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% Author Package file for use with AMS-LaTeX 1.2
\dateposted{December 9, 1999}
\PII{S 1079-6762(99)00073-6}

\newcommand{\lk}{\mbox{$\ell k$}} %%%%% new command 

\copyrightinfo{1999}%            % copyright year
  {American Mathematical Society}% copyright holder






{State-sum invariants of knotted curves and surfaces
 from quandle cohomology}

\author[Carter]{J. Scott Carter}
\address{Department of Mathematics, University of South Alabama,
Mobile, AL 36688 }

\author[Jelsovsky]{Daniel Jelsovsky}
\address{Department of Mathematics, University of South Florida,
 Tampa, FL 33620 }

\author[Kamada]{Seiichi Kamada}
\address{Department of Mathematics,  Osaka City University,
Osaka 558-8585, JAPAN}
\curraddr{Department of Mathematics, University of South Alabama,
Mobile, AL 36688 }
\thanks{The third author was supported  by a Fellowship from the 
Japan Society for the Promotion of Science.}

\author[Langford]{Laurel Langford}
\address{Department of Mathematics, University of Wisconsin at River Falls,
River Falls, WI 54022}
\email{laurel.langford@uwrf.edu }

\author[Saito]{Masahico Saito }
\address{Department of Mathematics, University of South Florida,
 Tampa, FL 33620}

\commby{Walter Neumann}

\subjclass{Primary 57M25, 57Q45; Secondary 55N99, 18G99}

\date{May 28, 1999}

\keywords{Knots, knotted surfaces, quandle cohomology, state-sum invariants}

State-sum invariants 
for classical knots and knotted surfaces in $4$-space are developed  
via the  cohomology theory of  quandles.
Cohomology groups of quandles are computed to evaluate the invariants.
Some twist spun torus knots are shown to be 
noninvertible using the invariants.

A cohomology theory for racks (self-distributive groupoids, defined below)
was defined 
and the general framework for defining invariants 
of codimension 2 embeddings was  outlined in  \cite{FRS1} and \cite{FRS2}
algebro-topological viewpoint. 
The present paper announces state-sum invariants, 
defined diagrammatically using knot diagrams
and quandle cocycles, for both 
classical knots in $3$-space and knotted surfaces in $4$-space.
The invariant is used to give a proof that 
$2$-twist spun torus
knots are noninvertible (not equivalent to 
the same knot 
orientation reversed). 
Details of proofs and computations 
can be found in \cite{CJKLS} and \cite{CJKS}. 

This cocycle invariant can be seen as 
an analog of 
 the Dijkgraaf-Witten 
invariants for $3$-manifolds \cite{DW} in that colorings and 
cocycles are used to define state-sum invariants. 
Our inspiration for the definition of these invariants is found
 in Neuchl's paper 
\cite{Neu} where related cocycles (in  %%%%space added
quantum doubles of finite groups) are used to 
show  representations of a Hopf category
form a braided monoidal 2-category.
Our definition was derived from 
an attempt to construct a $2$-functor from 
the braided $2$-category of knotted surfaces 
 as summarized in \cite{BL1} 
and presented in detail in \cite{BL},
to another $2$-category constructed from quandles.

The noninvertibility for certain classical knots 
been presumed 
since the  1920's  
but proved first 
by Trotter \cite{Trotter},
and subsequently by 
Kawauchi \cite{Kawa}
and Hartley \cite{Hart} 
(see also \cite{Kawabook}). 
Fox \cite{FoxTrip} presented a  noninvertible knotted sphere 
using Alexander modules,
which, however, fail to detect  the noninvertibility of 
the 2-twist spin trefoil.
D. Ruberman 
informed us
that  Levine pairings
and Casson-Gordon invariants
detect noninvertibility of some twist spun knots \cite{Hill,Ruber}.
Thus relations between these invariants and the state-sum invariants deserve
Furthermore, since state-sums can be  
used to define Jones polynomials \cite{Jones} 
and their generalizations, and quandles encode 
fundamental group information,
relations of the invariants defined herein 
to both quantum and classical knot invariants are expected.

The paper is organized as follows. 
The cohomology 
theory of quandles is defined in Section~\ref{quandsec}.
In Section~\ref{invsec} the invariants are defined.
Summaries of computations of quandle cocycles for some quandles 
and evaluations of the invariants
are presented in  Section~\ref{compsec}.
Properties of the invariant and applications are presented
 in Section~\ref{applsec}.

\section{Quandles, racks, and their cohomology} \label{quandsec}

A \textit{quandle}, $X$, is a set with a binary operation $(a, b) \mapsto a * b$
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 
$a= c*b$.

For any $a,b,c \in X$, we have
$ (a*b)*c=(a*c)*(b*c). $
A \textit{rack} is a set with a binary operation that satisfies 
(II) and (III).

A typical example of a quandle is a group $X=G$ with 
$n$-fold conjugation 
as the quandle operation: $a*b=b^{-n} a b^n$. 
Racks and quandles have been studied in
FR, Joyce, KP} and \cite{Matveev}, for example.
The axioms for a quandle correspond respectively to the 
Reidemeister moves of type I, II, and III
(see  also \cite{FR, KP}). 
Indeed, knot diagrams were one of the motivations 
 to define such an algebraic structure. 

Let $X$ be a rack, 
and let 
 $A$ be an abelian group,  
written additively. 
The cochain group $C^n=C^n(X;A)$ is the 
abelian group 
of functions $f: \text{FA}(X^n) \rightarrow A$
from the free abelian group generated by $n$-tuples of elements 
of $X$ to the 
abelian group $A$. 
The \textit{coboundary homomorphism} 
$ \delta : C^{n} \rightarrow C^{n+1}$
is defined by
\begin{align*} (\delta f)( x_0, \dots, x_n )
&= \sum_{i=1}^{n}  
f( x_0, \dots, \hat{x}_i , \dots, x_n)\\ 
&\quad+\sum_{j=1}^n (-1)^j f( x_0 * x_j , \dots, x_{j-1} * x_j , x_{j+1}, 
\dots, x_n ).
(Note: Neither sum includes a $0\/$th term as these terms cancel.)

A routine calculation shows that 
The cochain group and the boundary homomorphism 
a cochain complex.

The cohomology groups of the above complex 
called \textit{the rack cohomology groups} and 
denoted by $H_{\text{rack}}^n(X, A)$.
Also, the groups of cocycles and coboundaries are denoted by 
and $B_{\text{rack}}^n(X,A)$ respectively. 
Their elements are called $n$-\textit{cocycles} and $n$-\textit{coboundaries}, 
This definition coincides with the 
cohomology theory 
in \cite{FRS1} and \cite{FRS2}.

For applications, 
we are interested in the case 
when $X$ is a quandle, so we will 
intersect the cocycles and coboundaries 
with a subset that  captures axiom (I) and
 its consequences in higher dimensions.
Thus we modify the above definition
as follows.
 $P^n= \{ f \in C^n: f(\vec{x})=0 \text{ for all } \vec{x} \text{ such that }
x_j=x_{j+1} \text{ for some }j\,\}$.
Let $Z^n = Z^n_{\text{rack}} \cap P^n$,  and
$B^n = B^n_{\text{rack}} \cap P^n$. A straightforward calculation gives:
if $f \in P^n$, then 
$\delta f \in P^{n+1}$
if $X$ is a quandle.

 \begin{equation*}H^n_Q(X,A) = H^n(X,A)=
 (P^n \cap Z_{\text{rack}}^n)/ (\delta P^{n-1}).\end{equation*}
This group is called the \textit{quandle cohomology group}. 
The elements $f\in Z^n(X,A)$ are called \textit{quandle $n$-cocycles} or simply 

Throughout this paper we assume that our quandles are finite
and an $n$-cocycle means a quandle $n$-cocycle. 

\caption{Coloring condition and weights  for  crossings.}

\section{Cocycle invariants of knottings} 

\subsection{Cocycle invariants of classical knots}

A \textit{color} (or \textit{coloring}) 
on an oriented  classical knot diagram is a
function ${\mathcal C} : R \rightarrow X$, where $X$ is a fixed 
and $R$ is the set of over-arcs in the diagram,
satisfying the  condition
depicted in
In the figure, a 
crossing with
over-arc, $r$, has color ${\mathcal C}(r)= y \in X$. 
The under-arcs 
$r_1$ and $r_2$ 
are colored 
${\mathcal C}(r_1)= x$ and ${\mathcal C}(r_2)=x*y$,
where $r_1$ is the arc away from which the co-orientation arrow of the over-arc points.
Note that locally the colors do not depend on the 
orientation of the under-arc.
If the pair of the co-orientation of the over-arc and  that of the under-arc
matches the (right-hand) orientation of the plane, then the 
crossing is called \textit{positive}; otherwise it is \textit{negative}.

Let $X$ be a finite quandle. Pick a 
$\phi \in  Z^2(X, A),$ 
and write the coefficient 
group, $A$, multiplicatively. 
Consider a crossing in the diagram.
For each coloring of the diagram, evaluate 
the 2-cocycle on two of the three quandle colors that 
appear near the crossing. One such color is the color on the over-arc and 
is the second argument of the 2-cocycle. 
The other color should be chosen to be the color on 
the under-arc away from which the normal 
arrow points; this is the first argument of the cocycle. 

In Figure~\ref{twocrossings}, 
the two possible oriented and co-oriented 
crossings are depicted. The left is a positive crossing,
and the right is negative.
Let $\tau$ denote a crossing, and ${\mathcal  C}$ denote a coloring.
When the colors of 
the segments are as indicated, the 
\textit{(Boltzmann) weights 
of the crossing},
$B(\tau, {\mathcal C}) = \phi(x,y)^{\epsilon (\tau)}$,
 are as shown. These weights are assignments of cocycle
to the colored crossings where the arguments are as 
defined in the previous paragraph.

The \textit{partition function}, or a \textit{state-sum}, 
is the expression 
\sum_{{\mathcal C}}  \prod_{\tau}  B( \tau, {\mathcal C}).
The product is taken over all crossings of the given diagram,
and the sum is taken over all possible colorings.
The values of the partition function 
are  taken to be in  the group ring $\mathbf{Z}
[A]$, where $A$ is the coefficient 

Reidemeister moves are checked to prove

The partition function is invariant under Reidemeister moves, 
so that it defines an invariant of knots and links.
Thus it will be  denoted by $\Phi (K)$
(or $\Phi_{\phi}(K)$ to specify the $2$-cocycle $\phi$ used).

\caption{The 2-cocycle condition and the Reidemeister type III move.}

\begin{proposition} \label{coblemma3}
If $\Phi_{\phi}$ and $\Phi_{\phi '} $ denote the state-sum invariants 
defined from cohomologous cocycles  $\phi$ and $\phi'$,
then $\Phi_{\phi} =\Phi_{\phi '} $ (so that $\Phi_{\phi} (K)=\Phi_{\phi '}(K)$
 for any link $K$). 
In particular, the state-sum is equal to the number of
colorings of
a given 
knot diagram    
if the $2$-cocycle used for the Boltzmann weight is a coboundary.

\subsection{Cocycle invariants for  knotted surfaces}

First we recall the 
notion of 
knotted surface diagrams. 
See \cite{CSbook} for details and examples. 
Let $f:F \rightarrow \mathbf{R}^4$ denote a smooth embedding of a closed
surface $F$ into 4-dimensional space.
By deforming the map $f$ slightly by an ambient isotopy of $\mathbf{R}^4$
if necessary,         
we may assume that
$p \,\circ f$ is a general position map, 
where  $p: \mathbf{R}^4 \rightarrow \mathbf{R}^3$
denotes the 
projection onto an affine subspace.

Along the double curves, one of the sheets (called 
the \textit{over-sheet}) lies farther than the other (\textit{under-sheet})
with respect to the projection direction.
The \textit{under-sheets}
 are coherently broken in the projection,
and such broken surfaces are called \textit{knotted surface diagrams}.

When the surface is oriented, we take normal vectors $\vec{n}$
to the projection of the surface such that the triple
$(\vec{v}_1, \vec{v}_2, \vec{n})$ matches the orientaion of 3-space,
where $(\vec{v}_1, \vec{v}_2)$ defines the orientation of the surface.
Such normal vectors are defined on the projection at all points other than
the isolated branch points.

\begin{definition} \label{4dcolor} 
A \textit{color} on an oriented  (broken) knotted surface diagram is a
function ${\mathcal C} : R \rightarrow X$, where $X$ is a fixed 
and  where $R$ is the set of regions in the broken surface diagram,
satisfying the following condition at the double point set.  

At a double point curve, two coordinate planes intersect locally.
One is the 
over-sheet $r$, the other is the under-sheet, and the under-sheet is 
broken into two components, say $r_1$ and $r_2$.
 A normal of the over-sheet $r$ points to
one of the components, say $r_2$. 
If ${\mathcal C}(r_1) = x \in X$, ${\mathcal C} (r) = y$, then we require that 
${\mathcal C} (r_2) = x*y$. 

It is shown that the above colorings are consistent near each triple point.
Note that when three sheets form a triple point, they have relative
positions \textit{top, middle, bottom} 
with respect to the projection
direction of $p: \mathbf{R}^4 \rightarrow \mathbf{R}^3$.
The \textit{sign of a triple point} 
is positive
if the normals of top, middle, bottom sheets in this order 
match the
 orientation of the $3$-space. Otherwise 
the sign is negative.

A (Boltzmann) weight at a triple point, $\tau$, 
is defined as follows.
Let $R$ be the octant from which all normal vectors of the 
three sheets point outwards; let
a  coloring ${\mathcal C}$ be given.
Let $p$, $q$, $r$ be colors of the 
bottom, middle, and top 
sheets respectively, that bound the region $R$. 
Let $\epsilon (\tau) $ be the sign of the triple point,
and $\theta$ be a 
Then the Boltzmann weight $ B( \tau, {\mathcal C})$ 
assigned to $\tau$ with respect to ${\mathcal C}$
is defined to be $\theta (p,q,r) ^{ \epsilon (\tau) }$,
where $p$, $q$, $r$ are colors described above.

The \textit{partition function}, or a \textit{state-sum}, 
is the expression 
\begin{equation*} \sum_{\mathcal C}  \prod_{\tau}  B( \tau, {\mathcal C} ), \end{equation*} 
where $B ( \tau, {\mathcal C})$ 
the Boltzmann weight assigned to $\tau$.
As in the classical case, 
the value is taken to be in
the group ring $\mathbf{Z}[A]$, where $A$ is the coefficient 
group written multiplicatively.

Roseman \cite{Rose} generalized Reidemeister moves to 
dimension $4$. By showing that the invariant remains unchanged under
Roseman moves, we have 

The partition function does not depend on the choice of 
knotted surface diagram.
Thus it is an invariant of knotted surfaces $F$,
and denoted by $\Phi (F)$
(or $\Phi_{\theta} (F) $ to specify the $3$-cocycle $\theta$ used).

\begin{proposition} \label{coblemma4}
If $\Phi_{\theta}$ and $\Phi_{\theta '}$ denote the state-sum invariants 
defined from cohomologous cocycles  $\theta$ and $\theta '$,
then $\Phi_{\theta} =\Phi_{\theta '} $ 
(so that $\Phi_{\theta} (K)=\Phi_{\theta '}(K)$ for any knotted surface $K$).
In particular,
if $\theta$ is a $3$-coboundary,
then the state-sum defined above
 is equal to the number of

\section{Cocycles  of quandles and evaluations of invariants} \label{compsec}

\subsection{Computing cohomology}

 Suppose that the coefficient group $A$ is either a cyclic group,
$\mathbf{Z}$, $\mathbf{Z}_n$, or the rational numbers, $ \mathbf{Q}$.
Define a \textit{characteristic function}
\begin{equation*}\chi_x(y) = \left\{ \begin{array}{lr} 1 &\text{if} \  \ x=y,
                                               0   &\text{if} \ \  x\ne
y, \end{array} \right.\end{equation*} 
from the free abelian group
to the group $A.$
The set $\{ \chi_x: x \in X^n 
\}$ of such
spans the group $C^n_{\text{rack}}(X,A)$ of cochains.
Thus if  $f \in C^n_{\text{rack}}(X,A)$ is a cochain, then for some integers $C_x$,
\begin{equation*}f = \sum_{x \in X^n}  C_x \chi_x.\end{equation*}
We are interested in those $f$'s 
in $P^n$; \textit{i.e.\/} those
homomorphisms that vanish on
$S= \{ (x_1, \ldots , x_n) \in X^n: x_j = x_{j+1} \ 
\text{for some} \ 
j \}$.
So we can write
\begin{equation*}f = \sum_{x \in X^n \setminus S} \ \ \  
C_x \chi_x.\end{equation*}
We used these characteristic functions to compute cohomology groups.
We turn now to examples.

\begin{definition} [{\cite{FR}}]
A rack is called \textit{trivial} if $x*y=x$ for any $x,y$.

The \textit{dihedral quandle} $R_n$ of order $n$ is the quandle consisting of
reflections of the regular $n$-gon with the conjugation as operation.
The dihedral group $D_{2n}$  has a presentation 
\begin{equation*} \langle x, y \mid 
x^2 = 1 = y^n, xyx=y^{-1} \rangle, \end{equation*}
where $x$ is a reflection and $y$ is a rotation of a regular $n$-gon.
The set of reflections $R_n$ in this presentation 
is $\{ a_i = xy^i : i=0, \dots, n-1 \} $,
where we use the subscripts from $\mathbf{Z}_n$ in the following computations. 
The operation is 
\begin{equation*}a_i * a_j = a_j^{-1} a_i a_j =x y^{j} x y^i x y^j 
= x y^j y^{-i} y^j = a_{2j-i}.\end{equation*}
Hence $R_n = \mathbf{Z}_n$ as a set, 
	with quandle operation $i*j=2j-i \pmod{n}$.
Compare with 
the well known $n$-coloring of
 knot diagrams \cite{FoxTrip}.

Let $\Lambda = \mathbf{Z}[T, T^{-1}]$ be the Laurent polynomial ring
over the integers. Then any $\Lambda$-module $M$
 has a quandle structure defined by
$a*b= Ta + (1-T) b$ for $a, b \in M$. 
For any Laurent polynomial $h(T)$,
 $\mathbf{Z}_n[ T, T^{-1} ]  / (h(T)) $  
is  a 
We call such quandles \textit{(mod $n$)-Alexander quandles.}
We denote
the $4$-element quandle
$\mathbf{Z}_2[T, T^{-1}]/(T^2 +T +1)$ 
by $S_4$. 

We used \textit{Maple} and \textit{Mathematica} 
to compute some of the following results.
More computations can be found in \cite{CJKS}.

$H_Q^2( R_4, \mathbf{Z} ) \cong  \mathbf{Z} \times \mathbf{Z}$.
$H_Q^2( R_6, \mathbf{Z}_q ) \cong  \mathbf{Z}_q \times \mathbf{Z}_q$
for prime numbers $q$.
$H_Q^2( \mathbf{Z}_8[T, T^{-1}]/(T-5), \mathbf{Z}_q ) \cong  (\mathbf{Z}_q)^n$
where $n=16$ for $q=2$ and $n=12$ for all other primes $21$, there exists a
link $L$ whose cocycle invariant is nontrivial with 
the Alexander quandle $\mathbf{Z}_n[T, T^{-1}]/(T^{2m}-1)$.  

In particular, $H^2( \mathbf{Z}_n[T, T^{-1}]/(T^{2m}-1), \mathbf{Z}) \neq 0$ 
for any $n,m>1$. 

Recall that the linking number of a $2$-component classical
link $L=K_1 \cup K_2$ can be defined by counting the crossing
 number with signs
($\pm 1$) where the component $K_1$ crosses over $K_2$ (\cite{Rolf},
see also the preceding section). 
This definition is generalized  to 
linked surfaces.
Let $F=K_1 \cup \dots \cup K_n$ be a linked surface,
$K_i$, $i=1, \dots, n$, are 

Let $T_{\pm}(i,j,k)$ denote the number of 
positive and negative, respectively, triple points 
such that the top, middle, and bottom sheets are from 
$K_i$, $K_j$, and $K_k$ respectively.
Such a triple point is called of type $(i,j,k)$.
Then define $T(i,j,k)=T_+(i,j,k) - T_-(i,j,k)$.

The numbers $T(i,j,k)$ are invariants of isotopy classes of $F$
if $i\neq j$ and $j \neq k$. 

Using the above triple linking numbers, nontriviality 
of the invariant with $T_n$ is obtained.

Let $X=\{x,y,z \} $ be the trivial quandle of
three elements and $\theta \in Z^3(X, \mathbf{Z})$ 
be the cocycle 
$\chi_{(x,y,z)}$ which is the characteristic function:
\begin{equation*}\chi_{(x,y,z)}(a,b,c)= \left\{\begin{array}{ll} t &
\text{if \ }
 (a,b,c) = (x,y,z), \\ 1  &\text{otherwise.}
\end{array} \right. \end{equation*}
for any integers $p$ and $q$, 
there is a linked surface $F$ such that
\begin{equation*} \Phi_{\theta}(F)=
 t^p + t^{-p} + t^q + t^{-q} + t^{p+q} + t^{-p-q} + 21 .\end{equation*}

\subsection{Topological application: noninvertibility}

Surface braid theory was used in \cite{CJKLS} and \cite{CJKS} 
to evaluate the invariants for some  knotted surfaces.
The theory is a generalization of braid theory in classical knot theory,
and the fundamental theorems such as Alexander's, Artin's 
and Markov's theorems have been generalized to dimension $4$ 
Planar graphs called braid charts are used to obtain
the following topological application of the invariant.

For a certain cocycle
in $Z^3(R_3, \mathbf{Z}_3)$, the $2$-twist spun trefoil 
and its orientation reversed counterpart have the 
invariants $6+12t$ and $6+12t^2$ respectively.
In particular, the cocycle invariant detects 
the noninvertibility of 
the $2$-twist spun trefoil.

In \cite{CJKS}, we also show how to compute 
the invariants using movie descriptions of knotted surfaces.


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