,k),E2=1,D3=-(2+cn(2τ,k))2cn2(3τ,k)+({6∂∂τ}-sn(3τ,k)cn2(2τ,k)+(-4sn(3τ,k)-3∂∂τam(3τ,k))cn(2τ,k)+2∂∂τam(2τ,k)sn(2τ,k)-6∂∂τam(3τ,k)-4sn(3τ,k))×cn(3τ,k)+3∂∂τam(3τ,k)sn(3τ,k)cn(2τ,k)+(6∂∂τam(3τ,k)+2∂∂τam(2τ,k)sn(2τ,k))×sn(3τ,k)-sn2(4τ,k),E3=1.
Its plot we give in Figure 21.Figure 21
The plot of the EoS (4.17), t∈[0,2π], k=1/3, ρ(red), p1(blue), p2(green), p3(black).The scalar curvature has the form(4.19)R=2(2+cn(2τ,k))2cn(3τ,k)2+(2(2+cn(2τ,k))2sn(3τ,k)+(6dn(3τ,k)+2sn(4τ,k))cn(2τ,k)+12dn(3τ,k)-4dn(2τ,k)sn(2τ,k)+4sn(4τ,k)(2+cn(2τ,k))2)×cn(3τ,k)+2(2+cn(2τ,k))2sn(3τ,k)2+((-6dn(3τ,k)+2sn(4τ,k))cn(2τ,k)+4sn(4τ,k)-4dn(2τ,k)sn(2τ,k)-12dn(3τ,k))*sn(3τ,k)+2sn(4τ,k)2+8cn(4τ,k)dn(4τ,k).
In Figure 22 we plot the evolution of the R with respect of the cosmic time τ.Figure 22
The evolution of theR with respect of the cosmic time τ for (4.19).


## 5. Other Unknotted Models of the Universe

In this section we would like to present some unknotted but closed curve solutions of the Einstein equation for the Bianchi I type metric. As an examples we consider the spiky and Mobious strip universe solutions.


### 5.1. Spiky Universe Solutions

Our aim in this subsection is to present some unknotted closed curve solutions namely the spiky universe solutions.


#### 5.1.1. Example 1

Let our universe be filled by the fluid with the following parametric EoS:(5.1)ρ=D8E8,p1=-D9E9,p2=-D10E10,p3=-D11E11,
where
(5.2)D8=-[αsin((n-1)τ)(n-1)+α(n-1)sin(τ)]×[αcos((n-1)τ)(n-1)-α(n-1)cos(τ)]sin(τ)+[αcos((n-1)τ)(n-1)-α(n-1)cos(τ)]cos(τ)×[αcos((n-1)τ)+α(n-1)cos(τ)]-[αsin((n-1)τ)(n-1)+α(n-1)sin(τ)]×[αsin((n-1)τ)-α(n-1)sin(τ)]cos(τ),E8=[αcos((n-1)τ)+α(n-1)cos(τ)]×[αsin((n-1)τ)-α(n-1)sin(τ)]sin(τ),D9=[-αsin((n-1)τ)(n-1)2+α(n-1)sin(τ)]sin(τ)-[αsin((n-1)τ)-α(n-1)sin(τ)]sin(τ)+[αcos((n-1)τ)(n-1)-α(n-1)cos(τ)]cos(τ),E9=[αsin((n-1)τ)-α(n-1)sin(τ)]sin(τ),D10=[αcos((n-1)τ)+α(n-1)cos(τ)]sin(τ)+sin(τ)×[αcos((n-1)τ)(n-1)2+α(n-1)cos(τ)]+[αsin((n-1)τ)(n-1)+α(n-1)sin(τ)]cos(τ),E10=-[αcos((n-1)τ)+α(n-1)cos(τ)]sin(τ),D11=[αsin((n-1)τ)-α(n-1)sin(τ)]×[-αcos((n-1)τ)(n-1)2-α(n-1)cos(τ)]+[αcos((n-1)τ)+α(n-1)cos(τ)]×[-αsin((n-1)τ)(n-1)2+α(n-1)sin(τ)]-[αsin((n-1)τ)(n-1)+α(n-1)sin(τ)]×[αcos((n-1)τ)(n-1)-α(n-1)cos(τ)],E11=[αcos((n-1)τ)+α(n-1)cos(τ)]×[αsin((n-1)τ)-α(n-1)sin(τ)].
Substituting these expressions for the pressuries and the density of energy into the system (2.18), we obtain the following solution:
(5.3)A=αcos[(n-1)τ]+α(n-1)cos[τ],B=αsin[(n-1)τ]-α(n-1)sin[τ],C=sin(τ).
It is the spiky-like solution so that such solutions we call the spike universe. Its plot is presented in Figure 23 for the initial conditions A(0)=αn=10,B(0)=0,C(0)=0. Let us calculate the volume of this universe. It is given by
(5.4)V=α2[cos[(n-1)τ]+(n-1)cos[τ]][sin[(n-1)τ]-(n-1)sin[τ]]sin(τ).
In Figure 24 the evolution of the volume for (5.4) is shown, n=10,α=1. The scalar curvature has the form
(5.5)R=(((6(43-2n+n2))sin(τ)sin((n-1)τ)-2cos(τ)(n-1)cos2((n-1)τ)+((6(43-2n+n2))sin(τ)sin((n-1)τ)-(2((n-2)cos(τ)2+sin2(τ)(n2-3n+4)))(n-1)(6(43-2n+n2)))cos((n-1)τ)+2cos(τ)(sin(τ)(n2-4n+6)sin((n-1)τ)+(cos(τ)2-5sin2(τ))(n-1))sin2((n-1)τ)+sin(τ)(n2-4n+6)sin((n-1)τ)+(cos(τ)2-5sin2(τ))(n-1))(n-1)((6(43-2n+n2))sin(τ)sin((n-1)τ))/((cos((n-1)τ)+cos(τ)(n-1))(-sin((n-1)τ)+(n-1)sin(τ))sin(τ)).
In Figure 25 we plot the evolution of the R with respect of the cosmic time τ. In this example, we have shown that the Einstein equations admit the spike-like solution. We can show that this solution describes the accelerated and decelerated expansion phases of the universe.Figure 23
The spiky universe for (5.3), n=10,α=1.Figure 24
The evolution of the volume for (5.4), n=10,  α=1.Figure 25
The evolution of theR with respect of the cosmic time τ for (5.5).


#### 5.1.2. Example 2

The system (2.19) admits the following solution:
(5.6)H1=αcos[(n-1)τ]+α(n-1)cos[τ],H2=αsin[(n-1)τ]-α(n-1)sin[τ],H3=sin(τ).
The corresponding EoS takes the form
(5.7)ρ=D12E12,p1=-D13E13,p2=-D14E14,p3=-D15E15,
where
(5.8)D12=[αcos((n-1)τ)+α(n-1)cos(τ)]×[αsin((n-1)τ)+[1-α(n-1)]sin(τ)]+[αsin((n-1)τ)-α(n-1)sin(τ)]sin(τ),E12=1,D13=α(n-1)[cos((n-1)τ)-cos(τ)]+cos(τ)+[αsin((n-1)τ)-α(n-1)sin(τ)]2+[αsin((n-1)τ)+[1-α(n-1)]sin(τ)]sin(τ),E13=1,D14=-αsin((n-1)τ)(n-1)-α(n-1)sin(τ)+cos(τ)+[αcos((n-1)τ)+α(n-1)cos(τ)]2+sin(τ)2+[αcos((n-1)τ)+α*(n-1)cos(τ)]sin(τ),E14=1,D15=α(n-1)[cos((n-1)τ)-cos(τ)-sin((n-1)τ)-sin(τ)]+[αsin((n-1)τ)-α(n-1)sin(τ)]2+[αcos((n-1)τ)+α(n-1)cos(τ)]2+[αcos((n-1)τ)+α(n-1)cos(τ)]×[αsin((n-1)τ)-α(n-1)sin(τ)],E15=1.The scalar curvature has the form(5.9)R=2α2cos((n-1)τ)2+4α((12)αsin((n-1)τ)+(12+(-(12)n+12)α)sin(τ)+(n-1)(αcos(τ)+12))cos((n-1)τ)+2α2sin((n-1)τ)2+2α((1+(2-2n)α)sin(τ)+(αcos(τ)-1)(n-1))sin((n-1)τ)+(2+2α2(n-1)2+(2-2n)α)sin(τ)2-(2(n-1))α*(1+(-1+α(n-1))cos(τ))sin(τ)+(2(2α2(n-1)2cos(τ)+1+α(-n+1)))cos(τ).
In Figure 26 we plot the evolution of the R with respect of the cosmic time τ.Figure 26
The evolution of theR with respect of the cosmic time τ for (5.9).


#### 5.1.3. Example 3

Our next solution for the system (2.19) is given by
(5.10)H1=αcos[(n-1)τ]-α(n-1)cos[τ],H2=αsin[(n-1)τ]-α(n-1)sin[τ],H3=sin(τ).
In Figure 27 we plot this spiky type solution. The corresponding EoS takes the form
(5.11)ρ=D16E16,p1=-D17E17,p2=-D18E18,p3=-D19E19,
where
(5.12)D16=[αcos((n-1)τ)-α(n-1)cos(τ)]×[αsin((n-1)τ)+[1-α(n-1)]sin(τ)]+[αsin((n-1)τ)-α(n-1)sin(τ)]sin(τ),E16=1,D17=α(n-1)[cos((n-1)τ)-cos(τ)]+cos(τ)+[αsin((n-1)τ)-α(n-1)sin(τ)]2+[αsin((n-1)τ)+[1-α(n-1)]sin(τ)]sin(τ),E17=1,D18=-αsin((n-1)τ)(n-1)+α(n-1)sin(τ)+cos(τ)+[αcos((n-1)τ)-α(n-1)cos(τ)]2+sin(τ)2+[αcos((n-1)τ)-α*(n-1)cos(τ)]sin(τ),E18=1,D19=α(n-1)[cos((n-1)τ)-cos(τ)-sin((n-1)τ)+sin(τ)]+[αsin((n-1)τ)-α(n-1)sin(τ)]2+[αcos((n-1)τ)-α(n-1)cos(τ)]2+[αcos((n-1)τ)-α(n-1)cos(τ)]×[αsin((n-1)τ)-α(n-1)sin(τ)],E19=1.
The scalar curvature has the form
(5.13)R=2α2cos((n-1)τ)2-(4(-(12)αsin((n-1)τ)+(-12+((12)n-12)α)sin(τ)+(n-1)(-12+αcos(τ))))αcos((n-1)τ)+2α2sin((n-1)τ)2-(2((-1+(-2+2n)α)sin(τ)+(αcos(τ)+1)(n-1)))αsin((n-1)τ)+(2+2α2(n-1)2+(-2n+2)α)sin(τ)2+(2(n-1))×(1+(-1+α(n-1))*cos(τ))αsin(τ)+2cos(τ)(α2(n-1)2cos(τ)+1+(1-n)α).
In Figure 28 we plot the evolution of the R with respect of the cosmic time τ.Figure 27
The evolution of the spiky type solution (5.10) with n=10,α=1.Figure 28
The evolution of theR with respect of the cosmic time τ for (5.13).


### 5.2. Möbius Strip Universe Solutions

If we consider the model with the “cosmological constant”, then the systems (2.18) and (2.19) take the form, respectively,
(5.14)A˙B˙AB+B˙C˙BC+C˙A˙CA-ρ-Λ=0,B¨B+C¨C+B˙C˙BC+p1-Λ=0,C¨C+A¨A+C˙A˙CA+p2-Λ=0,A¨A+B¨B+A˙B˙AB+p3-Λ=0,(5.15)H1H2+H2H3+H1H3-ρ-Λ=0,H˙2+H˙3+H22+H32+H2H3+p1-Λ=0,H˙3+H˙1+H32+H12+H3H1+p2-Λ=0,H˙1+H˙2+H12+H22+H1H2+p3-Λ=0.
Now we want to present some solutions of these systems. Consider the following examples.


#### 5.2.1. Example 1

One of the simplest solutions of (5.14) is given by
(5.16)A=A0+(1+12Λcosτ2)cosτ,B=B0+(1+12Λcosτ2)sinτ,C=C0+12Λsinτ2.
It is the parametric equation of the Möbius strip and, hence, such model we call the Möbius strip universe. Its plot was presented in Figure 29.   The evolution of the volume of the Möbius strip universe for (5.16) with (3.4) reads as
(5.17)V=0.5Λ(1+12Λcosτ2)2cosτsinτsinτ2.
The evolution of the volume with (3.4) and α=Λ=1 is presented in Figure 30.Figure 29
The plot of the Möbius strip universe for (5.16) with (3.4) and τ=0→2π and Λ=[-1.1].Figure 30
The evolution of the volume of the Möbius strip universe for (5.16) with (3.4) and α=Λ=1.The corresponding EoS takes the form(5.18)ρ=D20E20,p1=-D21E21,p2=-D22E22,p3=-D23E23,
where
(5.19)D20=[14Λsin(τ2)cos(τ)+(1+12Λcos(τ2))sin(τ)]×[14Λsin(τ2)sin(τ)-(1+12Λcos(τ2))cos(τ)]×[C0+12Λsin(τ2)]+Λ4[-14Λsin(τ2)sin(τ)+(1+12Λcos(τ2))cos(τ)]cos(τ2)×[A0+(1+12)Λcos(τ2)cos(τ)]+Λ4[-14Λsin(τ2)cos(τ)-(1+12Λcos(τ2))sin(τ)]cos(τ2)×[B0+(1+12)Λcos(τ2)sin(τ)]-Λ[A0+(1+12)Λcos(τ2)cos(τ)]×[B0+(1+12Λcos(τ2))sin(τ)]×[C0+12Λsin(τ2)],E20=[A0+(1+12Λcos(τ2))cos(τ)]×[B0+(1+12Λcos(τ2))sin(τ)][C0+12Λsin(τ2)],D21=[C0+12Λsin(τ2)]×[-18sin(τ)Λcos(τ2)-12Λsin(τ2)cos(τ)-(1+12Λcos(τ2))sin(τ)]-Λ8[B0+(1+12Λcos(τ2))sin(τ)]sin(τ2)+Λ4[-14Λsin(τ2)sin(τ)+(1+12Λcos(τ2))×cos(τ)]cos(τ2)-Λ[B0+(1+12Λcos(τ2))sin(τ)][C0+12Λsin(τ2)],E21=[B0+(1+12Λcos(τ2))sin(τ)][C0+12Λsin(τ2)],D22=-Λ8[A0+(1+12cos(τ2))cos(τ)]Λsin(τ2)+[C0+12Λsin(τ2)]×[-Λ8cos(τ)cos(τ2)+12Λsin(τ2)sin(τ)-(1+12Λcos(τ2))cos(τ)]+14[-14Λsin(τ2)cos(τ)-(1+12Λcos(τ2))sin(τ)]Λcos(τ2)-Λ[A0+(1+12Λcos(τ2))cos(τ)]×[C0+12Λsin(τ2)],E22=[A0+(1+12Λcos(τ2))cos(τ)][C0+12Λsin(τ2)],D23=[B0+(1+12Λcos(τ2))sin(τ)]×[-Λ8cos(τ)cos(τ2)+12Λsin(τ2)sin(τ)-(1+12Λcos(τ2))cos(τ)]+[A0+(1+12Λcos(τ2))cos(τ)]×[-Λ8sin(τ)cos(τ2)-12Λsin(τ2)cos(τ)-(1+12Λcos(τ2))sin(τ)]-[14Λsin(τ2)cos(τ)+(1+12Λcos(τ2))sin(τ)]×[-14Λsin(τ2)sin(τ)+(1+12Λcos(τ2))cos(τ)]-Λ[A0+(1+12Λcos(τ2))cos(τ)]×[B0+(1+12Λcos(τ2))sin(τ)],E23=[B0+(1+12Λcos(τ2))sin(τ)][A0+(1+12Λcos(τ2))cos(τ)].
The scalar curvature has the form
(5.20)R=((-2sin(τ)2Λ2+2cos(τ)2Λ2)cos((12)τ)3+(-8sin(τ)2Λ-17cos(τ)sin(τ)sin((12)τ)Λ2+8cos(τ)2Λ)cos((12)τ)2+((6sin(τ)2Λ2-6cos(τ)2Λ2)sin((12)τ)2-60cos(τ)sin(τ)sin((12)τ)Λ+8cos(τ)2-8sin(τ)2((6sin(τ)2Λ2-6cos(τ)2Λ2)sin((12)τ)2-60cos(τ)sin(τ)sin((12)τ)Λ((12)τ))cos((12)τ)+(sin((12)τ)2Λ2cos(τ)sin(τ)+(12sin(τ)2Λ-12cos(τ)2Λ)sin((12)τ)-52cos(τ)sin(τ)+8cos(τ)2-8sin(τ)2((6sin(τ)2Λ2-6cos(τ)2Λ2)sin((12)τ)2-60cos(τ)sin(τ)sin((12)τ)Λ((12)τ))((12)τ))sin((12)τ))/(sin(τ)cos(τ)(2+Λcos((12)τ))2sin((12)τ)).
In Figure 31 we plot the evolution of the R with respect of the cosmic time τ. In this subsubsection, we have shown that the Einstein equations have the Möbius strip universe solution. Again we can show that this solution describes the accelerated and decelerated expansion phases of the universe.Figure 31
The evolution of theR with respect of the cosmic time τ for (5.20).


#### 5.2.2. Example  2

For the system (5.15) the Möbious solution reads as
(5.21)H1=(1+12Λcos(τ2))cos(τ),H2=(1+12Λcos(τ2))sin(τ),H3=12Λsin(τ2).The corresponding EoS takes the form(5.22)ρ=D24E24,p1=-D25E25,p2=-D26E26,p3=-D27E27,
where
(5.23)D24=[1+12Λcos(τ2)]2cos(τ)sin(τ)+Λ2[1+Λ2cos(τ2)]×[sin(τ)+cos(τ)]sin(τ2)-Λ,E24=1,D25=-14Λsin(τ2)sin(τ)+[1+12Λcos(τ2)]×[cos(τ)+Λ2sin(τ)sin(τ2)]+14Λcos(τ2)+[1+12Λcos(τ2)]2sin2(τ)+Λ24sin2(τ2)-Λ,E25=1,D26=-14Λsin(τ2)cos(τ)-[1+12Λcos(τ2)]×[sin(τ)+Λ2cos(τ)sin(τ2)]+14Λcos(τ2)+[1+12Λcos(τ2)]2cos2(τ)+Λ24sin2(τ2)-Λ,E26=1,D27=[1+Λ2cos(τ2)-Λ4sin(τ2)][cos(τ)+sin(τ)]+[1+Λ2cos(τ2)]2[1+cos(τ)sin(τ)]-Λ,E27=1.
The scalar curvature has the form
(5.24)R=(12)Λ2(cos2(τ)+sin2(τ)+cos(τ)sin(τ))cos2((12)τ)+(12(𝔄+1-2sin(τ)+4sin2(τ)))×Λcos((12)τ)+(12)Λ2sin2((12)τ)+(12(cos(τ)+sin(τ)))Λsin((12)τ)+2cos2(τ)+(12(4+4sin(τ)))×cos(τ)-2sin(τ)+2sin2(τ),
where 𝔄  denotes (cos(τ)+sin(τ))Λsin((1/2)τ)+4cos2(τ)+(2+4sin(τ))cos(τ).In Figure32 we plot the evolution of the R with respect of the cosmic time τ.Figure 32
The evolution of theR with respect of the cosmic time τ for (5.24).


### 5.3. Other Examples of Möbius Strip Like Universes Induced by Jacobian Elliptic Functions

#### 5.3.1. Example 1

Now we want to present some solutions in terms of the Jacobian elliptic functions. In fact, the system (5.14) has the following particular solution:
(5.25)A=A0+(1+12Λcnτ2)cnτ,B=B0+(1+12Λcnτ2)snτ,C=C0+12Λsnτ2The corresponding EoS takes the form(5.26)ρ=D28E28,p1=-D29E29,p2=-D30E30,p3=-D31E31,
where
(5.27)D28=[14Λdnτ2snτ2cnτ+(1+12Λcnτ2)dnτsnτ]×[14Λdnτ2snτ2snτ-(1+12Λcnτ2)cnτdnτ]×[C0+12Λsnτ2]+Λ4[-14Λdnτ2snτ2snτ+(1+12Λcnτ2)cnτdnτ]×cnτ2dnτ2×[A0+(1+12Λcnτ2)cnτ]-14[14Λdnτ2snτ2cnτ+(1+12Λcnτ2)dnτsnτ]Λcnτ2dnτ2×[B0+(1+12Λcnτ2)snτ]-Λ[A0+(1+12Λcnτ2)cnτ]×[B0+(1+12Λcnτ2)snτ]×[C0+12Λsnτ2],E28=[A0+(1+12Λcnτ2)cnτ]×[B0+(1+12Λcnτ2)snτ][C0+12Λsnτ2],D29=[C0+12Λsnτ2]×[18Λcnτ2sn2τ2snτ-18Λdn2τ2cnτ2snτ-12Λdnτ2snτ2cnτdnτ-(1+12Λcnτ2)dn2τsnτ-[1+12Λcnτ2]cn2τsnτ]+[B0+(1+12Λcnτ2)snτ]×[-18Λdn2τ2snτ2-18Λcn2τ2snτ2]-Λ4[14Λdnτ2snτ2snτ-(1+12Λcnτ2)cnτdnτ]×cnτ2dnτ2-Λ[B0+(1+12Λcnτ2)snτ]×[C0+12Λsnτ2],E29=[B0+(1+12Λcnτ2)snτ][C0+12Λsnτ2],D30=-[A0+(1+12Λcnτ2)cnτ]×[18Λdn2τ2snτ2+18Λcn2τ2snτ2]+[C0+12Λsnτ2]×[18Λcnτ2sn2τ2cnτ-18Λdn2τ2cnτ2cnτ+12Λdnτ2snτ2dnτsnτ+(1+12Λcnτ2)(cnτsn2τ-dn2τcnτ)]-Λ4[14Λdnτ2snτ2cnτ+(1+12Λcnτ2)×dnτsnτ]cnτ2dnτ2-Λ[A0+(1+12Λcnτ2)cnτ][C0+12Λsnτ2],E30=[A0+(1+12Λcnτ2)cnτ][C0+12Λsnτ2],D31=[B0+(1+12Λcnτ2)snτ]×[18Λcnτ2sn2τ2cnτ-18Λdn2τ2cnτ2cnτ+12Λdnτ2snτ2dnτsnτ+(1+12Λcnτ2)(sn2τ-dn2τ)cnτ]+[A0+(1+12Λcnτ2)cnτ]×[18Λcnτ2snτ(sn2τ2-dn2τ)-12Λdnτ2snτ2cnτdnτ-(1+12Λcnτ2)dn2τsnτ-(1+12Λcnτ2)cn2τsnτ]+[14Λdnτ2snτ2cnτ+(1+12Λcnτ2)dnτsnτ]×[14Λdnτ2snτ2snτ-(1+12Λcnτ2)cnτdnτ]-Λ[A0+(1+12Λcnτ2)cnτ]×[B0+(1+12Λcnτ2)snτ],E31=[B0+(1+12Λcnτ2)snτ][A0+(1+12Λcnτ2)cnτ].
The evolution of the volume of the universe for (3.4) reads as (A0=B0=C0=0)
(5.28)V=12Λ(1+12Λcnτ2)2cnτsnτsnτ2.
The evolution of the volume with (3.4) and Λ=1 is presented in Figure 33.Figure 33
The evolution of the volume of the trefoil knot universe with respect to the cosmic timeτ for (5.28).The scalar curvature has the form(5.29)R=((2+Λcn((12)τ,k))2)2Λ(2k2cn((12)τ,k)+Λk2cn2((12)τ,k)+(12)Λdn2((12)τ,k))cn(τ,k)sn(τ,k)sn((12)τ,k)3-6Λdn(τ,k)dn((12)τ,k)(cn(τ,k)-sn(τ,k))×(cn(τ,k)+sn(τ,k))×(2+Λcn((12)τ,k))sn2((12)τ,k)-(4(2+Λcn((12)τ,k)))×((14)cn3((12)τ,k)k2Λ+(12)cn2((12)τ,k)k2+Λ(+(54)dn2((12)τ,k)-sn2(τ,k)k2)3dn2(τ,k)+cn2(τ,k)k2+(54)dn2((12)τ,k)-sn2(τ,k)k2)×cn((12)τ,k)+6dn2(τ,k)+(12)dn2((12)τ,k)-2sn2(τ,k)k2+2cn2(τ,k)k2((12)τ,k))×cn(τ,k)sn(τ,k)sn((12)τ,k)+2cn((12)τ,k)dn((12)τ,k)dn(τ,k)(cn(τ,k)-sn(τ,k))×(cn(τ,k)+sn(τ,k))×(2+Λcn((12)τ,k))2)/(((12)τ,k)cn(τ,k)sn(τ,k)*(2+Λcn2((12)τ,k))sn((12)τ,k)).
In Figure 34 we plot the evolution of the R with respect of the cosmic time τ.Figure 34
The evolution of theR with respect of the cosmic time τ for (5.29).


#### 5.3.2. Example2

Similarly, we can show that the system (5.15) has the following solution:
(5.30)H1=(1+12Λcnτ2)cnτ,H2=(1+12Λcnτ2)snτ,H3=12Λsnτ2.The corresponding EoS takes the form(5.31)ρ=D32E32,p1=-D33E33,p2=-D34E34,p3=-D35E35,
where
(5.32)D32=[1+12Λcnτ2]2cnτsnτ+Λ2×[1+12Λcnτ2][snτ+cnτ]snτ2-Λ,E32=1,D33=14Λdnτ2snτ2[1-snτ]+[1+12Λcnτ2]×[cnτdnτ+Λ2snτsnτ2]+[1+12Λcnτ2]2sn2τ+14Λ2sn2τ2-Λ,E33=1,D34=-14Λdnτ2snτ2cnτ-(1+12Λcnτ2)×[dnτsnτ+Λ2cnτsnτ2]+14Λcnτ2dnτ2+[1+12Λcnτ2]2cn2τ+14Λ2sn2τ2-Λ,E34=1,D35=-14Λdnτ2snτ2[cnτ+snτ]+[1+12Λcnτ2][cnτ-snτ]dnτ+[1+12Λcnτ2]2[sn2τ+cn2τ+cnτsnτ]-Λ,E35=1.The scalar curvature has the form(5.33)R=(12)Λ2(cn2(τ,k)+sn2(τ,k)+cn(τ,k)sn(τ,k))cn2((12)τ,k)+(12(Λ(sn(τ,k)+cn(τ,k))sn((12)τ,k)+dn((12)τ,k)+4cn2(τ,k)+(4sn(τ,k)+2dn(τ,k))cn(τ,k)+4sn2(τ,k)-2dn(τ,k)sn(τ,k)((12)τ,k))((12)τ,k))Λcn((12)τ,k)+(12)Λ2sn2((12)τ,k)-(12)Λ(dn((12)τ,k)-2)×(sn(τ,k)+cn(τ,k))sn((12)τ,k)+2cn2(τ,k)+(12(4dn(τ,k)+4sn(τ,k)))cn(τ,k)-2sn(τ,k)(-sn(τ,k)+dn(τ,k)).
In Figure 35 we plot the evolution of the R with respect of the cosmic time τ.Figure 35
The evolution of theR with respect of the cosmic time τ for (5.33).


## 5.1. Spiky Universe Solutions

Our aim in this subsection is to present some unknotted closed curve solutions namely the spiky universe solutions.


### 5.1.1. Example 1

Let our universe be filled by the fluid with the following parametric EoS:(5.1)ρ=D8E8,p1=-D9E9,p2=-D10E10,p3=-D11E11,
where
(5.2)D8=-[αsin((n-1)τ)(n-1)+α(n-1)sin(τ)]×[αcos((n-1)τ)(n-1)-α(n-1)cos(τ)]sin(τ)+[αcos((n-1)τ)(n-1)-α(n-1)cos(τ)]cos(τ)×[αcos((n-1)τ)+α(n-1)cos(τ)]-[αsin((n-1)τ)(n-1)+α(n-1)sin(τ)]×[αsin((n-1)τ)-α(n-1)sin(τ)]cos(τ),E8=[αcos((n-1)τ)+α(n-1)cos(τ)]×[αsin((n-1)τ)-α(n-1)sin(τ)]sin(τ),D9=[-αsin((n-1)τ)(n-1)2+α(n-1)sin(τ)]sin(τ)-[αsin((n-1)τ)-α(n-1)sin(τ)]sin(τ)+[αcos((n-1)τ)(n-1)-α(n-1)cos(τ)]cos(τ),E9=[αsin((n-1)τ)-α(n-1)sin(τ)]sin(τ),D10=[αcos((n-1)τ)+α(n-1)cos(τ)]sin(τ)+sin(τ)×[αcos((n-1)τ)(n-1)2+α(n-1)cos(τ)]+[αsin((n-1)τ)(n-1)+α(n-1)sin(τ)]cos(τ),E10=-[αcos((n-1)τ)+α(n-1)cos(τ)]sin(τ),D11=[αsin((n-1)τ)-α(n-1)sin(τ)]×[-αcos((n-1)τ)(n-1)2-α(n-1)cos(τ)]+[αcos((n-1)τ)+α(n-1)cos(τ)]×[-αsin((n-1)τ)(n-1)2+α(n-1)sin(τ)]-[αsin((n-1)τ)(n-1)+α(n-1)sin(τ)]×[αcos((n-1)τ)(n-1)-α(n-1)cos(τ)],E11=[αcos((n-1)τ)+α(n-1)cos(τ)]×[αsin((n-1)τ)-α(n-1)sin(τ)].
Substituting these expressions for the pressuries and the density of energy into the system (2.18), we obtain the following solution:
(5.3)A=αcos[(n-1)τ]+α(n-1)cos[τ],B=αsin[(n-1)τ]-α(n-1)sin[τ],C=sin(τ).
It is the spiky-like solution so that such solutions we call the spike universe. Its plot is presented in Figure 23 for the initial conditions A(0)=αn=10,B(0)=0,C(0)=0. Let us calculate the volume of this universe. It is given by
(5.4)V=α2[cos[(n-1)τ]+(n-1)cos[τ]][sin[(n-1)τ]-(n-1)sin[τ]]sin(τ).
In Figure 24 the evolution of the volume for (5.4) is shown, n=10,α=1. The scalar curvature has the form
(5.5)R=(((6(43-2n+n2))sin(τ)sin((n-1)τ)-2cos(τ)(n-1)cos2((n-1)τ)+((6(43-2n+n2))sin(τ)sin((n-1)τ)-(2((n-2)cos(τ)2+sin2(τ)(n2-3n+4)))(n-1)(6(43-2n+n2)))cos((n-1)τ)+2cos(τ)(sin(τ)(n2-4n+6)sin((n-1)τ)+(cos(τ)2-5sin2(τ))(n-1))sin2((n-1)τ)+sin(τ)(n2-4n+6)sin((n-1)τ)+(cos(τ)2-5sin2(τ))(n-1))(n-1)((6(43-2n+n2))sin(τ)sin((n-1)τ))/((cos((n-1)τ)+cos(τ)(n-1))(-sin((n-1)τ)+(n-1)sin(τ))sin(τ)).
In Figure 25 we plot the evolution of the R with respect of the cosmic time τ. In this example, we have shown that the Einstein equations admit the spike-like solution. We can show that this solution describes the accelerated and decelerated expansion phases of the universe.Figure 23
The spiky universe for (5.3), n=10,α=1.Figure 24
The evolution of the volume for (5.4), n=10,  α=1.Figure 25
The evolution of theR with respect of the cosmic time τ for (5.5).


### 5.1.2. Example 2

The system (2.19) admits the following solution:
(5.6)H1=αcos[(n-1)τ]+α(n-1)cos[τ],H2=αsin[(n-1)τ]-α(n-1)sin[τ],H3=sin(τ).
The corresponding EoS takes the form
(5.7)ρ=D12E12,p1=-D13E13,p2=-D14E14,p3=-D15E15,
where
(5.8)D12=[αcos((n-1)τ)+α(n-1)cos(τ)]×[αsin((n-1)τ)+[1-α(n-1)]sin(τ)]+[αsin((n-1)τ)-α(n-1)sin(τ)]sin(τ),E12=1,D13=α(n-1)[cos((n-1)τ)-cos(τ)]+cos(τ)+[αsin((n-1)τ)-α(n-1)sin(τ)]2+[αsin((n-1)τ)+[1-α(n-1)]sin(τ)]sin(τ),E13=1,D14=-αsin((n-1)τ)(n-1)-α(n-1)sin(τ)+cos(τ)+[αcos((n-1)τ)+α(n-1)cos(τ)]2+sin(τ)2+[αcos((n-1)τ)+α*(n-1)cos(τ)]sin(τ),E14=1,D15=α(n-1)[cos((n-1)τ)-cos(τ)-sin((n-1)τ)-sin(τ)]+[αsin((n-1)τ)-α(n-1)sin(τ)]2+[αcos((n-1)τ)+α(n-1)cos(τ)]2+[αcos((n-1)τ)+α(n-1)cos(τ)]×[αsin((n-1)τ)-α(n-1)sin(τ)],E15=1.The scalar curvature has the form(5.9)R=2α2cos((n-1)τ)2+4α((12)αsin((n-1)τ)+(12+(-(12)n+12)α)sin(τ)+(n-1)(αcos(τ)+12))cos((n-1)τ)+2α2sin((n-1)τ)2+2α((1+(2-2n)α)sin(τ)+(αcos(τ)-1)(n-1))sin((n-1)τ)+(2+2α2(n-1)2+(2-2n)α)sin(τ)2-(2(n-1))α*(1+(-1+α(n-1))cos(τ))sin(τ)+(2(2α2(n-1)2cos(τ)+1+α(-n+1)))cos(τ).
In Figure 26 we plot the evolution of the R with respect of the cosmic time τ.Figure 26
The evolution of theR with respect of the cosmic time τ for (5.9).


### 5.1.3. Example 3

Our next solution for the system (2.19) is given by
(5.10)H1=αcos[(n-1)τ]-α(n-1)cos[τ],H2=αsin[(n-1)τ]-α(n-1)sin[τ],H3=sin(τ).
In Figure 27 we plot this spiky type solution. The corresponding EoS takes the form
(5.11)ρ=D16E16,p1=-D17E17,p2=-D18E18,p3=-D19E19,
where
(5.12)D16=[αcos((n-1)τ)-α(n-1)cos(τ)]×[αsin((n-1)τ)+[1-α(n-1)]sin(τ)]+[αsin((n-1)τ)-α(n-1)sin(τ)]sin(τ),E16=1,D17=α(n-1)[cos((n-1)τ)-cos(τ)]+cos(τ)+[αsin((n-1)τ)-α(n-1)sin(τ)]2+[αsin((n-1)τ)+[1-α(n-1)]sin(τ)]sin(τ),E17=1,D18=-αsin((n-1)τ)(n-1)+α(n-1)sin(τ)+cos(τ)+[αcos((n-1)τ)-α(n-1)cos(τ)]2+sin(τ)2+[αcos((n-1)τ)-α*(n-1)cos(τ)]sin(τ),E18=1,D19=α(n-1)[cos((n-1)τ)-cos(τ)-sin((n-1)τ)+sin(τ)]+[αsin((n-1)τ)-α(n-1)sin(τ)]2+[αcos((n-1)τ)-α(n-1)cos(τ)]2+[αcos((n-1)τ)-α(n-1)cos(τ)]×[αsin((n-1)τ)-α(n-1)sin(τ)],E19=1.
The scalar curvature has the form
(5.13)R=2α2cos((n-1)τ)2-(4(-(12)αsin((n-1)τ)+(-12+((12)n-12)α)sin(τ)+(n-1)(-12+αcos(τ))))αcos((n-1)τ)+2α2sin((n-1)τ)2-(2((-1+(-2+2n)α)sin(τ)+(αcos(τ)+1)(n-1)))αsin((n-1)τ)+(2+2α2(n-1)2+(-2n+2)α)sin(τ)2+(2(n-1))×(1+(-1+α(n-1))*cos(τ))αsin(τ)+2cos(τ)(α2(n-1)2cos(τ)+1+(1-n)α).
In Figure 28 we plot the evolution of the R with respect of the cosmic time τ.Figure 27
The evolution of the spiky type solution (5.10) with n=10,α=1.Figure 28
The evolution of theR with respect of the cosmic time τ for (5.13).


## 5.1.1. Example 1

Let our universe be filled by the fluid with the following parametric EoS:(5.1)ρ=D8E8,p1=-D9E9,p2=-D10E10,p3=-D11E11,
where
(5.2)D8=-[αsin((n-1)τ)(n-1)+α(n-1)sin(τ)]×[αcos((n-1)τ)(n-1)-α(n-1)cos(τ)]sin(τ)+[αcos((n-1)τ)(n-1)-α(n-1)cos(τ)]cos(τ)×[αcos((n-1)τ)+α(n-1)cos(τ)]-[αsin((n-1)τ)(n-1)+α(n-1)sin(τ)]×[αsin((n-1)τ)-α(n-1)sin(τ)]cos(τ),E8=[αcos((n-1)τ)+α(n-1)cos(τ)]×[αsin((n-1)τ)-α(n-1)sin(τ)]sin(τ),D9=[-αsin((n-1)τ)(n-1)2+α(n-1)sin(τ)]sin(τ)-[αsin((n-1)τ)-α(n-1)sin(τ)]sin(τ)+[αcos((n-1)τ)(n-1)-α(n-1)cos(τ)]cos(τ),E9=[αsin((n-1)τ)-α(n-1)sin(τ)]sin(τ),D10=[αcos((n-1)τ)+α(n-1)cos(τ)]sin(τ)+sin(τ)×[αcos((n-1)τ)(n-1)2+α(n-1)cos(τ)]+[αsin((n-1)τ)(n-1)+α(n-1)sin(τ)]cos(τ),E10=-[αcos((n-1)τ)+α(n-1)cos(τ)]sin(τ),D11=[αsin((n-1)τ)-α(n-1)sin(τ)]×[-αcos((n-1)τ)(n-1)2-α(n-1)cos(τ)]+[αcos((n-1)τ)+α(n-1)cos(τ)]×[-αsin((n-1)τ)(n-1)2+α(n-1)sin(τ)]-[αsin((n-1)τ)(n-1)+α(n-1)sin(τ)]×[αcos((n-1)τ)(n-1)-α(n-1)cos(τ)],E11=[αcos((n-1)τ)+α(n-1)cos(τ)]×[αsin((n-1)τ)-α(n-1)sin(τ)].
Substituting these expressions for the pressuries and the density of energy into the system (2.18), we obtain the following solution:
(5.3)A=αcos[(n-1)τ]+α(n-1)cos[τ],B=αsin[(n-1)τ]-α(n-1)sin[τ],C=sin(τ).
It is the spiky-like solution so that such solutions we call the spike universe. Its plot is presented in Figure 23 for the initial conditions A(0)=αn=10,B(0)=0,C(0)=0. Let us calculate the volume of this universe. It is given by
(5.4)V=α2[cos[(n-1)τ]+(n-1)cos[τ]][sin[(n-1)τ]-(n-1)sin[τ]]sin(τ).
In Figure 24 the evolution of the volume for (5.4) is shown, n=10,α=1. The scalar curvature has the form
(5.5)R=(((6(43-2n+n2))sin(τ)sin((n-1)τ)-2cos(τ)(n-1)cos2((n-1)τ)+((6(43-2n+n2))sin(τ)sin((n-1)τ)-(2((n-2)cos(τ)2+sin2(τ)(n2-3n+4)))(n-1)(6(43-2n+n2)))cos((n-1)τ)+2cos(τ)(sin(τ)(n2-4n+6)sin((n-1)τ)+(cos(τ)2-5sin2(τ))(n-1))sin2((n-1)τ)+sin(τ)(n2-4n+6)sin((n-1)τ)+(cos(τ)2-5sin2(τ))(n-1))(n-1)((6(43-2n+n2))sin(τ)sin((n-1)τ))/((cos((n-1)τ)+cos(τ)(n-1))(-sin((n-1)τ)+(n-1)sin(τ))sin(τ)).
In Figure 25 we plot the evolution of the R with respect of the cosmic time τ. In this example, we have shown that the Einstein equations admit the spike-like solution. We can show that this solution describes the accelerated and decelerated expansion phases of the universe.Figure 23
The spiky universe for (5.3), n=10,α=1.Figure 24
The evolution of the volume for (5.4), n=10,  α=1.Figure 25
The evolution of theR with respect of the cosmic time τ for (5.5).


## 5.1.2. Example 2

The system (2.19) admits the following solution:
(5.6)H1=αcos[(n-1)τ]+α(n-1)cos[τ],H2=αsin[(n-1)τ]-α(n-1)sin[τ],H3=sin(τ).
The corresponding EoS takes the form
(5.7)ρ=D12E12,p1=-D13E13,p2=-D14E14,p3=-D15E15,
where
(5.8)D12=[αcos((n-1)τ)+α(n-1)cos(τ)]×[αsin((n-1)τ)+[1-α(n-1)]sin(τ)]+[αsin((n-1)τ)-α(n-1)sin(τ)]sin(τ),E12=1,D13=α(n-1)[cos((n-1)τ)-cos(τ)]+cos(τ)+[αsin((n-1)τ)-α(n-1)sin(τ)]2+[αsin((n-1)τ)+[1-α(n-1)]sin(τ)]sin(τ),E13=1,D14=-αsin((n-1)τ)(n-1)-α(n-1)sin(τ)+cos(τ)+[αcos((n-1)τ)+α(n-1)cos(τ)]2+sin(τ)2+[αcos((n-1)τ)+α*(n-1)cos(τ)]sin(τ),E14=1,D15=α(n-1)[cos((n-1)τ)-cos(τ)-sin((n-1)τ)-sin(τ)]+[αsin((n-1)τ)-α(n-1)sin(τ)]2+[αcos((n-1)τ)+α(n-1)cos(τ)]2+[αcos((n-1)τ)+α(n-1)cos(τ)]×[αsin((n-1)τ)-α(n-1)sin(τ)],E15=1.The scalar curvature has the form(5.9)R=2α2cos((n-1)τ)2+4α((12)αsin((n-1)τ)+(12+(-(12)n+12)α)sin(τ)+(n-1)(αcos(τ)+12))cos((n-1)τ)+2α2sin((n-1)τ)2+2α((1+(2-2n)α)sin(τ)+(αcos(τ)-1)(n-1))sin((n-1)τ)+(2+2α2(n-1)2+(2-2n)α)sin(τ)2-(2(n-1))α*(1+(-1+α(n-1))cos(τ))sin(τ)+(2(2α2(n-1)2cos(τ)+1+α(-n+1)))cos(τ).
In Figure 26 we plot the evolution of the R with respect of the cosmic time τ.Figure 26
The evolution of theR with respect of the cosmic time τ for (5.9).


## 5.1.3. Example 3

Our next solution for the system (2.19) is given by
(5.10)H1=αcos[(n-1)τ]-α(n-1)cos[τ],H2=αsin[(n-1)τ]-α(n-1)sin[τ],H3=sin(τ).
In Figure 27 we plot this spiky type solution. The corresponding EoS takes the form
(5.11)ρ=D16E16,p1=-D17E17,p2=-D18E18,p3=-D19E19,
where
(5.12)D16=[αcos((n-1)τ)-α(n-1)cos(τ)]×[αsin((n-1)τ)+[1-α(n-1)]sin(τ)]+[αsin((n-1)τ)-α(n-1)sin(τ)]sin(τ),E16=1,D17=α(n-1)[cos((n-1)τ)-cos(τ)]+cos(τ)+[αsin((n-1)τ)-α(n-1)sin(τ)]2+[αsin((n-1)τ)+[1-α(n-1)]sin(τ)]sin(τ),E17=1,D18=-αsin((n-1)τ)(n-1)+α(n-1)sin(τ)+cos(τ)+[αcos((n-1)τ)-α(n-1)cos(τ)]2+sin(τ)2+[αcos((n-1)τ)-α*(n-1)cos(τ)]sin(τ),E18=1,D19=α(n-1)[cos((n-1)τ)-cos(τ)-sin((n-1)τ)+sin(τ)]+[αsin((n-1)τ)-α(n-1)sin(τ)]2+[αcos((n-1)τ)-α(n-1)cos(τ)]2+[αcos((n-1)τ)-α(n-1)cos(τ)]×[αsin((n-1)τ)-α(n-1)sin(τ)],E19=1.
The scalar curvature has the form
(5.13)R=2α2cos((n-1)τ)2-(4(-(12)αsin((n-1)τ)+(-12+((12)n-12)α)sin(τ)+(n-1)(-12+αcos(τ))))αcos((n-1)τ)+2α2sin((n-1)τ)2-(2((-1+(-2+2n)α)sin(τ)+(αcos(τ)+1)(n-1)))αsin((n-1)τ)+(2+2α2(n-1)2+(-2n+2)α)sin(τ)2+(2(n-1))×(1+(-1+α(n-1))*cos(τ))αsin(τ)+2cos(τ)(α2(n-1)2cos(τ)+1+(1-n)α).
In Figure 28 we plot the evolution of the R with respect of the cosmic time τ.Figure 27
The evolution of the spiky type solution (5.10) with n=10,α=1.Figure 28
The evolution of theR with respect of the cosmic time τ for (5.13).


## 5.2. Möbius Strip Universe Solutions

If we consider the model with the “cosmological constant”, then the systems (2.18) and (2.19) take the form, respectively,
(5.14)A˙B˙AB+B˙C˙BC+C˙A˙CA-ρ-Λ=0,B¨B+C¨C+B˙C˙BC+p1-Λ=0,C¨C+A¨A+C˙A˙CA+p2-Λ=0,A¨A+B¨B+A˙B˙AB+p3-Λ=0,(5.15)H1H2+H2H3+H1H3-ρ-Λ=0,H˙2+H˙3+H22+H32+H2H3+p1-Λ=0,H˙3+H˙1+H32+H12+H3H1+p2-Λ=0,H˙1+H˙2+H12+H22+H1H2+p3-Λ=0.
Now we want to present some solutions of these systems. Consider the following examples.


### 5.2.1. Example 1

One of the simplest solutions of (5.14) is given by
(5.16)A=A0+(1+12Λcosτ2)cosτ,B=B0+(1+12Λcosτ2)sinτ,C=C0+12Λsinτ2.
It is the parametric equation of the Möbius strip and, hence, such model we call the Möbius strip universe. Its plot was presented in Figure 29.   The evolution of the volume of the Möbius strip universe for (5.16) with (3.4) reads as
(5.17)V=0.5Λ(1+12Λcosτ2)2cosτsinτsinτ2.
The evolution of the volume with (3.4) and α=Λ=1 is presented in Figure 30.Figure 29
The plot of the Möbius strip universe for (5.16) with (3.4) and τ=0→2π and Λ=[-1.1].Figure 30
The evolution of the volume of the Möbius strip universe for (5.16) with (3.4) and α=Λ=1.The corresponding EoS takes the form(5.18)ρ=D20E20,p1=-D21E21,p2=-D22E22,p3=-D23E23,
where
(5.19)D20=[14Λsin(τ2)cos(τ)+(1+12Λcos(τ2))sin(τ)]×[14Λsin(τ2)sin(τ)-(1+12Λcos(τ2))cos(τ)]×[C0+12Λsin(τ2)]+Λ4[-14Λsin(τ2)sin(τ)+(1+12Λcos(τ2))cos(τ)]cos(τ2)×[A0+(1+12)Λcos(τ2)cos(τ)]+Λ4[-14Λsin(τ2)cos(τ)-(1+12Λcos(τ2))sin(τ)]cos(τ2)×[B0+(1+12)Λcos(τ2)sin(τ)]-Λ[A0+(1+12)Λcos(τ2)cos(τ)]×[B0+(1+12Λcos(τ2))sin(τ)]×[C0+12Λsin(τ2)],E20=[A0+(1+12Λcos(τ2))cos(τ)]×[B0+(1+12Λcos(τ2))sin(τ)][C0+12Λsin(τ2)],D21=[C0+12Λsin(τ2)]×[-18sin(τ)Λcos(τ2)-12Λsin(τ2)cos(τ)-(1+12Λcos(τ2))sin(τ)]-Λ8[B0+(1+12Λcos(τ2))sin(τ)]sin(τ2)+Λ4[-14Λsin(τ2)sin(τ)+(1+12Λcos(τ2))×cos(τ)]cos(τ2)-Λ[B0+(1+12Λcos(τ2))sin(τ)][C0+12Λsin(τ2)],E21=[B0+(1+12Λcos(τ2))sin(τ)][C0+12Λsin(τ2)],D22=-Λ8[A0+(1+12cos(τ2))cos(τ)]Λsin(τ2)+[C0+12Λsin(τ2)]×[-Λ8cos(τ)cos(τ2)+12Λsin(τ2)sin(τ)-(1+12Λcos(τ2))cos(τ)]+14[-14Λsin(τ2)cos(τ)-(1+12Λcos(τ2))sin(τ)]Λcos(τ2)-Λ[A0+(1+12Λcos(τ2))cos(τ)]×[C0+12Λsin(τ2)],E22=[A0+(1+12Λcos(τ2))cos(τ)][C0+12Λsin(τ2)],D23=[B0+(1+12Λcos(τ2))sin(τ)]×[-Λ8cos(τ)cos(τ2)+12Λsin(τ2)sin(τ)-(1+12Λcos(τ2))cos(τ)]+[A0+(1+12Λcos(τ2))cos(τ)]×[-Λ8sin(τ)cos(τ2)-12Λsin(τ2)cos(τ)-(1+12Λcos(τ2))sin(τ)]-[14Λsin(τ2)cos(τ)+(1+12Λcos(τ2))sin(τ)]×[-14Λsin(τ2)sin(τ)+(1+12Λcos(τ2))cos(τ)]-Λ[A0+(1+12Λcos(τ2))cos(τ)]×[B0+(1+12Λcos(τ2))sin(τ)],E23=[B0+(1+12Λcos(τ2))sin(τ)][A0+(1+12Λcos(τ2))cos(τ)].
The scalar curvature has the form
(5.20)R=((-2sin(τ)2Λ2+2cos(τ)2Λ2)cos((12)τ)3+(-8sin(τ)2Λ-17cos(τ)sin(τ)sin((12)τ)Λ2+8cos(τ)2Λ)cos((12)τ)2+((6sin(τ)2Λ2-6cos(τ)2Λ2)sin((12)τ)2-60cos(τ)sin(τ)sin((12)τ)Λ+8cos(τ)2-8sin(τ)2((6sin(τ)2Λ2-6cos(τ)2Λ2)sin((12)τ)2-60cos(τ)sin(τ)sin((12)τ)Λ((12)τ))cos((12)τ)+(sin((12)τ)2Λ2cos(τ)sin(τ)+(12sin(τ)2Λ-12cos(τ)2Λ)sin((12)τ)-52cos(τ)sin(τ)+8cos(τ)2-8sin(τ)2((6sin(τ)2Λ2-6cos(τ)2Λ2)sin((12)τ)2-60cos(τ)sin(τ)sin((12)τ)Λ((12)τ))((12)τ))sin((12)τ))/(sin(τ)cos(τ)(2+Λcos((12)τ))2sin((12)τ)).
In Figure 31 we plot the evolution of the R with respect of the cosmic time τ. In this subsubsection, we have shown that the Einstein equations have the Möbius strip universe solution. Again we can show that this solution describes the accelerated and decelerated expansion phases of the universe.Figure 31
The evolution of theR with respect of the cosmic time τ for (5.20).


### 5.2.2. Example  2

For the system (5.15) the Möbious solution reads as
(5.21)H1=(1+12Λcos(τ2))cos(τ),H2=(1+12Λcos(τ2))sin(τ),H3=12Λsin(τ2).The corresponding EoS takes the form(5.22)ρ=D24E24,p1=-D25E25,p2=-D26E26,p3=-D27E27,
where
(5.23)D24=[1+12Λcos(τ2)]2cos(τ)sin(τ)+Λ2[1+Λ2cos(τ2)]×[sin(τ)+cos(τ)]sin(τ2)-Λ,E24=1,D25=-14Λsin(τ2)sin(τ)+[1+12Λcos(τ2)]×[cos(τ)+Λ2sin(τ)sin(τ2)]+14Λcos(τ2)+[1+12Λcos(τ2)]2sin2(τ)+Λ24sin2(τ2)-Λ,E25=1,D26=-14Λsin(τ2)cos(τ)-[1+12Λcos(τ2)]×[sin(τ)+Λ2cos(τ)sin(τ2)]+14Λcos(τ2)+[1+12Λcos(τ2)]2cos2(τ)+Λ24sin2(τ2)-Λ,E26=1,D27=[1+Λ2cos(τ2)-Λ4sin(τ2)][cos(τ)+sin(τ)]+[1+Λ2cos(τ2)]2[1+cos(τ)sin(τ)]-Λ,E27=1.
The scalar curvature has the form
(5.24)R=(12)Λ2(cos2(τ)+sin2(τ)+cos(τ)sin(τ))cos2((12)τ)+(12(𝔄+1-2sin(τ)+4sin2(τ)))×Λcos((12)τ)+(12)Λ2sin2((12)τ)+(12(cos(τ)+sin(τ)))Λsin((12)τ)+2cos2(τ)+(12(4+4sin(τ)))×cos(τ)-2sin(τ)+2sin2(τ),
where 𝔄  denotes (cos(τ)+sin(τ))Λsin((1/2)τ)+4cos2(τ)+(2+4sin(τ))cos(τ).In Figure32 we plot the evolution of the R with respect of the cosmic time τ.Figure 32
The evolution of theR with respect of the cosmic time τ for (5.24).


## 5.2.1. Example 1

One of the simplest solutions of (5.14) is given by
(5.16)A=A0+(1+12Λcosτ2)cosτ,B=B0+(1+12Λcosτ2)sinτ,C=C0+12Λsinτ2.
It is the parametric equation of the Möbius strip and, hence, such model we call the Möbius strip universe. Its plot was presented in Figure 29.   The evolution of the volume of the Möbius strip universe for (5.16) with (3.4) reads as
(5.17)V=0.5Λ(1+12Λcosτ2)2cosτsinτsinτ2.
The evolution of the volume with (3.4) and α=Λ=1 is presented in Figure 30.Figure 29
The plot of the Möbius strip universe for (5.16) with (3.4) and τ=0→2π and Λ=[-1.1].Figure 30
The evolution of the volume of the Möbius strip universe for (5.16) with (3.4) and α=Λ=1.The corresponding EoS takes the form(5.18)ρ=D20E20,p1=-D21E21,p2=-D22E22,p3=-D23E23,
where
(5.19)D20=[14Λsin(τ2)cos(τ)+(1+12Λcos(τ2))sin(τ)]×[14Λsin(τ2)sin(τ)-(1+12Λcos(τ2))cos(τ)]×[C0+12Λsin(τ2)]+Λ4[-14Λsin(τ2)sin(τ)+(1+12Λcos(τ2))cos(τ)]cos(τ2)×[A0+(1+12)Λcos(τ2)cos(τ)]+Λ4[-14Λsin(τ2)cos(τ)-(1+12Λcos(τ2))sin(τ)]cos(τ2)×[B0+(1+12)Λcos(τ2)sin(τ)]-Λ[A0+(1+12)Λcos(τ2)cos(τ)]×[B0+(1+12Λcos(τ2))sin(τ)]×[C0+12Λsin(τ2)],E20=[A0+(1+12Λcos(τ2))cos(τ)]×[B0+(1+12Λcos(τ2))sin(τ)][C0+12Λsin(τ2)],D21=[C0+12Λsin(τ2)]×[-18sin(τ)Λcos(τ2)-12Λsin(τ2)cos(τ)-(1+12Λcos(τ2))sin(τ)]-Λ8[B0+(1+12Λcos(τ2))sin(τ)]sin(τ2)+Λ4[-14Λsin(τ2)sin(τ)+(1+12Λcos(τ2))×cos(τ)]cos(τ2)-Λ[B0+(1+12Λcos(τ2))sin(τ)][C0+12Λsin(τ2)],E21=[B0+(1+12Λcos(τ2))sin(τ)][C0+12Λsin(τ2)],D22=-Λ8[A0+(1+12cos(τ2))cos(τ)]Λsin(τ2)+[C0+12Λsin(τ2)]×[-Λ8cos(τ)cos(τ2)+12Λsin(τ2)sin(τ)-(1+12Λcos(τ2))cos(τ)]+14[-14Λsin(τ2)cos(τ)-(1+12Λcos(τ2))sin(τ)]Λcos(τ2)-Λ[A0+(1+12Λcos(τ2))cos(τ)]×[C0+12Λsin(τ2)],E22=[A0+(1+12Λcos(τ2))cos(τ)][C0+12Λsin(τ2)],D23=[B0+(1+12Λcos(τ2))sin(τ)]×[-Λ8cos(τ)cos(τ2)+12Λsin(τ2)sin(τ)-(1+12Λcos(τ2))cos(τ)]+[A0+(1+12Λcos(τ2))cos(τ)]×[-Λ8sin(τ)cos(τ2)-12Λsin(τ2)cos(τ)-(1+12Λcos(τ2))sin(τ)]-[14Λsin(τ2)cos(τ)+(1+12Λcos(τ2))sin(τ)]×[-14Λsin(τ2)sin(τ)+(1+12Λcos(τ2))cos(τ)]-Λ[A0+(1+12Λcos(τ2))cos(τ)]×[B0+(1+12Λcos(τ2))sin(τ)],E23=[B0+(1+12Λcos(τ2))sin(τ)][A0+(1+12Λcos(τ2))cos(τ)].
The scalar curvature has the form
(5.20)R=((-2sin(τ)2Λ2+2cos(τ)2Λ2)cos((12)τ)3+(-8sin(τ)2Λ-17cos(τ)sin(τ)sin((12)τ)Λ2+8cos(τ)2Λ)cos((12)τ)2+((6sin(τ)2Λ2-6cos(τ)2Λ2)sin((12)τ)2-60cos(τ)sin(τ)sin((12)τ)Λ+8cos(τ)2-8sin(τ)2((6sin(τ)2Λ2-6cos(τ)2Λ2)sin((12)τ)2-60cos(τ)sin(τ)sin((12)τ)Λ((12)τ))cos((12)τ)+(sin((12)τ)2Λ2cos(τ)sin(τ)+(12sin(τ)2Λ-12cos(τ)2Λ)sin((12)τ)-52cos(τ)sin(τ)+8cos(τ)2-8sin(τ)2((6sin(τ)2Λ2-6cos(τ)2Λ2)sin((12)τ)2-60cos(τ)sin(τ)sin((12)τ)Λ((12)τ))((12)τ))sin((12)τ))/(sin(τ)cos(τ)(2+Λcos((12)τ))2sin((12)τ)).
In Figure 31 we plot the evolution of the R with respect of the cosmic time τ. In this subsubsection, we have shown that the Einstein equations have the Möbius strip universe solution. Again we can show that this solution describes the accelerated and decelerated expansion phases of the universe.Figure 31
The evolution of theR with respect of the cosmic time τ for (5.20).


## 5.2.2. Example  2

For the system (5.15) the Möbious solution reads as
(5.21)H1=(1+12Λcos(τ2))cos(τ),H2=(1+12Λcos(τ2))sin(τ),H3=12Λsin(τ2).The corresponding EoS takes the form(5.22)ρ=D24E24,p1=-D25E25,p2=-D26E26,p3=-D27E27,
where
(5.23)D24=[1+12Λcos(τ2)]2cos(τ)sin(τ)+Λ2[1+Λ2cos(τ2)]×[sin(τ)+cos(τ)]sin(τ2)-Λ,E24=1,D25=-14Λsin(τ2)sin(τ)+[1+12Λcos(τ2)]×[cos(τ)+Λ2sin(τ)sin(τ2)]+14Λcos(τ2)+[1+12Λcos(τ2)]2sin2(τ)+Λ24sin2(τ2)-Λ,E25=1,D26=-14Λsin(τ2)cos(τ)-[1+12Λcos(τ2)]×[sin(τ)+Λ2cos(τ)sin(τ2)]+14Λcos(τ2)+[1+12Λcos(τ2)]2cos2(τ)+Λ24sin2(τ2)-Λ,E26=1,D27=[1+Λ2cos(τ2)-Λ4sin(τ2)][cos(τ)+sin(τ)]+[1+Λ2cos(τ2)]2[1+cos(τ)sin(τ)]-Λ,E27=1.
The scalar curvature has the form
(5.24)R=(12)Λ2(cos2(τ)+sin2(τ)+cos(τ)sin(τ))cos2((12)τ)+(12(𝔄+1-2sin(τ)+4sin2(τ)))×Λcos((12)τ)+(12)Λ2sin2((12)τ)+(12(cos(τ)+sin(τ)))Λsin((12)τ)+2cos2(τ)+(12(4+4sin(τ)))×cos(τ)-2sin(τ)+2sin2(τ),
where 𝔄  denotes (cos(τ)+sin(τ))Λsin((1/2)τ)+4cos2(τ)+(2+4sin(τ))cos(τ).In Figure32 we plot the evolution of the R with respect of the cosmic time τ.Figure 32
The evolution of theR with respect of the cosmic time τ for (5.24).


## 5.3. Other Examples of Möbius Strip Like Universes Induced by Jacobian Elliptic Functions

### 5.3.1. Example 1

Now we want to present some solutions in terms of the Jacobian elliptic functions. In fact, the system (5.14) has the following particular solution:
(5.25)A=A0+(1+12Λcnτ2)cnτ,B=B0+(1+12Λcnτ2)snτ,C=C0+12Λsnτ2The corresponding EoS takes the form(5.26)ρ=D28E28,p1=-D29E29,p2=-D30E30,p3=-D31E31,
where
(5.27)D28=[14Λdnτ2snτ2cnτ+(1+12Λcnτ2)dnτsnτ]×[14Λdnτ2snτ2snτ-(1+12Λcnτ2)cnτdnτ]×[C0+12Λsnτ2]+Λ4[-14Λdnτ2snτ2snτ+(1+12Λcnτ2)cnτdnτ]×cnτ2dnτ2×[A0+(1+12Λcnτ2)cnτ]-14[14Λdnτ2snτ2cnτ+(1+12Λcnτ2)dnτsnτ]Λcnτ2dnτ2×[B0+(1+12Λcnτ2)snτ]-Λ[A0+(1+12Λcnτ2)cnτ]×[B0+(1+12Λcnτ2)snτ]×[C0+12Λsnτ2],E28=[A0+(1+12Λcnτ2)cnτ]×[B0+(1+12Λcnτ2)snτ][C0+12Λsnτ2],D29=[C0+12Λsnτ2]×[18Λcnτ2sn2τ2snτ-18Λdn2τ2cnτ2snτ-12Λdnτ2snτ2cnτdnτ-(1+12Λcnτ2)dn2τsnτ-[1+12Λcnτ2]cn2τsnτ]+[B0+(1+12Λcnτ2)snτ]×[-18Λdn2τ2snτ2-18Λcn2τ2snτ2]-Λ4[14Λdnτ2snτ2snτ-(1+12Λcnτ2)cnτdnτ]×cnτ2dnτ2-Λ[B0+(1+12Λcnτ2)snτ]×[C0+12Λsnτ2],E29=[B0+(1+12Λcnτ2)snτ][C0+12Λsnτ2],D30=-[A0+(1+12Λcnτ2)cnτ]×[18Λdn2τ2snτ2+18Λcn2τ2snτ2]+[C0+12Λsnτ2]×[18Λcnτ2sn2τ2cnτ-18Λdn2τ2cnτ2cnτ+12Λdnτ2snτ2dnτsnτ+(1+12Λcnτ2)(cnτsn2τ-dn2τcnτ)]-Λ4[14Λdnτ2snτ2cnτ+(1+12Λcnτ2)×dnτsnτ]cnτ2dnτ2-Λ[A0+(1+12Λcnτ2)cnτ][C0+12Λsnτ2],E30=[A0+(1+12Λcnτ2)cnτ][C0+12Λsnτ2],D31=[B0+(1+12Λcnτ2)snτ]×[18Λcnτ2sn2τ2cnτ-18Λdn2τ2cnτ2cnτ+12Λdnτ2snτ2dnτsnτ+(1+12Λcnτ2)(sn2τ-dn2τ)cnτ]+[A0+(1+12Λcnτ2)cnτ]×[18Λcnτ2snτ(sn2τ2-dn2τ)-12Λdnτ2snτ2cnτdnτ-(1+12Λcnτ2)dn2τsnτ-(1+12Λcnτ2)cn2τsnτ]+[14Λdnτ2snτ2cnτ+(1+12Λcnτ2)dnτsnτ]×[14Λdnτ2snτ2snτ-(1+12Λcnτ2)cnτdnτ]-Λ[A0+(1+12Λcnτ2)cnτ]×[B0+(1+12Λcnτ2)snτ],E31=[B0+(1+12Λcnτ2)snτ][A0+(1+12Λcnτ2)cnτ].
The evolution of the volume of the universe for (3.4) reads as (A0=B0=C0=0)
(5.28)V=12Λ(1+12Λcnτ2)2cnτsnτsnτ2.
The evolution of the volume with (3.4) and Λ=1 is presented in Figure 33.Figure 33
The evolution of the volume of the trefoil knot universe with respect to the cosmic timeτ for (5.28).The scalar curvature has the form(5.29)R=((2+Λcn((12)τ,k))2)2Λ(2k2cn((12)τ,k)+Λk2cn2((12)τ,k)+(12)Λdn2((12)τ,k))cn(τ,k)sn(τ,k)sn((12)τ,k)3-6Λdn(τ,k)dn((12)τ,k)(cn(τ,k)-sn(τ,k))×(cn(τ,k)+sn(τ,k))×(2+Λcn((12)τ,k))sn2((12)τ,k)-(4(2+Λcn((12)τ,k)))×((14)cn3((12)τ,k)k2Λ+(12)cn2((12)τ,k)k2+Λ(+(54)dn2((12)τ,k)-sn2(τ,k)k2)3dn2(τ,k)+cn2(τ,k)k2+(54)dn2((12)τ,k)-sn2(τ,k)k2)×cn((12)τ,k)+6dn2(τ,k)+(12)dn2((12)τ,k)-2sn2(τ,k)k2+2cn2(τ,k)k2((12)τ,k))×cn(τ,k)sn(τ,k)sn((12)τ,k)+2cn((12)τ,k)dn((12)τ,k)dn(τ,k)(cn(τ,k)-sn(τ,k))×(cn(τ,k)+sn(τ,k))×(2+Λcn((12)τ,k))2)/(((12)τ,k)cn(τ,k)sn(τ,k)*(2+Λcn2((12)τ,k))sn((12)τ,k)).
In Figure 34 we plot the evolution of the R with respect of the cosmic time τ.Figure 34
The evolution of theR with respect of the cosmic time τ for (5.29).


### 5.3.2. Example2

Similarly, we can show that the system (5.15) has the following solution:
(5.30)H1=(1+12Λcnτ2)cnτ,H2=(1+12Λcnτ2)snτ,H3=12Λsnτ2.The corresponding EoS takes the form(5.31)ρ=D32E32,p1=-D33E33,p2=-D34E34,p3=-D35E35,
where
(5.32)D32=[1+12Λcnτ2]2cnτsnτ+Λ2×[1+12Λcnτ2][snτ+cnτ]snτ2-Λ,E32=1,D33=14Λdnτ2snτ2[1-snτ]+[1+12Λcnτ2]×[cnτdnτ+Λ2snτsnτ2]+[1+12Λcnτ2]2sn2τ+14Λ2sn2τ2-Λ,E33=1,D34=-14Λdnτ2snτ2cnτ-(1+12Λcnτ2)×[dnτsnτ+Λ2cnτsnτ2]+14Λcnτ2dnτ2+[1+12Λcnτ2]2cn2τ+14Λ2sn2τ2-Λ,E34=1,D35=-14Λdnτ2snτ2[cnτ+snτ]+[1+12Λcnτ2][cnτ-snτ]dnτ+[1+12Λcnτ2]2[sn2τ+cn2τ+cnτsnτ]-Λ,E35=1.The scalar curvature has the form(5.33)R=(12)Λ2(cn2(τ,k)+sn2(τ,k)+cn(τ,k)sn(τ,k))cn2((12)τ,k)+(12(Λ(sn(τ,k)+cn(τ,k))sn((12)τ,k)+dn((12)τ,k)+4cn2(τ,k)+(4sn(τ,k)+2dn(τ,k))cn(τ,k)+4sn2(τ,k)-2dn(τ,k)sn(τ,k)((12)τ,k))((12)τ,k))Λcn((12)τ,k)+(12)Λ2sn2((12)τ,k)-(12)Λ(dn((12)τ,k)-2)×(sn(τ,k)+cn(τ,k))sn((12)τ,k)+2cn2(τ,k)+(12(4dn(τ,k)+4sn(τ,k)))cn(τ,k)-2sn(τ,k)(-sn(τ,k)+dn(τ,k)).
In Figure 35 we plot the evolution of the R with respect of the cosmic time τ.Figure 35
The evolution of theR with respect of the cosmic time τ for (5.33).


## 5.3.1. Example 1

Now we want to present some solutions in terms of the Jacobian elliptic functions. In fact, the system (5.14) has the following particular solution:
(5.25)A=A0+(1+12Λcnτ2)cnτ,B=B0+(1+12Λcnτ2)snτ,C=C0+12Λsnτ2The corresponding EoS takes the form(5.26)ρ=D28E28,p1=-D29E29,p2=-D30E30,p3=-D31E31,
where
(5.27)D28=[14Λdnτ2snτ2cnτ+(1+12Λcnτ2)dnτsnτ]×[14Λdnτ2snτ2snτ-(1+12Λcnτ2)cnτdnτ]×[C0+12Λsnτ2]+Λ4[-14Λdnτ2snτ2snτ+(1+12Λcnτ2)cnτdnτ]×cnτ2dnτ2×[A0+(1+12Λcnτ2)cnτ]-14[14Λdnτ2snτ2cnτ+(1+12Λcnτ2)dnτsnτ]Λcnτ2dnτ2×[B0+(1+12Λcnτ2)snτ]-Λ[A0+(1+12Λcnτ2)cnτ]×[B0+(1+12Λcnτ2)snτ]×[C0+12Λsnτ2],E28=[A0+(1+12Λcnτ2)cnτ]×[B0+(1+12Λcnτ2)snτ][C0+12Λsnτ2],D29=[C0+12Λsnτ2]×[18Λcnτ2sn2τ2snτ-18Λdn2τ2cnτ2snτ-12Λdnτ2snτ2cnτdnτ-(1+12Λcnτ2)dn2τsnτ-[1+12Λcnτ2]cn2τsnτ]+[B0+(1+12Λcnτ2)snτ]×[-18Λdn2τ2snτ2-18Λcn2τ2snτ2]-Λ4[14Λdnτ2snτ2snτ-(1+12Λcnτ2)cnτdnτ]×cnτ2dnτ2-Λ[B0+(1+12Λcnτ2)snτ]×[C0+12Λsnτ2],E29=[B0+(1+12Λcnτ2)snτ][C0+12Λsnτ2],D30=-[A0+(1+12Λcnτ2)cnτ]×[18Λdn2τ2snτ2+18Λcn2τ2snτ2]+[C0+12Λsnτ2]×[18Λcnτ2sn2τ2cnτ-18Λdn2τ2cnτ2cnτ+12Λdnτ2snτ2dnτsnτ+(1+12Λcnτ2)(cnτsn2τ-dn2τcnτ)]-Λ4[14Λdnτ2snτ2cnτ+(1+12Λcnτ2)×dnτsnτ]cnτ2dnτ2-Λ[A0+(1+12Λcnτ2)cnτ][C0+12Λsnτ2],E30=[A0+(1+12Λcnτ2)cnτ][C0+12Λsnτ2],D31=[B0+(1+12Λcnτ2)snτ]×[18Λcnτ2sn2τ2cnτ-18Λdn2τ2cnτ2cnτ+12Λdnτ2snτ2dnτsnτ+(1+12Λcnτ2)(sn2τ-dn2τ)cnτ]+[A0+(1+12Λcnτ2)cnτ]×[18Λcnτ2snτ(sn2τ2-dn2τ)-12Λdnτ2snτ2cnτdnτ-(1+12Λcnτ2)dn2τsnτ-(1+12Λcnτ2)cn2τsnτ]+[14Λdnτ2snτ2cnτ+(1+12Λcnτ2)dnτsnτ]×[14Λdnτ2snτ2snτ-(1+12Λcnτ2)cnτdnτ]-Λ[A0+(1+12Λcnτ2)cnτ]×[B0+(1+12Λcnτ2)snτ],E31=[B0+(1+12Λcnτ2)snτ][A0+(1+12Λcnτ2)cnτ].
The evolution of the volume of the universe for (3.4) reads as (A0=B0=C0=0)
(5.28)V=12Λ(1+12Λcnτ2)2cnτsnτsnτ2.
The evolution of the volume with (3.4) and Λ=1 is presented in Figure 33.Figure 33
The evolution of the volume of the trefoil knot universe with respect to the cosmic timeτ for (5.28).The scalar curvature has the form(5.29)R=((2+Λcn((12)τ,k))2)2Λ(2k2cn((12)τ,k)+Λk2cn2((12)τ,k)+(12)Λdn2((12)τ,k))cn(τ,k)sn(τ,k)sn((12)τ,k)3-6Λdn(τ,k)dn((12)τ,k)(cn(τ,k)-sn(τ,k))×(cn(τ,k)+sn(τ,k))×(2+Λcn((12)τ,k))sn2((12)τ,k)-(4(2+Λcn((12)τ,k)))×((14)cn3((12)τ,k)k2Λ+(12)cn2((12)τ,k)k2+Λ(+(54)dn2((12)τ,k)-sn2(τ,k)k2)3dn2(τ,k)+cn2(τ,k)k2+(54)dn2((12)τ,k)-sn2(τ,k)k2)×cn((12)τ,k)+6dn2(τ,k)+(12)dn2((12)τ,k)-2sn2(τ,k)k2+2cn2(τ,k)k2((12)τ,k))×cn(τ,k)sn(τ,k)sn((12)τ,k)+2cn((12)τ,k)dn((12)τ,k)dn(τ,k)(cn(τ,k)-sn(τ,k))×(cn(τ,k)+sn(τ,k))×(2+Λcn((12)τ,k))2)/(((12)τ,k)cn(τ,k)sn(τ,k)*(2+Λcn2((12)τ,k))sn((12)τ,k)).
In Figure 34 we plot the evolution of the R with respect of the cosmic time τ.Figure 34
The evolution of theR with respect of the cosmic time τ for (5.29).


## 5.3.2. Example2

Similarly, we can show that the system (5.15) has the following solution:
(5.30)H1=(1+12Λcnτ2)cnτ,H2=(1+12Λcnτ2)snτ,H3=12Λsnτ2.The corresponding EoS takes the form(5.31)ρ=D32E32,p1=-D33E33,p2=-D34E34,p3=-D35E35,
where
(5.32)D32=[1+12Λcnτ2]2cnτsnτ+Λ2×[1+12Λcnτ2][snτ+cnτ]snτ2-Λ,E32=1,D33=14Λdnτ2snτ2[1-snτ]+[1+12Λcnτ2]×[cnτdnτ+Λ2snτsnτ2]+[1+12Λcnτ2]2sn2τ+14Λ2sn2τ2-Λ,E33=1,D34=-14Λdnτ2snτ2cnτ-(1+12Λcnτ2)×[dnτsnτ+Λ2cnτsnτ2]+14Λcnτ2dnτ2+[1+12Λcnτ2]2cn2τ+14Λ2sn2τ2-Λ,E34=1,D35=-14Λdnτ2snτ2[cnτ+snτ]+[1+12Λcnτ2][cnτ-snτ]dnτ+[1+12Λcnτ2]2[sn2τ+cn2τ+cnτsnτ]-Λ,E35=1.The scalar curvature has the form(5.33)R=(12)Λ2(cn2(τ,k)+sn2(τ,k)+cn(τ,k)sn(τ,k))cn2((12)τ,k)+(12(Λ(sn(τ,k)+cn(τ,k))sn((12)τ,k)+dn((12)τ,k)+4cn2(τ,k)+(4sn(τ,k)+2dn(τ,k))cn(τ,k)+4sn2(τ,k)-2dn(τ,k)sn(τ,k)((12)τ,k))((12)τ,k))Λcn((12)τ,k)+(12)Λ2sn2((12)τ,k)-(12)Λ(dn((12)τ,k)-2)×(sn(τ,k)+cn(τ,k))sn((12)τ,k)+2cn2(τ,k)+(12(4dn(τ,k)+4sn(τ,k)))cn(τ,k)-2sn(τ,k)(-sn(τ,k)+dn(τ,k)).
In Figure 35 we plot the evolution of the R with respect of the cosmic time τ.Figure 35
The evolution of theR with respect of the cosmic time τ for (5.33).


## 6. Conclusion

In the present paper, we have constructed several concrete models describing the trefoil and figure-eight knot universes from Bianchi-type I cosmology and examined the cosmological features and properties in detail.To realize the cyclic universes, it is necessary to a noncanonical scalar field with ill-defined vacuum in the context of the quantum field theory or extended gravity, for example, with adding higher order derivative terms andf(R) gravity [79]. Indeed, however, these modified gravity theories have to satisfy the tests on the solar system scale as well as cosmological constraints so that those can be alternative gravitational theories to general relativity. The significant cosmological consequence of our approach is that we have shown the possibility to obtain the knot universes related to the cyclic universes from Bianchi-type I spacetime within general relativity.Furthermore, recently it has been pointed out that the asymmetry of the EoS for the universe can lead to cosmological hysteresis [80]. On the other hand, Bianchi-type I spacetime describes the spatially anisotropic cosmology and hence the EoS for the universe has the asymmetry in the oscillating process through the expanding and contracting behaviors. Accordingly, it is considered that in the constructed models of the knot universes cosmological hysteresis could occur. The observation of this phenomenon in our models is one of our future works on the knot universes.Finally, it should be remarked that by summarizing the results of our previous [98–101, 103] and this works, the knot universes describing the cyclic universes can be realized from the homogeneous and isotropic FLRW spacetime as well as the homogeneous and anisotropic Bianchi-type I cosmology. In these series of works, the formulations of model construction method of the knot universes have been established. Thus, it can be expected that the presented formalism is useful to realize the universes with other features from both the isotropic and anisotropic spacetimes.Finally we would like to note that all solutions presented above describe the accelerated and decelerated expansion phases of the universe.



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*Source: 868203-2012-12-27.xml*