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\title{Quandle Cocycle Invariants of Knots}
\author{Linked from: http://shell.cas.usf.edu/quandle}

\begin{document}


\maketitle


\section{Introduction}

\subsection{What are these web pages about}

\begin{sloppypar}
This article is
 written as web pages posted at 
 http://shell.cas.usf.edu/quandle under the title 
{\it Background}.
\end{sloppypar}
The purpose of this paper is to give an overview of 
{\it quandle cocycle invariants} of knots, including definitions of the invariants.
The main goal is to provide the reader with  necessary background 
information quickly, in an effective way.


\subsection{Target readers}

These pages are  intended for undergraduate students, as well as  professionals 
who want to get a quick overview. 
The knowledge we assume are groups, and laymen's knowledge 
in knot theory, such as Reidemeister moves of knot diagrams.
For such basics, we refer the reader to 
any introductory book on knots, such as \cite{Ad94,Liv93,Mura96}.

\subsection{Other sources of data used in our calculations}

\begin{sloppypar}
We used the knot table available from Livingston's web site:
http://www.indiana.edu/~knotinfo.
For quandles, we used the table of quandles in Appendix 
of \cite{CKS00}. 
\end{sloppypar}

\subsection{Comments on history}

Racks and quandles are fundamental algebraic structures. 
Quandles have been rediscovered and studied extensively over the past sixty years
\cite{Br88,Deh,GhoKep,Joy82,Kauff91,Mat82,Taka42}.
It appears that a special case of quandles (involutory quandles), called {\it Kei},
 first appeared in literature 
in \cite{Taka42}. The name {\it quandle} is due to \cite{Joy82}. 
It is beyond our scope here to review such a rich history of quandles in details.

In \cite{CJKLS},  quandle cohomology theory was constructed
as a modification of  rack cohomology theory~\cite{FRS95,FRS}, and 
quandle cocycle invariants were also defined in a state-sum form. 
It is our goal here to briefly review this invariant.


\section{Definitions}

\subsection{Quandles}

A {\it quandle}, $X$, is a set with a binary operation 
$X \times X \rightarrow X$ denoted by 
$(a, b) \mapsto a * b$
such that

(I) For any $a \in X$,
$a* a =a$.

(II) For any $a,b \in X$, there is a unique $c \in X$ such that
$a= c*b$.

(III)
For any $a,b,c \in X$, we have
$ (a*b)*c=(a*c)*(b*c). $

\noindent
A {\it rack} is a set with a binary operation that satisfies
(II) and (III).
Racks and quandles have been studied in, for example,
\cite{Br88,FennRou92,Joy82,Mat82}.

The following are typical examples of quandles.
A group $G$ 
with
conjugation
as the quandle operation:
$a*b=b a b^{-1}$, denoted by $X=$Conj$(G)$,
is a quandle.
Any subset of $G$ that is closed under such conjugation
is also a quandle.
Another example is the set of  polynomials 
$M={\Z_p }[t] / f(t)$
with mod $p$ integers, for a prime $p$, and modulo a polynomial $f(t)$.
Then  the elements of $M$ are represented by remainders of 
polynomials mod $p$ when divided by $f$, so that $M$ consists of 
$p^d$ elements. A quandle structure is defined by 
$a*b=ta+(1-t)b$, $a,b \in M$, and 
it is  called an {\it  Alexander  quandle}.
(In general, any $\Lambda (={\Z }[t, t^{-1}])$-module $M$
is a quandle with
$a*b=ta+(1-t)b$, $a,b \in M$,
that is
called an {\it  Alexander  quandle}.)
Classification of Alexander quandles are discussed in \cite{Nel03}.

Let $n$ be a positive integer, and
for elements  $i, j \in \{ 0, 1, \ldots , n-1 \}$, define
$i\ast j \equiv 2j-i \pmod{n}$.
Then $\ast$ defines a quandle
structure  called the {\it dihedral quandle},
  $R_n$.
  This is an Alexander quandle $R_n=\Z_n[t]/(t+1)$. 

A map $f: X \rightarrow Y$ between quandles $X, Y$ is 
a quandle {\it homomorphism} if $f(a,b)=f(a)*f(b)$ for any $a, b \in X$,
where we abuse notation and use $*$ for quandle operations on different quandles
unless confusion arises.
A quandle homomorphism is an isomorphism if it is surjective and injective.
If there is an isomorphism between two quandles, they are called 
{\it isomorphic}.

If  $G$ is a group and $s$ is an automorphism,
$a*b=s(ab^{-1})b$ defines a quandle structure on $G$,
 and if $s(h)=h$ for for all $h$ in a subgroup $H$, then $G/H$ inherits the operation:
$Ha * Hb = Hs(ab^{-1})b$. This example is due to Joyce~\cite{Joy82}, where it is proved
that 
any homogeneous quandle, $X$,  is isomorphic to such a quandle: 
Let $G={\rm Aut}(X)$, let an element $x\in X$  be fixed, let
 $H= \{ f \in {\rm Aut}(X): xf=x \}$. For any element $z \in H$,  $z \in Z(H)$ (centralizer). Then the evaluation map $e(g)= xg$ provides the isomorphism.


\subsection{Colorings of knot diagrams}

Let $X$ be a fixed quandle.
Let $K$ be a given oriented classical knot or link diagram,
and let ${\cal R}$ be the set of (over-)arcs.
The normals (normal vectors) are given in such a way that
the ordered pair
(tangent, normal) agrees with
the orientation of the plane (the ordered pair of the standard $x$- and $y$-axes), 
see Fig.~\ref{qcolor}.

\begin{figure}[htb]
\begin{center}
\mbox{
\epsfxsize=2.2in
\epsfbox{qcolor.eps}
}
\end{center}
\caption{ Quandle relation at a crossing  }
\label{qcolor}
\end{figure}

A (quandle) {\it coloring} ${\cal C}$ is a map
${\cal C} : {\cal R} \rightarrow X$ such that at every crossing,
the relation depicted in Fig.~\ref{qcolor} holds.
If the normal to the over-arc $\beta$ points from the 
arc $\alpha$ to $\gamma$, then it is required that 
 ${\cal C}(\alpha)*{\cal C}(\beta)={\cal C}(\gamma)$.
 The pair $(a,b)$ is called the {\it ordered pair} of colors at the crossing.
Let ${\rm Col}_X(K)$ denote the set of colorings of a knot diagram $K$
by a quandle $X$.
It is known~\cite{FennRou92} that there is a unique coloring after any  Reidemeister move 
given a coloring before the move, and therefore, 
the cardinality
$|{\rm Col}_X(K)|$ is a knot invariant.

Note that any knot diagram is colored by a single element of a given quandle.
Such a monochromatic coloring is called {\it trivial}.
Hence for any knot $K$ and any quandle $X$, $|{\rm Col}_X(K)|$ is at least 
the number of quandle, $|X|$. 

\begin{sloppypar}
When we use Alexander quandles to color diagrams, 
Inoue's Theorem~\cite{Ino01} is helpful.
There is a knot invariant called Alexander polynomial, 
and an implication of Inoue's Theorem is that 
a knot is colored only trivially by an Alexander quandle $\Z_p[t, t^{-1}]/g(t)$
if its Alexander polynomial is coprime mod $p$ to $g(t)$. 
\end{sloppypar}



\subsection{Colorings of regions}

For a coloring $ {\mathcal C}$ of a knot diagram $K$,
there is a coloring of regions
that extend $ {\mathcal C}$ as depicted in Fig.~\ref{Foxcolors}.
A coloring of regions is an assignment of quandle elements to regions
that satisfies the following requirement: Suppose the two regions $R_1$ and $R_2$
are divided by an arc $\alpha$ such that the normal to $\alpha $ points from 
$R_1$ to $R_2$, and let 
$x$ be a color of $R_1$.  Then the color of $R_2$ is required to be  
$x * {\mathcal C}(\alpha)$.
At a crossing, there are four regions. There is a unique region among the four 
from which all  the normals of the arcs point to other regions.
Such a region is called the {\it source} region at a crossing.


\begin{figure}[htb]
\begin{center}
\mbox{
\epsfxsize=2.5in
\epsfbox{Foxcolors.eps}
}
\end{center}
\caption{ Quandle colorings of regions }
\label{Foxcolors}
\end{figure}

 In Fig.~\ref{Foxcolors}, it is seen that colorings of regions 
 (region colors) are well-defined.
In the left of the figure, a {\it positive} crossing is depicted. There are two ways 
to go from the source region (leftmost region)
to the rightmost region, through upper arcs and lower arcs.
Through upper arcs we obtain the color $(x*y)*z$, 
and through the lower arcs $(x*z)*(y*z)$, that coincide by a quandle condition.
The triple $(x,y,z)$ is called the {\it ordered triple} of colors at the crossing.
The situation is similar at a negative crossing (the right of Fig.~\ref{Foxcolors}.
The triple $(x,y,z)$ is called the ordered triple of colors at the crossing.
Specifically, $x$ is the color of the source region, $y$ is the color of the under-arc
from which the normal of the over-arc points, and $z$ is the color of the over-arc.


\subsection{$2$-Cocycle invariants}

The cocycle invariant for classical knots \cite{CJKLS} using quandle $2$-cocycles
was defined as follows. 
Let $X$ be a finite quandle and $A$ an abelian group, typically a finite cyclic 
group $\Z_n$, $n \in \Z$, $n>1$.

A function $\phi : X \times X \rightarrow A$
is called a {\it quandle $2$-cocycle}  if it  satisfies the {\it $2$-cocycle condition}
$$\phi(x,y) - \phi(x,z) + \phi(x*y, z) - \phi(x*z, y*z)=0, \quad \forall x,y,z \in X$$
and $\phi(x,x)=0, \forall x \in X$.
Let ${\cal C}$ be a coloring of a given knot diagram $K$ by $X$.

The {\em Boltzmann weight} $B( {\cal C}, \tau)$ at a crossing $\tau$ of $K$
is then defined by  $B( {\cal C}, \tau)=\phi(x_{\tau}, y_{\tau})^{\epsilon(\tau)}$,
where $(x_{\tau}$, $y_{\tau})$ is  the ordered pair of  colors at $\tau$ and
$\epsilon(\tau)$ is the sign ($\pm 1$) of $\tau$.
Here $B( {\cal C}, \tau)$ is an element of $A$ written multiplicatively.
The formal sum (called a state-sum) in the group ring $\Z[A]$
$$\Phi_{\phi} (K) = \sum_{{\cal C}\in {\rm Col_X(K)}} \prod_{\tau}
B( {\cal C}, \tau ) $$
is called the quandle $2$-cocycle invariant.
A value (an element of $\Z[A]$) is written as $\sum_{g \in G} a_g g $,
so that if $A=\Z_n=\{ u^k \ | \  0 \leq k < n \}$, a value is written as 
a polynomial $\sum_{k=0}^{n-1} a_k u^k$. 
That 
$\Phi_{\kappa} (K) $ is a knot invariant is proved easily by checking 
Reidemeister moves~\cite{CJKLS}.

The cocycle invariant can be also written as
a family (multi-set, a set with repetition allowed) of weight sums~\cite{Lop03}
$$ \left\{ \left. \sum_{\tau} B( {\cal C}, \tau ) \ \right|
 \ {\cal C}\in {\rm Col_X(K)} \right\} $$
where now the values of $B$ in $A$ are denoted by additive notation.
Then this is well-defined for any quandle of any cardinality.



\subsection{$3$-Cocycle invariants}

Let $X$ be a finite quandle and $A$ an abelian group.

A function $\theta : X \times X \times X \rightarrow A$
is called a {\it quandle $3$-cocycle}  if it  satisfies the {\it $3$-cocycle condition}
\begin{eqnarray*}
\lefteqn{\theta (x,z,w)- \theta(x,y,w)+\theta(x,y,z)
-\theta(x*y,z,w)} \\ & & +\theta(x*z,y*z,w)-\theta(x*w,y*w,z*w)=0, 
 \quad \forall x,y,z, w \in X,
 \end{eqnarray*}
and $\theta(x,x,y)=0=\theta(x,y,y), \forall x, y \in X$.


Let ${\cal C}$ be a coloring of arcs and regions of a given diagram $K$.
Let $(x,y,z) (=(x_{\tau}, y_{\tau}, z_{\tau}))$ be  the ordered triple of colors 
at a crossing  $\tau$, see Fig.~\ref{Foxcolors}.
Let $\theta$ be a $3$-cocycle.
Then the weight in this case is defined by
 $B( {\cal C}, \tau)=\phi(x_{\tau}, y_{\tau}, z_{\tau})^{\epsilon(\tau)}$
 where $\epsilon(\tau)$ is $\pm 1$  for positive and negative
crossing, respectively.
Then the ($3$-)cocycle invariant is defined in a similar way to $2$-cocycle invariants by
$ \Phi_{\phi}(K) =\{ \sum_{\tau} B( {\cal C}, \tau ) \ | \ {\cal C}\in {\rm Col_X(K)} \} $.
The multiset version is defined similarly.




\subsection{Polynomial (Mochizuki) cocycles}

We consider Alexander quandles and their quandle cocycles
with the coefficient group of the form
$X=\mathbb{Z}_p[t,t^{-1}]/h(t)=A$.
In this case some $2$- and $3$-cocycles are constructed by 
polynomial functions. These constructions were given in \cite{Mochi}.

In general,  functions of the form 
$$f(x_1, \ldots, x_n)=(x_1-x_2)^{p^{m_1}} \cdots (x_{n-1}-x_n)^{p^{m_{n-1}}}
x_n^{a_n} $$
are $n$-cocycles for a variety of Alexander quandles mod $p$, 
where $p$ is a prime and $a_n$ is either $0$ or a power of $p$.
In our case, we used the following cocycles.

\begin{itemize}
\setlength{\itemsep}{-2pt}
\item $f(x,y)=(x-y)^{p^{n}}$ is a $2$-cocycle for any Alexander quandle mod $p$.

\item $f(x,y)=(x-y)^{p^{m_1}}y^{p^{m_2}}$ is a $2$-cocycle for an 
Alexander quandle $\Z_p[t, t^{-1}]/g(t)$ if $g(t)$ divides $(t^{p^{m_1}+p^{m_2}}-1)$.

\item $f(x,y, z)=(x-y)^{p^{m_1}} (y-z)^{p^{m_2}}$ is a $3$-cocycle for any Alexander quandle mod $p$.

\item $f(x,y, z)=(x-y)^{p^{m_1}} (y-z)^{p^{m_2}} z^{p^{m_3}}$ is a $3$-cocycle for an 
Alexander quandle $\Z_p[t, t^{-1}]/g(t)$ if $g(t)$ divides $(t^{p^{m_1}+p^{m_2}+p^{m_3}}-1)$.

\end{itemize}

There is another type of $3$-cocycles Mochizuki constructed for dihedral quandles $R_p$ for prime $p$.
It is given by the formula
$$ \theta (x,y,z)= (x-y) [(2 z^p - y^p) - (2z-y)^p ]/p \quad \mbox{mod} \; p,$$
where the numerator computed in $\Z$ is divisible by $p$, and then after 
dividing it by $p$, the value is taken as an integer modulo $p$.


\subsection{Twisted cocycle invariants}

In \cite{CES02}, a cocycle invariant using twisted quandle 
cohomology theory was defined. 
Let $X$ be a quandle, and $A$ be an Alexander quandle with variable $t$
(typically $A=\Z_p[t, t^{-1}]/h(t)$ for a prime $p$ and a polynomial $h(t)$).
A function $\phi : X \times X \rightarrow A$
is called a {\it twisted quandle $2$-cocycle}  if it  satisfies 
the {\it twisted $2$-cocycle condition}
$$t [ \phi(y,z) - \phi(x,z)+ \phi(x,y) ]-
[\phi(y,z) - \phi(x*y, z) + \phi(x*z, y*z)]=0,$$
 for all $x,y,z \in X$, 
and $\phi(x,x)=0, \forall x \in X$.

Let $K$ be an oriented knot diagram with orientation normal vectors. 
The underlying projection  of the 
diagram divides the $3$-space into regions.
For a crossing $\tau$, among four regions adjacent to $\tau$, 
there is a  unique region $R$ 
such that  all normal vectors of arcs near $\tau$ point from $R$
to other regions. Such a region $R$ is called 
the {\it source} region of $\tau$.
Take an oriented arc $\ell$ from  the region at infinity to 
a region 
$H$ such that $\ell$  intersects the 
arcs (missing crossing points)  of 
the diagram transversely in  finitely many points.
The {\it Alexander numbering} 
${\cal L}(H)$ of a region $H$ is the number 
of such intersections counted with signs. 
If the orientation of $\ell$ agrees with the normal vector of the arc 
where $\ell$ intersects, then the intersection is positive 
(counted as $1$), otherwise negative (counted as $-1$).
The {\it Alexander numbering} ${\cal L}(\tau)$ of a crossing $\tau$ 
is defined to be 
${\cal L}(R) $ 
where $R$ is the source region of $\tau$. 



For a coloring ${\cal C}$ of $K$ by $X$, the {\it twisted $($Boltzmann$)$ weight}
\index{twisted (Boltzmann) weight}
at a triple point $\tau$ is defined by 
$B_T(\tau, {\cal C})=
[\phi(x,y)^{\epsilon (\tau)} ]^{t^{-{\cal L}(\tau)}} , $ 
where $\phi$ is a twisted $2$-cocycle, 
 and  
$\epsilon(\tau)$ is the sign of the crossing  $\tau$. 
This is in multiplicative notation, so that the action of $t$ 
on $a \in A$ is written by $a^t$. 
The colors $x,y$ assigned near $\tau$ are chosen in
the same manner as in the ordinary $2$-cocycle invariant,
and ${\cal L}(\tau)$ is the Alexander numbering of $\tau$. 
The {\it twisted quandle $2$-cocycle invariant}
is the state-sum
$$
\Phi_T (K) = \sum_{{\cal C}}  \prod_{\tau}  B_T( \tau, {\cal C}).
$$  
The value of the weight  $B_T( \tau, {\cal C})$
is in the coefficient group $A$ written multiplicatively, and 
the value of the state-sum is 
again 
in  the group ring ${\bf Z}[A]$.
It was proved  \cite{CES02}
by checking Reidemeister  moves that
 $\Phi_T (K)$ is an invariant of knots.




\subsection{Invariants for knotted surfaces}

A diagram of knotted surfaces in $4$-space is defined in a similar manner
as classical knot diagrams. A projection of a surface embedded in $4$-space to
$3$-space has, locally, embedded sheets, 
double point curves, isolated branch points and triple points.
According to crossing information along the double curves, under-sheets are broken.
See \cite{CKS00,CarSai98} for details.

We specify the orientation of a knotted surface diagram 
by orientation normal vectors to broken sheets. 
A {\it coloring} 
for a knotted surface diagram $D$ by a quandle $X$
is an assignment of an element (called a color) of a quandle $X$ 
to each broken sheet such that 
$p*q=r$ holds at every double point, 
where $p$ (or $r$, respectively) 
is the color of the under-sheet 
behind (or in front of) the over-sheet 
with the color $q$, 
where the normal of the over-sheet points from the under-sheet behind
it to the front. 
The pair $(p,q)\in X\times X$ is called 
the {\it color of the double point}. 
The property whether a knotted surface $K$ 
admits a non-trivial coloring or not 
is independent of 
the particular choice of a diagram of $K$. 
The coloring rule is depicted in Fig.~\ref{triplept}.


\begin{figure}[htb]
\begin{center}
\mbox{
\epsfxsize=3in
\epsfbox{triplept.eps} 
}
\end{center}
\caption{Colors at double curves and $3$-cocycle at a triple point }
\label{triplept} 
\end{figure}



For a fixed finite quandle $X$ and 
a $3$-cocycle $\theta$, we define a  
knotted surface invariant as follows: 
 Let $K$ be 
a knotted surface diagram and  
let ${\cal C}: {\cal R} \rightarrow X$ be a coloring 
of $K$, where ${\cal R}$ is the set of sheets of $K$.  
Recall that a function $\theta:X\times X\times X \rightarrow A$
 is a quandle $3$-cocycle 
if, for any $p,q,r, s \in X$, 
\begin{eqnarray*}
\lefteqn{
\theta (p,r,s)  +\theta(p*r,q*r,s) +\theta(p,q,r) } \\
  &=& \theta(p*q,r, s) + \theta(p,q,s) + \theta(p*s,q*s,r*s), \end{eqnarray*}
and 
$\theta(p,p,q)  =  0$, 
$\theta(p,q,q) = 0$.  
The Boltzman weight at a triple point $\tau$ is defined by 
$ B(\tau, {\cal C}) = \theta(x,y,z)^{\epsilon (\tau)}, $  
where $\epsilon(\tau)$ is the sign of the  triple point $\tau$ 
(see  \cite{CarSai98}),  
and  $x, y, z$ are the colors of the bottom, middle, and 
top sheets, respectively, 
 around the source region of $\tau$.  
The {\it source region} 
 is the region from which 
normals of top, middle and bottom sheets point.   
In the right of Fig.~\ref{triplept}, 
the triple point $\tau$ is positive, 
so that $B(\tau, {\cal C})=\theta(p,q,r)$.  
The state-sum is defined by 
$$\Phi(K)=  
\sum_{{\cal C}}  \prod_{\tau}  B( \tau, {\cal C}). $$
it was shown in \cite{CJKLS} that 
$\Phi(K)$ is an invariant, 
called  the {\it $($quandle$)$ cocycle invariant}
of knotted surfaces. 

The value of the state-sum invariant depends
only on the cohomology class represented by the defining cocycle
\cite{CJKLS}
for both  classical knots and knotted surfaces. 



\section{Applications}

\subsection{Chirality of knots and graphs}

A knot or a spatial graph $K$ is called {\it achiral} or {\it amphichaeral}
if it is equivalent to its mirror image $K^*$. 
Otherwise $K$ is called {\it chiral}.
It is well known, for example, that trefoil ($3_1$) is chiral and 
the figure-eight knot ($4_1$) is achiral.

In \cite{RouSand00*}, a new proof of the chirality of  trefoil 
was given using quandle homology theory.
This can be detected by the difference in values of  the $3$-cocycle invariant with
the dihedral quandle $R_3$.

In \cite{ShinGraph}, infinitely many  spatial graphs called Suzuki's $\theta_n$-curves
were proven to be chiral using $3$-cocycle invariants of dihedral quandles $R_p$.
Suzuki's $\theta_n$-curves has the property that any proper subgraph is 
trivially embedded, but itself is non-trivial, so that any method that uses 
proper subgraphs can not be applied effectively.


\subsection{Colored chirality of knots}

For a given knot diagram $K$ with a (Fox) $n$-coloring ${\mathcal C}$,
its mirror $K^*$ has the $n$-coloring ${\mathcal C}^*$ 
(called the {\it mirror coloring}) that is the mirror image of 
${\mathcal C}$.
Specifically, let ${\mathcal A}$ be the set of (over-)arcs of a diagram $K$,
then the mirror image $K^*$ has the set of arcs ${\mathcal A}^*$ that 
is in a natural bijection with ${\mathcal A}$, such that each arc $a \in {\mathcal A}$
has its mirror $a^* \in {\mathcal A}^*$, and vice-versa 
(for any arc $\alpha \in  {\mathcal A}^*$,
there is a unique arc $a \in {\mathcal A}$ such that $\alpha=a^*$). 
Then ${\mathcal C}^* :  {\mathcal A}^* \rightarrow \Z_n$ is defined by 
${\mathcal C}^* (a^*)={\mathcal C} (a)$. 

  
\begin{figure}[htb]
\begin{center}
\mbox{
\epsfxsize=2in
\epsfbox{fig8colored.eps} 
}
\end{center}
\caption{Is figure-eight amphicheiral with $5$-colorings?}
\label{fig8colored} 
\end{figure}


 In Fig.~\ref{fig8colored} a $5$-colored figure-eight knot diagram 
and its mirror  with its mirror coloring are depicted.
It is, then, natural to ask if an $n$-colored knot diagram 
(a pair $(K, {\mathcal C})$ of a diagram $K$ and a coloring ${\mathcal C}$)
is equivalent 
to its mirror with its mirror coloring $(K^*, {\mathcal C}^*)$. If this is the case,
we call the colored diagram  $(K, {\mathcal C})$
{\it amphicheiral (or achiral) with (respect to) the $n$-coloring ${\mathcal C}$}.
Otherwise we say a diagram is {\it chiral with (respect to) the $n$-coloring}.

The answer to the above question is NO, and it is seen from the contributions
of the $3$-cocycle invariant with the dihedral quandle $R_5$
are distinct between the left and right of Fig.~\ref{fig8colored}.
Many other achiral knots have colorings with which they become chiral.

\subsection{Minimal number of type III Reidemeister moves}

The idea of quandle cocycle invariants was used to determine 
the minimal number of Reidemeister moves needed to move one diagram
to another diagram of the same knot for some examples
in \cite{CESS}.


\subsection{Tangle embeddings}

 The number of  Fox colorings,
 as well as branched coverings and quantum invariants, 
  was used as
obstructions to tangle embeddings
(see, for example,  \cite{PSW}). 
Cocycle invariants can be used as obstructions as well. 
We illustrate the method by an example.
Consider the weight sum (the sum of cocycle values) of a tangle $T$
for a coloring such that 
 the boundary points of the tangle are required to have the same color.
 If the tangle $T$ is a subtangle of a link $L$, such a coloring 
extends to a coloring of $L$ by choosing the color of the boundary points of $T$ 
for all arcs of $L$ outside of $T$. 
Then the weight sum for $L$ is the same as that of $T$ for these colorings.
Hence the cocycle invariant of $L$ must have the same contribution.

\begin{figure}[h]
\begin{center}
\mbox{
\epsfxsize=1.5in
\epsfbox{tangle.eps} 
}
\end{center}
\caption{A colored tangle}
\label{tangle} 
\end{figure}

For example, see the colored tangle in Fig.~\ref{tangle},
which  has the weight sum $2 \in \Z_3$.
Hence this does not embed in  any link that does not have $2$
in its cocycle invariant with $R_3$.

\subsection{Non-invertibility of knotted surfaces}


A (classical or higher dimensional) knot 
is called {\it invertible} (or {\it reversible}) if it is equivalent to its
orientation reversed counterpart, with the orientation of the 
ambient space  fixed. 
A knot is called {\it non-invertible} (or {\it irreversible})
if it is not invertible.

The cocycle invariant provided a diagrammatic method
(by a state-sum)  of detecting non-invertibility 
of knotted surfaces. In 
\cite{CJKLS}, it was proved that 
the 
$2$-twist spun trefoil $K$ and its orientation-reversed counterpart $-K$
have distinct cocycle invariants
$(6 +12u$ and $6+12 u^2)$ 
with a cocycle of the dihedral quandle $R_3$, and therefore,  
$K$ is non-invertible.
This result is extended to many new examples in  \cite{AS,CEGS,Iwakiri}.


The higher genus surfaces (called stabilized surfaces)
obtained from $K$ by adding an arbitrary
number of trivial $1$-handles are also non-invertible, since 
such handle additions do not alter the cocycle invariant.
This property that the conclusion applied right away to stabilized surfaces 
is characteristic and advantage of the cocycle invariant.



\subsection{Minimal number of triple points on knotted surface projections}

One of the fundamental problems in classical knot theory is 
to determine the crossing number of a given knot.
An analog of the crossing number for knotted surfaces is the minimal number of 
triple points on projections, called the  {\it triple point number}.
The quandle cocycle invariants have been used to obtain 
 lower bounds for the triple point number.
 
  In \cite{SatShi04}, the triple point number of the $2$-twist spun trefoil 
was determined to be $4$  using cocycle invariants.
It was the first time that the triple point number was determined 
for a specific knot (earlier, only inequalities were known).
See \cite{Hata04,SatShi01b*} for further results on triple point numbers.



\subsection{Minimal number of broken sheets in knotted surface diagrams}

For a classical knot, the number of crossings and the number of arcs coincide.
For knotted surfaces, then, another analog of the crossing number is
the minimal number of  broken sheets  that are needed 
to form a diagram of a given knotted surface.

In \cite{SaiSat}, a quandle defined by a $2$-cocycle was used 
to show that the minimal number of sheets for the untwisted spun trefoil 
is four. 


\subsection{Ribbon concordance}

Let $F_0$ and $F_1$ be knotted surfaces of the same genus. 
We say that {\it $F_1$ is ribbon concordant to $F_0$} 
if there is a concordance $C$ (a properly embedded orientable submanifold
diffeomorphic to $F_0\times I$)
in ${\R}^4\times [0,1]$ between 
$F_1\subset{\R}^4\times\{1\}$ and 
$F_0\subset{\R}^4\times\{0\}$ 
such that the restriction to $C$ 
of the projection ${\R}^4\times[0,1]\rightarrow[0,1]$ 
is a Morse function with critical points 
of index $0$ and $1$ only. 
We write $F_1\geq F_0$. 


\begin{figure}[htb]
\begin{center}
\mbox{
\epsfxsize=2in
\epsfbox{ribbonidea.eps} 
}
\end{center}
\caption{Ribbon concordance}
\label{ribbonidea} 
\end{figure}


Note that if $F_1\geq F_0$, 
then there is a set of $n$ $1$-handles 
on a split union of $F_0$ and $n$ trivial sphere-knots, 
for some $n\geq 0$, 
such that $F_1$ is obtained by surgeries 
along these handles (Fig.~\ref{ribbonidea}). 
Ribbon concordance was first defined in \cite{Gor81}.
It is defined in general for knots in any dimension.
In \cite{CSSribbon}, quandle cocycle invariants were used as obstructions
to ribbon concordance for surfaces, that is, to give examples $F_0$ and $F_1$ that are 
not related by ribbon concordance.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{thebibliography}{[CarSai99]}%{aaaaaaaaa}
\setlength{\itemsep}{-3pt} %%%
\def\by#1{#1,}
\def\paper#1{{\em #1},}
\def\jour#1{#1}
\def\vol#1{{\bf #1}}
\def\yr#1{(#1)}
\def\pages#1{#1}
\def\book#1{{\em #1},}
\def\publ#1{#1}
\def\publaddr#1{#1}
\def\inbook#1{in ``#1''}
\def\bookinfo#1{#1}
\def\comma{,}
\def\endref{.}

\bibitem[Ad94]{Ad94}
\by{C. C. Adams}
\book{The knot book, An elementary introduction to the mathematical theory of knots.}
\publ{W. H. Freeman and Company, New York}
\yr{1994}\endref



%\bibitem[Alex28]{Alex28}
%\by{J. W. Alexander}
%\paper{Topological invariants of knots and links}
%\jour{Trans. Amer. Math. Soc.}
%\vol{30} \yr{1928} \pages{275--306}\endref

\bibitem[AG02]{AG02*} 
\by{N. Andruskiewitsch and M. Gra\~{n}a}
\paper{From racks to pointed Hopf algebras}
\jour{Adv. in Math.}
\vol{178} \yr{2003} \pages{177--243}\endref

\bibitem[AS03*]{AS}
\by{M. Asami, S. Satoh}
\paper{An infinite family of non-invertible surfaces in $4$-space}
\jour{to appear in: Bull. London Math. Soc.}\endref


%\bibitem[BL98]{BL98}
%\by{J. C. Baez and L. Langford}
%\paper{$2$-tangles}
%\jour{Lett. Math. Phys.}
%\vol{43} \yr{1998} \pages{187--197}\endref

\bibitem[Br88]{Br88} 
\by{E. Brieskorn}
\paper{Automorphic sets and singularities}
\inbook{Braids (Santa Cruz, CA, 1986),}
\bookinfo{Contemp. Math.} \vol{78}, 
\publ{Amer. Math. Soc.}
 \yr{1988} \pages{45--115}\endref


\bibitem[Brown82]{Brown82} 
\by{K. S. Brown}
\book{Cohomology of Groups}
\bookinfo{Graduate Texts in Mathematics} \vol{87},
 \publ{Springer-Verlag} \yr{1982}\endref


\bibitem[CEGS05]{CEGS}  
\by{J.S. Carter;  M. Elhamdadi;  M. Gra\~{n}a; M. Saito}
\paper{Cocycle knot invariants from quandle modules and generalized 
quandle cohomology}
\jour{to appear in  Osaka J. Math.}\endref
 

\bibitem[CENS01]{CENS01*}
\by{J. S. Carter, M.  Elhamdadi,  M. A. Nikiforou, and M. Saito}
\paper{Extensions of quandles and cocycle knot invariants}  
\jour{J. Knot Theory Ramifications} \vol{12} \yr{2001} \pages{725--738}\endref


\bibitem[CES02]{CES02}
\by{J. S. Carter, M.   Elhamdadi and M.   Saito}
\paper{Twisted Quandle homology theory and cocycle knot invariants} 
\jour{Algebraic and Geometric Topology}
\vol{2} \yr{2002} \pages{95--135}\endref 


 
\bibitem[CESS05*]{CESS} 
\by{J.S. Carter; M. Elhamdadi; M. Saito; S. Satoh}
\paper{A lower bound for the number of Reidemeister moves of type III}
\jour{Preprint, arXive.math.GT/0501490}\endref

\bibitem[CHNS02]{CHNS02*}
\by{J. S. Carter, A. Harris, M. A.  Nikiforou and M. Saito}
\paper{Cocycle knot invariants, quandle extensions, and Alexander matrices}
\jour{Suurikaisekikenkyusho Koukyuroku,
(Seminar note at RIMS, Kyoto)}
\vol{1272} \yr{2002} \pages{12--35},
\jour{available at
 http://xxx.lanl.gov/math/abs/GT0204113}\endref

 
%\bibitem[CJKLS99]{CJKLS99}
%\by{J. S. Carter, D. Jelsovsky, S. Kamada, L. Langford and M. Saito}
%\paper{State-sum invariants of knotted curves and surfaces from quandle 
%cohomology}
%\jour{Electron. Res. Announc. Amer. Math. Soc.}
%\vol{5} \yr{1999} \pages{146--156}\endref


\bibitem[CJKLS03]{CJKLS}
\by{J. S. Carter, D. Jelsovsky, S. Kamada, L. Langford and M. Saito}
\paper{Quandle cohomology and state-sum invariants of knotted curves and 
surfaces} \jour{Trans. Amer. Math. Soc.}
\vol{355} \yr{2003} \pages{3947--3989}\endref


\bibitem[CJKS01a]{CJKS01a}
\by{J. S. Carter, D. Jelsovsky, S. Kamada and M. Saito}
\paper{Computations of quandle cocycle invariants of knotted curves
and surfaces}
\jour{Adv. in Math.}
\vol{157} \yr{2001} \pages{36--94}\endref


\bibitem[CJKS01b]{CJKS01b}
\by{J. S. Carter, D. Jelsovsky, S. Kamada and M. Saito}
\paper {Quandle homology groups, their Betti numbers, and virtual knots}
\jour{J. Pure Appl. Algebra}
\vol{157} \yr{2001} \pages{135--155}\endref

\bibitem[CJKS01c]{CJKS01c}
\by{J. S. Carter, D. Jelsovsky, S. Kamada and M. Saito}
\paper{Shifting homomorphisms in quandle cohomology and skeins
of cocycle knot invariants}
\jour{J. Knot Theory Ramifications}
\vol{10} \yr{2001} \pages{579--596}\endref




\bibitem[CKS00]{CKS00}  
\by{J. S. Carter, S. Kamada and M. Saito}
\book{Surfaces in 4-space}
\bookinfo{Encyclopedia of Mathematical Sciences}\vol{142}, 
\publ{Springer Verlag} \yr{2000}\endref 



\bibitem[CKS01]{CKS01a}  
\by{J. S. Carter, S. Kamada and M. Saito}
\paper{Geometric interpretations of quandle homology}
\jour{J. Knot Theory Ramifications}
\vol{10} \yr{2001} \pages{345--386}\endref

\bibitem[CKS03]{CKS01b}  
\by{J. S. Carter, S. Kamada and M. Saito}
\paper{Diagrammatic computations for quandles and
cocycle knot invariants}
\jour{Contemp. Math.}\vol{318}\yr{2003}
\publ{Amer. Math. Soc.} \pages{51--74}\endref 



\bibitem[CarSai98a]{CarSai98}
\by{J. S. Carter and M. Saito}
\book{Knotted surfaces and their diagrams}
\bookinfo{Surveys and monographs} \vol{55}, \publ{Amer. Math. Soc.}
\yr{1998}\endref


%\bibitem[CarSai98b]{CarSai98b}
%\by{J. S. Carter and M. Saito}
%\paper{Surfaces in $3$-space that do not lift to embeddings in
%$4$-space}
%\jour{Knot Theory (Warsaw, 1995), Banach Center Publ.}
%\vol{42} \publ{Polish Acad. Sci., Warsaw}
%\yr{1998}
%5\pages{29--47}\endref

\bibitem[CarSai03]{CarSai01*}
\by{J. S. Carter and M. Saito}
\paper{Quandle Homology Theory and Cocycle Knot Invariants}
\inbook{Proceedings of Georgia Topology Conference (2001)} 
\bookinfo{Proceedings of Symposia in Pure Math.} \vol{71} \yr{2003}
\pages{249--269}\endref  

 
\bibitem[CSS03*]{CSSribbon}
\by{J. S. Carter;  M. Saito; S. Satoh}
\paper{Ribbon concordance of surface-knots via quandle cocycle invariants}
\jour{J.  Australian Math. Soc., to appear,
 available at
 http://xxx.lanl.gov/abs/math.GT/0309150}
\endref


\bibitem[Deh00]{Deh}
�\by{P. Dehornoy}
\book{Braids and self-distributivity}
\publ{Progress in Mathematics, 192. Birkh�user Verlag, Basel}
 \yr{2000}\endref

%\bibitem[DW90]{DW}
%\by{R. Dijkgraaf; E. Witten}
%\paper{Topological
%gauge theories and group cohomology}
%\jour{Comm. Math. Phys.}
% \vol{129} \yr{1990} \pages{393--429}\endref




\bibitem[Eis02]{Eis02*} 
\by{M. Eisermann} 
\paper{Homological characterization of the unknot}
\jour{J. Pure Appl. Algebra} \vol{177} \yr{2003} \pages{131--157}\endref


\bibitem[EG03]{EG02*} 
\by{P. Etingof and M. Gra\~ na}
\paper{On rack cohomology}
\jour{J. Pure Appl. Algebra} \vol{177} \yr{2003} \pages{49--59}\endref



\bibitem[FeRou92]{FennRou92}   
\by{R. Fenn and C. Rourke}
\paper{Racks and links in codimension two}
\jour{J. Knot Theory  Ramifications} 
\vol{1} \yr{1992} \pages{343--406}\endref  



   \bibitem[FRS95]{FRS95}
\by{R. Fenn, C. Rourke, and B. Sanderson}
\paper{Trunks and classifying spaces}
\jour{Appl. Categ. Structures}
\vol{3} \yr{1995} \pages{321--356}\endref  


\bibitem[FRS*]{FRS} \by{R. Fenn; C. Rourke; B. Sanderson}
\paper{James bundles and applications}
\jour{Preprint}, available at:
http://www.maths.warwick.ac.uk/${}^{\sim}$bjs/\endref



\bibitem[Fl95]{Fl95} 
\by{J. Flower}
\paper{Cyclic Bordism and Rack Spaces} 
\jour{Ph.D.  Dissertation, Warwick}  \yr{1995}\endref  



\bibitem[Fox61a]{Fox61a}
\by{R. H. Fox} 
\paper{A quick trip through knot theory}
\inbook{Topology of 3-manifolds and related topics
(Georgia, 1961),} 
\publ{Prentice-Hall}
\yr{1962} \pages{120--167}\endref


%\bibitem[Fox61b]{Fox61b}
%\by{R. H. Fox} 
%\paper{Some problems in knot theory}
%\inbook{Topology of 3-manifolds and related topics (Georgia, 1961),}
%\bookinfo{Prentice-Hall}
%\yr{1962} \pages{168--176}\endref

\bibitem[GK03]{GhoKep} 
\by{S. Ghoneim; T. Kepka}
\paper{Left selfdistributive rings generated by idempotents}
\jour{Acta math. Hungar.}
\vol{102 (1-2)} \yr{2003} \pages{21--31}\endref


\bibitem[Gor81]{Gor81} 
\by{C. McA. Gordon}
\paper{Ribbon concordance of knots in the $3$-sphere}
\jour{Math. Ann.} 
\vol{257} \yr{1981} 
\pages{157--170}\endref



\bibitem[Gra02a]{Gra02a*}
\by{M. Gra\~ na} 
 \paper{Quandle knot invariants are quantum knot invariants}
 \jour{J.  Knot Theory Ramifications}
\vol{11} \yr{2002} \pages{673--681}\endref


\bibitem[Gra02b*]{Gra02b*}
\by{M. Gra\~ na} 
 \paper{Indecomposable racks of order $p^2$}
 \jour{Preprint, http://xxx.lanl.gov/abs/math.QA/0203157}\endref


\bibitem[GraPre02*]{GraPre02*}
\by{M. Gra\~ na and A. Preygel} 
 \paper{Computation of quandle cocycle invariants of knots 
using certain simple quandles}
 \jour{Talk given at the $982$nd AMS meeting 
in Orlando, FL, Nov. 9-10, 2002}\endref



\bibitem[Gre97]{Gre97} 
\by{M. T. Greene}
\paper{Some Results in Geometric Topology and Geometry} 
\jour{Ph.D. Dissertation, Warwick} \yr{1997}\endref




\bibitem[Har03]{Har03*} 
\by{A. Harris}
\paper{Quandle Colorings of Classical Knots and Knotted Surfaces}
\jour{Master's Thesis, University of South Alabama} \yr{2003}\endref



\bibitem[Hata04]{Hata04} 
\by{E. Hatakenaka}
\paper{An estimate of the triple point numbers of surface-knots
by quandle cocycle invariants}
\jour{Topology  Appl.}
\vol{139}  \yr{2004}  \pages{129--144}\endref



\bibitem[Ino01]{Ino01}
\by{A. Inoue}
\paper{Quandle homomorphisms of knot quandles to Alexander quandles}
\jour{J. Knot Theory Ramifications} 
\vol{10} \yr{2001} \pages{813--821}\endref

\bibitem[Iwa04]{Iwakiri}
\by{M. Iwakiri}
\jour{Calculation of dihedral quandle cocycle invariants of twist spun
$2$-bridge knots}
\inbook{Proceedings of the first East Asian School of Knots, Links, and Related Topics
(Feb. 16--20, 2004)},
\pages{85--94},
\jour{available at: http://knot.kaist.ac.kr/2004/proceedings.php}\endref



\bibitem[Joy82]{Joy82} 
\by{D. Joyce}
\paper{A classifying invariant of knots, the knot quandle}
\jour{J. Pure Appl. Alg.}
\vol{23} \yr{1982} \pages{37--65}\endref 


\bibitem[Kauff91]{Kauff91}
\by{L. H. Kauffman}
\book{Knots and Physics}
\bookinfo{Series on Knots and Everything} \vol{1}, 
\publ{World Sci. Publishing} \yr{1991}\endref



\bibitem[Kawa90a]{Kawa90a}
\by{A. Kawauchi}
\paper{The first Alexander modules of surfaces in 4-sphere}
\inbook{Algebra and Topology}
\bookinfo{(Taejon, 1990), Proc. KAIST Math. Workshop}
\vol{5}, KAIST, Taejon,
Korea
\yr{1990} \pages{81--89}\endref



\bibitem[Lith85]{Lith85}
\by{R. A. Litherland}
\paper{ Symmetries of twist-spun knots}
\inbook{Knot theory and manifolds (Vancouver, B.C.,
1983),}
\bookinfo{Lecture Notes in Math.} \vol{1144}, \publ{Springer-Verlag}
\yr{1985} \pages{97--107}\endref

\bibitem[Lith02*]{Lith02*}
\by{R. A. Litherland}
\paper{Quadratic quandles and their link invariants}
\jour{Preprint,
http://xxx.lanl.gov/abs/math.GT/0207099}\endref



\bibitem[LN03]{LN03} \by{R. A. Litherland and S. Nelson} 
\paper{The Betti numbers of some finite racks}
\jour{J. Pure Appl. Algebra} \vol{178} \yr{2003} \pages{187--202}\endref

\bibitem[Liv93]{Liv93}
\by{C. Livingston}
\book{Knot theory}
\bookinfo{Carus Mathematical Monographs}
\vol{24}
\publ{Mathematical Association of America, Washington, DC}
\yr{1993}\endref


\bibitem[Lop03]{Lop03} \by{P. Lopes} 
\paper{Quandles at finite temperatures I}
\jour{J. Knot Theory Ramifications},\vol{12}\yr{2003}\pages{159--186}\endref




\bibitem[Mat82]{Mat82} 
\by{S. Matveev}
\paper{Distributive groupoids in knot theory}
\jour{(in Russian), Math. USSR-Sbornik} \vol{47} \yr{1982} \pages{73--83}\endref

        
\bibitem[Mochi03]{Mochi} 
\by{T. Mochizuki}
\paper{Some calculations of cohomology groups of finite
Alexander quandles}
\jour{J. Pure Appl. Algebra} \vol{179} \yr{2003} \pages{287--330}\endref

\bibitem[Mochi05]{Mochi05}
\by{T. Mochizuki}
\paper{The 3-cocycles of the Alexander quandles $ F\sb  q[T]/(T-\omega)$}
  \jour{Algebr. Geom. Topol.}  \vol{5}  \yr{2005} \pages{183--205}\endref



\bibitem[Mura96]{Mura96}
�\by{K. Murasugi}
\book{Knot theory and its applications}
\bookinfo{Translated from the 1993 Japanese original by Bohdan Kurpita.}
\publ{Birkh�user Boston, Inc., Boston, MA}
\yr{1996}\endref

\bibitem[Nel03]{Nel03} 
\by{S. Nelson}
\paper{Classification of Finite Alexander Quandles}
\jour{Proceedings of the Spring Topology and Dynamical Systems  Conference.  Topology Proc.}  \vol{27}  \yr{2003} \pages{245--258}\endref


\bibitem[PSW04*]{PSW} 
\by{J. Przytycki; D. Silver; S. Williams}
\paper{$3$-manifolds, tangles, and persistent invariants}
\jour{Preprint, available at:
http://xxx.lanl.gov/abs/math.GT0405465}\endref



%\bibitem[Rolf76]{Rolf76} 
%\by{D. Rolfsen}
%\book{Knots and Links}  
%\publ{Publish or Perish Press (Berkley)}
%\yr{1976}\endref



\bibitem[Rose98]{Rose98}
\by{D. Roseman}
\paper{Reidemeister-type moves for surfaces in four-dimensional space}
 \inbook{Knot theory (Warsaw, 1995),}
\bookinfo{Banach Center Publ., Polish Acad. Sci., Warsaw,} 
\vol{42}  \yr{1998} \pages{347--380}\endref


\bibitem[RouSand00*]{RouSand00*}%03*
\by{C. Rourke and B. Sanderson}
\paper{A new classification of links and some calculations using it}
\jour{Preprint, http://xxx.lanl.gov/abs/math.GT/0006062}\endref


\bibitem[Rub83]{Rub83} 
\by{D. Ruberman}
\paper{Doubly slice knots and the Casson-Gordon invariants}
  \jour{Trans. Amer. Math. Soc.} 
\vol{279} \yr{1983} \pages{569--588}\endref



%\bibitem[Rub84]{Rub84}
%\by{D. Ruberman}
%\paper {Concordance of links in $S^4$}
%\inbook{Four-manifold theory (Durham,1982),}
%\bookinfo{Contemp. Math.} \vol{35}, 
%\publ{Amer. Math. Soc.}
%\yr{1984} \pages{481--483}\endref


%\bibitem[Sat00a]{Sat00a} 
%\by{S. Satoh} 
%\paper {On non-orientable surfaces in $4$-space which are projected 
%with at most one triple point}
%\jour{Proc. Amer. Math. Soc.}
%\vol{128} \yr{2000} \pages{2789--2793}\endref

%\bibitem[Sat01a]{Sat01}
%\by{S. Satoh} 
%\paper {Minimal triple numbers of some non-orientable surface-links}
%\jour{Pacific J. Math.}
%\vol{197} \yr{2001} \pages{213--221}\endref




\bibitem[Sat02]{Sat01a*}
\by{S. Satoh} 
\paper{Surface diagrams of twist-spun $2$-knots}
\inbook{Knots 2000 Korea, Vol. 1
(Yongpyong)}
\jour{J. Knot Theory Ramifications} \vol{11} \yr{2002}
\pages{413--430}\endref 

\bibitem[Sat04*]{ShinGraph}
\by{S. Satoh}
\paper{On chirality of Suzuki's $\theta_n$-curves}
\jour{Preprint}\endref



\bibitem[SaiSat03*]{SaiSat}
\by{M. Saito; S. Satoh}
\paper{The spun trefoil needs four broken sheets}
\jour{J. Knot Theory Remifications} \pages{to appear}\endref



\bibitem[SatShi01b*]{SatShi01b*}
\by{S. Satoh and A. Shima}
\paper{Triple point numbers of surface-knots and colorings by quandles}
\jour{Preprint} \yr{2001}\endref

%%
\bibitem[SatShi04]{SatShi04}
\by{S. Satoh and A. Shima}
\paper{The $2$-twist-spun trefoil has the triple point number four} 
\jour{Trans. Amer. Mat. Soc.} \vol{356} \yr{2004} \pages{1007--1024}\endref


%\bibitem[Shima00]{Shima00*}
%\by{A. Shima}
%\paper{Colorings and Alexander polynomials for ribbon 2-knots}
%\inbook{Knots 2000 Korea, Vol. 1 (Yongpyong),} 
%\jour{J. Knot Theory Ramifications} \vol{11} \yr{2002} \pages{403--412}\endref 

%\bibitem[Shima01*]{Shima01*}
%\by{A. Shima}
%\paper{Knotted $2$-spheres whose projections into $3$-space 
%contain at most two triple points}
%\jour{Preprint} \yr{2001}\endref


\bibitem[Tana*]{Tanaka}
\by{K. Tanaka}
\paper{On surface-links represented by diagrams with two or three triple points}
\jour{Preprint}\endref




\bibitem[Taka42]{Taka42} 
\by{M. Takasaki}
\paper{Abstraction of symmetric transformations}
\jour{(in Japanese), Tohoku Math. J.}
\vol{49} \yr{1942/43} \pages{145--207}\endref

%\bibitem[Wak92]{Wak92}
%\by{M. Wakui}
%\paper{On Dijkgraaf-Witten invariants for $3$-manifolds}
%\jour{Osaka J. Math.} \vol{29} \yr{1992} \pages{675--696}\endref


\bibitem[Win84]{Win84}
\by{S. Winker}
\paper{Quandles, knots invariants and the $N$-fold branched cover}
\jour{Ph.D. Thesis, University of Illinois at Chicago}
\yr{1984}\endref



%\bibitem [Yoshi94]{Yoshi94} 
%\by{K. Yoshikawa}
%\paper{An enumeration of surfaces in four-space}
%\jour{Osaka J. Math.}
%\vol{31}  \yr{1994} \pages{497--522}\endref


\bibitem[Ze65]{Ze65}
\by{E. C. Zeeman}
\paper {Twisting spun knots}
\jour{Trans. Amer. Math. Soc.}
\vol{115} \yr{1965} \pages{471--495}\endref


\end{thebibliography}

\end{document}


