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%TCIDATA{Created=Wed May 12 09:17:25 1999}
%TCIDATA{LastRevised=Wed Apr 16 21:15:35 2003}
%TCIDATA{Language=American English}

\begin{document}
\draft
\title{Knotlike Cosmic Strings in The Early Universe}
\author{Yi-shi Duan and Xin Liu\thanks{%
Corresponding author. Electronic address: liuxin@lzu.edu.cn}}
\address{{\it Institute of Theoretical Physics, Lanzhou University,}\\
{\it Lanzhou 730000, P. R. China}}
\maketitle

\begin{abstract}
In this paper, the knotlike cosmic strings in the early universe are
discussed. We derive the cosmic string structures from the non-zero torsion
tensor, and reveal that these strings are just created from the zero points
of complex scalar quintessence field. In these strings we mainly study the
knotlike configurations, and have constructed a new topological invariant
for them. It is also shown that this invariant is just the total sum of all
the self-linking and all the linking numbers of the knotlike strings family,
and that this invariant is preserved in the branch processes during the
evolution of cosmic strings.
\end{abstract}

\pacs{PACS number(s): 98.80.Cq, 02.40.-k, 11.15.-q}

\section{Introduction}

As we all know, torsion is a slight modification of the Einstein theory of
relativity (proposed in 1922-1923 by \'{E}. Cartan \cite{Cartan}), and is a
reasonable generalization that appears to be necessary when one considers
the early universe and tries to conciliate general relativity with quantum
theory \cite{Sabbata}. It is known that in the presence of torsion the
space-time is the Riemann-Cartan manifold $({\bf U}^4)$, hence in this paper
the Riemann-Cartan geometry of the early universe is studied \cite
{Sabbata,Turner,jiangying}.

The torsion in Riemann-Cartan space-time is defined as 
\begin{equation}
T_{\mu \nu }^a=D_\mu e_\nu ^a-D_\nu e_\mu ^a.  \label{Tuva}
\end{equation}
Here $e_\mu ^a$ is the vierbein, with $a$ being the local Lorentz group
index and $\mu $ denoting the $4$-dimensional space-time $(a,\;\mu
=0,\;1,\;2,\;3)$; $D_\mu e_\nu ^a$ is the covariant derivative of $e_\mu ^a:$%
\begin{equation}
D_\mu e_\nu ^a=\partial _\mu e_\nu ^a-\omega _\mu ^{ab}e_\nu ^b
\label{Dueva}
\end{equation}
where $\omega _\mu ^{ab}$ is the spin connection of Lorentz group. On the
analogy of 't Hooft's theory \cite{'tHooft}, the gauge invariant torsion
tensor is defined as 
\begin{equation}
T_{\mu \nu }=T_{\mu \nu }^aN^a-e_{[\mu }^aD_{\nu ]}N^a,  \label{Tuv1}
\end{equation}
where $N^a(x)$ is the unit vector field in the Lorentz group space and $%
D_\mu N^a$ is its covariant derivative: 
\begin{equation}
D_\mu N^a=\partial _\mu N^a-\omega _\mu ^{ab}N^b.
\end{equation}
From (\ref{Tuva}), (\ref{Dueva}) and (\ref{Tuv1}), the torsion tensor $%
T_{\mu \nu }$ can be expressed as 
\begin{equation}
T_{\mu \nu }=\partial _\mu A_\nu -\partial _\nu A_\mu ,  \label{TuvU1}
\end{equation}
where $A_\mu =e_\mu ^aN^a$ is a $U(1)$ gauge potential. The expression (\ref
{TuvU1}) shows that the antisymmetric $2$-order tensor $T_{\mu \nu }$ can be
just regarded as a $U(1)$ gauge field tensor.

Recently, the quintessence field has been introduced as the background field
of universe to account for the flatness and the accelerating expansion of
universe \cite{quintessence}. In this paper, we mainly consider the complex
scalar quintessence field $\psi (x)$ \cite{cmplxsclr}. For the purpose to
describe the interaction between $\psi $ and the $U(1)$ gauge potential $%
A_\mu $, one introduces the covariant derivative of $\psi $ as 
\begin{equation}
D_\mu \psi =\partial _\mu \psi -i\frac{2\pi }{L_p}A_\mu \psi ,
\label{quintcovder}
\end{equation}
where $L_p=\sqrt{\hbar G/c^3}$ is the Planck length. In the following
Sect.II, through (\ref{quintcovder}) and (\ref{TuvU1}) and using the $\phi $%
-mapping topological current theory, we will derive the cosmic string
structures from $T_{\mu \nu }$, and reveal that these cosmic strings are
just created from the zero points of quintessence field $\psi $. In
Sect.III, we will mainly study the knotlike configurations in these cosmic
string structures. Based on the integral of Chern-Simons $3$-form we
construct a new topological invariant for the knotlike strings. It is also
shown that this invariant is just the total sum of all the self-linking and
all the linking numbers of the knotlike strings family. In Sect.IV,
moreover, the conservation of this topological invariant in the branch
processes during the evolution of cosmic strings is simply discussed.

\section{The Cosmic Strings}

In this section we will show that there are cosmic string structures
inhering in the torsion tensor $T_{\mu \nu }.$

The complex scalar quintessence field $\psi (x)$ is a kind of order
parameter field, which is a section of complex line bundle, i.e., a section
of the $2$-dimensional real vector bundle on ${\bf U}^4$: 
\begin{equation}
\psi (x)=\phi ^1{\bf (}x)+i\phi ^2{\bf (}x).  \label{psiphi12}
\end{equation}
From (\ref{quintcovder}) and (\ref{psiphi12}) it can be proved that the $%
U(1) $ gauge potential $A_\mu $ can be expressed as \cite{U1decom} 
\begin{equation}
A_\mu =\frac{L_p}{2\pi }\epsilon _{ab}\frac{\phi ^a}{\left\| \phi \right\| ^2%
}\partial _\mu \phi ^b.\;\;(\left\| \phi \right\| ^2=\phi ^a\phi ^a)
\end{equation}
Then from (\ref{TuvU1}) we have 
\begin{equation}
T_{\mu \nu }=\frac{L_p}{2\pi }2\epsilon _{ab}\partial _\mu (\frac 1{\left\|
\phi \right\| ^2}\phi ^a)\partial _\nu \phi ^b.
\end{equation}
According to the $\phi $-mapping topological current theory \cite{topcurrent}%
, it can be proved that $T_{\mu \nu }$ can be expressed in a $\delta $%
-function form: 
\begin{equation}
\frac{2\pi }{L_p}\frac 1{8\pi }\epsilon ^{\mu \nu \rho \lambda }T_{\rho
\lambda }=\delta ^2(\vec{\phi})D^{\mu \nu }(\frac \phi x),  \label{deltav}
\end{equation}
where{\ }${D^{\mu \nu }(\phi /x)=\frac 12\epsilon }^{\mu \nu \rho \lambda }{%
\epsilon _{ab}\partial }_\rho {\phi }^a{\partial _\lambda \phi }^b.${\
Taking the }index $\mu $ in (\ref{deltav}) as $\mu =0,$ we obtain the $%
\delta $-function expression for the spatial components of $T_{\mu \nu }:$%
\begin{equation}
\frac{2\pi }{L_p}\frac 1{8\pi }\epsilon ^{ijk}T_{jk}=\delta ^2(\vec{\phi}%
)D^i(\frac \phi x),\;\;(i,j,k=1,2,3)  \label{jidel}
\end{equation}
where $D^i(\phi /x)=\frac 12{\epsilon }^{ijk}{\epsilon _{ab}\partial }_j{%
\phi }^a{\partial _k\phi }^b$ is the Jacobian vector.

Obviously the expression (\ref{jidel}) provides an important conclusion: 
\begin{equation}
T_{ij}\left\{ 
\begin{array}{l}
=0,\;iff\;\vec{\phi}\neq 0; \\ 
\neq 0,\;iff\;\vec{\phi}=0,
\end{array}
\right.
\end{equation}
so it is necessary to study the zero points of $\vec{\phi}$ to determine the
non-zero solutions of $T_{ij}.$ The implicit function theory shows \cite
{Goursat} that under the regular condition 
\begin{equation}
D^{\mu \nu }(\phi /x)\neq 0,  \label{regulcond}
\end{equation}
the general solutions of 
\begin{equation}
\phi ^1(t,\;x^1,\;x^2,\;x^3)=0,\;\phi ^2(t,\;x^1,\;x^2,\;x^3)=0
\end{equation}
can be expressed as 
\begin{equation}
x^1=x_k^1(s,\;t),\;x^2=x_k^2(s,\;t),\;x^3=x_k^3(s,\;t),
\end{equation}
which represent the world surfaces of a family of $N$ moving isolated
singular strings $L_k$ with string parameter $s$ $(k=1,\;2,\;...,\;N)$.
These singular string solutions are just the cosmic strings. Furthermore, we
have \cite{Schouton,topcurrent} 
\begin{equation}
\delta ^2(\vec{\phi})D^i(\frac \phi x)=\sum_{k=1}^NW_k\int_{L_k}\frac{dx^i}{%
ds}\delta ^3(\vec{x}-\vec{x}_k(s))ds,  \label{jitopcurr}
\end{equation}
where $W_k=\beta _k\eta _k$ is the winding number of $\vec{\phi}$ around $%
L_k,$ with the positive integer $\beta _k$ being the Hopf index and $\eta
_k=\pm 1$ the Brouwer degree. Hence the topological charge of the cosmic
string $L_k$ is 
\begin{equation}
\frac{2\pi }{L_p}\int_{\Sigma _k}\frac 1{8\pi }\epsilon ^{ijk}T_{jk}d\sigma
_i=W_k.
\end{equation}

In the end of this section, it should be addressed that in the above the
regular condition (\ref{regulcond}) has been used; when this condition
fails, the branch processes during the evolution of cosmic strings will
occur. This will be detailed in Sect.IV.

\section{The Topological Invariant for The Knotlike Cosmic Strings}

It has been pointed out, that for a string structure of finite energy, its
length must be finite, which is possible if its core forms a knot \cite
{FadNie}. Recently it is shown by Faddeev and Niemi that there are knotlike
structures appearing as stable finite solitons in a realistic $(3+1)$%
-dimensional model \cite{FadNie}. In this section, we will mainly study the
knotlike cosmic string structures, and construct a new topological invariant
for them. It is also shown that this invariant is just the self-linking and
the linking numbers of the knotlike cosmic strings family.

The expression (\ref{TuvU1}) shows that $T_{\mu \nu }$ can be regarded as a $%
U(1)$ gauge field tensor. In order to construct a topological invariant in
the space-time, one must pick a Lagrangian which does not require any choice
of metric $g_{\mu \nu }$. Precisely in $3$-dimensional space there is a
reasonable choice, namely, the integral of the Chern-Simons $3$-form \cite
{CSterm,Witten}: 
\begin{equation}
Q=(\frac{2\pi }{L_p})^2\frac 1{8\pi }\int_M\epsilon ^{ijk}A_iT_{jk}d^3x.
\label{intCSform}
\end{equation}
Hereinafter we just study (\ref{intCSform}) to get the topological invariant
for the knotlike cosmic strings. Using the above (\ref{jidel}) and (\ref
{jitopcurr}), the expression (\ref{intCSform}) can be written as 
\begin{equation}
Q=\frac{2\pi }{L_p}\int_MA_i\delta ^2(\vec{\phi})D^i(\frac \phi x)d^3x=\frac{%
2\pi }{L_p}\sum_{k=1}^NW_k\int_{L_k}A_idx^i.  \label{CS1}
\end{equation}
It can be seen that when the $N$ cosmic strings of (\ref{CS1}) are $N$
closed curves, i.e., a family of $N$ knotlike strings $\gamma
_k\;(k=1,...,N) $, (\ref{CS1}) leads to 
\begin{equation}
Q=\frac{2\pi }{L_p}\sum_{k=1}^NW_k\oint_{\gamma _k}A_idx^i.  \label{CSact3}
\end{equation}
This is a very important expression. Consider the $U(1)$ gauge
transformation of $A_i$ \cite{Nash}: 
\begin{equation}
A_i^{\prime }=A_i+i\frac{2\pi }{L_p}\partial _i\alpha ,  \label{U1trans}
\end{equation}
where $\alpha \in {\bf R}$ is a phase factor denoting the $U(1)$
transformation. It is seen that the $(i\frac{2\pi }{L_p}\partial _i\alpha )$
term in (\ref{U1trans}) contributes nothing to the integral $Q,$ hence the
expression (\ref{CSact3}) is invariant under the gauge transformation.
Therefore, from the fact that $Q$ of (\ref{CSact3}) is independent of the
choice of metric and is invariant under the $U(1)$ gauge transformation, one
can conclude that $Q$ is a topological invariant for the knotlike cosmic
strings, which can be used in the research of the topology of string
structures in the early universe.

In following we will show that $Q$ is just the total sum of all the
self-linking and all the linking numbers of the knotlike strings family.
Using (\ref{jidel}), the expression (\ref{CSact3}) can be reexpressed as 
\begin{equation}
Q=\frac{2\pi }{L_p}\sum_{k=1}^N\sum_{l=1}^NW_kW_l\oint_{\gamma
_k}\oint_{\gamma _l}\partial _iA_jdx^idy^j,  \label{CSact4}
\end{equation}
where $\vec{x}$ and $\vec{y}$ are two points respectively on knots $\gamma
_k $ and $\gamma _l$. Noticing that $\gamma _k$ and $\gamma _l$ can be the
same one knot, or two different knots, we should write (\ref{CSact4}) in two
parts ($k=l$ and $k\neq l$); furthermore the $k=l$ part includes both the $%
\vec{x}\neq \vec{y}$ and the $\vec{x}=\vec{y}$ cases. So $Q$ should be
written in three terms: $(k=l;\;\vec{x}\neq \vec{y}),$ $(k=l$ $;$ $\vec{x}=%
\vec{y})$ and $(k\neq l)$ terms. Defining a $3$-dimensional unit vector $%
\vec{m}=\frac{\vec{y}-\vec{x}}{\left\| \vec{y}-\vec{x}\right\| },$ and
another $2$-dimensional unit vector $\vec{e}$ on the $\vec{m}$-formed-sphere 
$S^2\;(\vec{e}\bot \vec{m})$, $A_i$ can be decomposed in terms of $e^a$ as 
\cite{U1decom} 
\begin{equation}
A_i=\frac{L_p}{2\pi }\epsilon _{ab}e^a\partial _ie^b.\;\;(a,\;b=1,2)
\label{decomsec}
\end{equation}
Then using (\ref{decomsec}) and the relation $2\epsilon _{ab}\partial
_ie^a\partial _je^b=\vec{m}\cdot (\partial _i\vec{m}\times \partial _j\vec{m}%
)$, the three terms of $Q$ can be expressed as 
\begin{eqnarray}
Q &=&2\pi [\sum_{k=1\;(\vec{x}\neq \vec{y})}^N\frac 1{4\pi }%
W_k^2\oint_{\gamma _k}\oint_{\gamma _k}\vec{m}^{*}(dS)+\frac 1{2\pi }%
\sum_{k=1}^NW_k^2\oint_{\gamma _k}\epsilon _{ab}e^a\partial _ie^bdx^i 
\nonumber \\
&&+\sum_{k,l=1\;(k\neq l)}^N\frac 1{4\pi }W_kW_l\oint_{\gamma
_k}\oint_{\gamma _l}\vec{m}^{*}(dS)].  \label{CSact8}
\end{eqnarray}
where $\vec{m}^{*}(dS)=$ $\vec{m}\cdot (\partial _i\vec{m}\times \partial _j%
\vec{m})dx^i\wedge dy^j\;(\vec{x}\neq \vec{y})$ denotes the pull-back of $%
S^2 $ surface element.

Let us discuss these three terms in detail. Firstly, the first term of (\ref
{CSact8}) is just related to the writhing number $Wr(\gamma _k)$ of $\gamma
_k:$%
\begin{equation}
Wr(\gamma _k)=\frac 1{4\pi }\oint_{\gamma _k}\oint_{\gamma _k}\vec{m}%
^{*}(dS).  \label{wrnum}
\end{equation}
For the second term of (\ref{CSact8}), since this is the $\vec{x}=\vec{y}$
term, one can prove that it is related to the twisting number $Tw(\gamma _k)$
of $\gamma _k$%
\begin{equation}
\frac 1{2\pi }\oint_{\gamma _k}\epsilon _{ab}e^a\partial _ie^bdx^i=\frac 1{%
2\pi }\oint_{\gamma _k}(\vec{T}\times \vec{V})\cdot d\vec{V}=Tw(\gamma _k),
\label{twnum}
\end{equation}
where $\vec{T}$ is the unit tangent vector of knot $\gamma _k$ at $\vec{x}$ (%
$\vec{m}=\vec{T}$ when $\vec{x}=\vec{y}$), and $\vec{V}$ is defined as $%
e^a=\epsilon ^{ab}V^b\;(\vec{V}\bot \vec{T},\;\vec{e}=\vec{T}\times \vec{V})$%
. From the White formula \cite{White} 
\begin{equation}
SL(\gamma _k)=Wr(\gamma _k)+Tw(\gamma _k),  \label{white}
\end{equation}
we see that the first and the second terms of (\ref{CSact8}) just compose
the self-linking numbers of knots. Secondly, for the third term, one can
prove 
\begin{equation}
\frac 1{4\pi }\oint_{\gamma _k}\oint_{\gamma _l}\vec{m}^{*}(dS)=\frac 1{4\pi 
}\epsilon ^{ijk}\oint_{\gamma _k}dx^i\oint_{\gamma _l}dy^j\frac{(x^k-y^k)}{%
\left\| \vec{x}-\vec{y}\right\| ^3}=Lk(\gamma _k,\gamma _l)\;\;(k\neq l)
\label{Gausslink}
\end{equation}
where $Lk(\gamma _k,\gamma _l)$ is the Gauss linking number between $\gamma
_k$ and $\gamma _l$\cite{Witten,Polyakov}. Therefore, thirdly, from (\ref
{wrnum}), (\ref{twnum}), (\ref{white}) and (\ref{Gausslink}) we arrive at
the important result: 
\begin{equation}
Q=2\pi [\sum_{k=1}^NW_k^2SL(\gamma _k)+\sum_{k,l=1\;(k\neq
l)}^NW_kW_lLk(\gamma _k,\gamma _l)].  \label{CSact9}
\end{equation}
This precise expression just reveals the relationship between $Q$ and the
self-linking and the linking numbers of the knots family. Since the
self-linking and the linking numbers are both the invariant characteristic
numbers of the knots family in topology, $Q$ is an important invariant
required to describe the topology of knotlike cosmic strings in early
universe.

\section{The Conservation of $Q$ in The Branch Processes of Knotlike Cosmic
Strings}

In our previous work \cite{jiangying} it has been pointed out that, during
the evolution of cosmic strings, when the regular condition (\ref{regulcond}%
) fails, the branch processes (i.e. the splitting, mergence and
intersection) will occur; and in these branch processes, the sum of the
topological charges of final cosmic string(s) is equal to that of the
initial cosmic string(s) at the bifurcation point, namely:

(a) for the case that one string $L$ split into two strings $L_1$ and $L_2,$
we have $W_L=W_{L_1}+W_{L_2};$

(b) the case that two strings $L_1$ and $L_2$ merge into one string $L:$ $%
W_{L_1}+W_{L_2}=W_L;$

(c) the case that two strings $L_1$ and $L_2$ meet, and then depart as two
other strings $L_3$ and $L_4:$ $W_{L_1}+W_{L_2}=W_{L_3}+W_{L_4}.$

In following we will show that when the branch processes of knotlike strings
occur, the topological invariant $Q$ of (\ref{CSact3}) (i.e. (\ref{CSact9}))
is preserved:

(i) The splitting case. We will consider one knot $\gamma $ split into two
knots $\gamma _1$ and $\gamma _2$ which are of the same self-linking number
as $\gamma \;(SL(\gamma )=SL(\gamma _1)=SL(\gamma _2))$, and will compare
the two numbers $Q_\gamma $ and $Q_{\gamma _1+\gamma _2},$ where $Q_\gamma $
is the contribution of $\gamma $ to $Q$ before splitting, and $Q_{\gamma
_1+\gamma _2}$ is the total contribution of $\gamma _1$ and $\gamma _2$ to $%
Q $ after splitting. Firstly, from the above text we have $W_\gamma
=W_{\gamma _1}+W_{\gamma _2}$ in the splitting process. Secondly, on the one
hand, noticing that in the neighborhood of bifurcation point $(\vec{x}%
^{*},\;t^{*}) $, $\gamma _1$ and $\gamma _2$ are infinitesimally displaced
from each other; on the other hand, for a knot $\gamma $ its self-linking
number $SL(\gamma )$ is defined as $SL(\gamma )=Lk(\gamma ,\;\gamma _V),$
where $\gamma _V$ is another knot obtained by infinitesimally displacing $%
\gamma $ in the normal direction $\vec{V}$ \cite{Witten}. Therefore $%
SL(\gamma )=SL(\gamma _1)=SL(\gamma _2)=Lk(\gamma _1,\;\gamma _2)=Lk(\gamma
_2,\;\gamma _1),$ and $Lk(\gamma ,\;\gamma _k^{\prime })=Lk(\gamma
_1,\;\gamma _k^{\prime })=Lk(\gamma _2,\;\gamma _k^{\prime })$ (here $\gamma
_k^{\prime }$ denotes another arbitrary knot in the family ($\gamma
_k^{\prime }\neq \gamma ,\;\gamma _k^{\prime }\neq \gamma _{1,2}$)). Then,
thirdly, we can compare $Q_\gamma $ and $Q_{\gamma _1+\gamma _2}$ as: before
splitting, from (\ref{CSact9}) we have 
\begin{equation}
Q_\gamma =2\pi [W_\gamma ^2SL(\gamma )+\sum_{k=1\;(\gamma _k^{\prime }\neq
\gamma )}^N2W_\gamma W_{\gamma _k^{\prime }}Lk(\gamma ,\gamma _k^{\prime })],
\label{before}
\end{equation}
where $Lk(\gamma ,\gamma _k^{\prime })=Lk(\gamma _k^{\prime },\gamma )$;
after splitting, 
\begin{eqnarray}
Q_{\gamma _1+\gamma _2} &=&2\pi [W_{\gamma _1}^2SL(\gamma _1)+W_{\gamma
_2}^2SL(\gamma _2)+2W_{\gamma _1}W_{\gamma _2}Lk(\gamma _1,\gamma _2) 
\nonumber \\
&&+\sum_{k=1\;(\gamma _k^{\prime }\neq \gamma _{1,2})}^N2W_{\gamma
_1}W_{\gamma _k^{\prime }}Lk(\gamma _1,\gamma _k^{\prime
})+\sum_{k=1\;(\gamma _k^{\prime }\neq \gamma _{1,2})}^N2W_{\gamma
_2}W_{\gamma _k^{\prime }}Lk(\gamma _2,\gamma _k^{\prime })].  \label{after}
\end{eqnarray}
Comparing (\ref{before}) and (\ref{after}), we just have 
\begin{equation}
Q_\gamma =Q_{\gamma _1+\gamma _2}.
\end{equation}
This means that in the splitting process $Q$ is preserved.

(ii) The mergence case. We consider two knots $\gamma _1$ and $\gamma _2,$
which are of the same self-linking number, merge into one knot $\gamma $
which is of the same self-linking number as $\gamma _1$ and $\gamma _2$.
This is obviously the inverse process of the above splitting case, therefore
we have 
\begin{equation}
Q_{\gamma _1+\gamma _2}=Q_\gamma .
\end{equation}

(iii) The intersection case. This case is related to the collision of two
knots \cite{collision}. We consider two knots $\gamma _1$ and $\gamma _2,$
which are of the same self-linking number, meet, and then depart as other
two knots $\gamma _3$ and $\gamma _4$ which are of the same self-linking
number as $\gamma _1$ and $\gamma _2$. This process can be identified to two
sub-processes: $\gamma _1$ and $\gamma _2$ merge into one knot $\gamma $,
and then $\gamma $ split into $\gamma _3$ and $\gamma _4.$ Thus from the
above two cases (ii) and (i) we have 
\begin{equation}
Q_{\gamma _1+\gamma _2}=Q_{\gamma _3+\gamma _4}.
\end{equation}

Therefore we obtain the result that, in the branch processes during the
evolution of knotlike cosmic strings (i.e., the splitting, mergence and
intersection), the topological invariant $Q$ is preserved.

\section{Conclusion}

In this paper, the early universe as a Riemann-Cartan space-time is
considered. In Sect.II, it is revealed that from the non-zero torsion tensor 
$T_{\mu \nu }$ one can derive the cosmic string structures, which are just
created from the zero points of complex scalar quintessence field. In
Sect.III, we emphasize on the knotlike configurations in these cosmic
strings. Based on the integral of Chern-Simons $3$-form, we construct a new
topological invariant $Q$ for the knotlike strings. It is also pointed out
that $Q$ is just the total sum of all the self-linking and all the linking
numbers of the knotlike strings family. In Sect.IV, it is shown that $Q$ is
preserved in the branch processes (i.e. the splitting, mergence and
intersection) during the evolution of knotlike cosmic strings.

\section{Acknowledgment}

One of the authors XL would like to thank Dr. P. M. Zhang for the useful
discussions and help. This work was supported by the National Natural
Science Foundation and the Doctor Education Fund of Educational Department
of the People's Republic of China.

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