Non injectivity of the ``hair'' map
Bertrand Patureau-Mirand
In \cite{GaKr} an invariant of knots $Z^{rat}$ with values in a space of diagrams with beads has been constructed. When composed with the so called ``hair'' map $H$, It gives the Kontsevich integral of the knot. We introduce a new grading on diagrams with beads and use it to show that a non trivial element constructed with Vogel's zero divisor in the algebra $\L$ is in the kernel of $H$. This shows that $H$ is not injective.
hair.tex
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\def\ZZ{\mathbb{Z}}  
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\begin{document}
\title{Non injectivity of the ``hair'' map}
\author{Bertrand Patureau-Mirand}
\address{Institut de Math{\'e}matiques de Jussieu, Equipe ``Topologie
et g{\'e}om{\'e}trie  alg{\'e}briques'', Case 7012, Universit{\'e}
Paris VII,   75251 Paris Cedex 05, France}
\email{patureau@math.jussieu.fr}
\urladdr{http://www.math.jussieu.fr/\~\,patureau}
\date{February 2002}

\begin{abstract}
In \cite{GaKr} an invariant of knots $Z^{rat}$ with values in a space of
diagrams with beads has been constructed. When composed with the so called
``hair'' map $H$, It gives the Kontsevich integral of the knot. We introduce a
new grading on diagrams with beads and use it to show that a non trivial
element constructed with Vogel's zero divisor in the algebra $\L$ is in the
kernel of $H$. This shows that $H$ is not injective.
\end{abstract}

\maketitle

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Introduction}
The Kontsevich integral $Z$ is the universal rational finite type invariant for
knots (see \cite{BN}). For a knot $K$, $Z(K)$ lives in the space of Chinese
diagrams isomorphic to $\hat\B(*)$ (see section 1.1). Rozansky conjectures
(\cite{Roz}) and Kricker proves (\cite{Kr}) that $Z$ could be organized into
series of ``lines'' which could be represented by finite $\QQ$-linear
combinations of diagrams whose edges were labelled, in an appropriate sense,
with rational functions. For knots with trivial Alexander polynomial, this means
that $Z(K)$ is in the image of the hair map $H$ (see section 1.3). In
\cite{GaKr}, It is directly showed that the map $Z^{rat}$ with values in a space
of diagrams with beads is an isotopy invariant and $Z$ factor through
$Z^{rat}$. Rozansky, Garoufalidis and Kricker conjecture (see \cite{Oth}
conjecture 3.18) that $H$ could be injective. The theorem 4 shows that this is
false.

\section{The hair map}
\subsection{Classical diagrams}
Let $X$ be a finite set. An $X$-diagram is an isomorphism class of finite
$1-3-$valent graph $K$ with the following data:
\begin{itemize}
\item A cyclic ordering on the three oriented edges starting from every
trivalent vertices $x$ of $K$.
\item An isomorphism between the set of univalent vertices of $K$ and the set
$X$.
\end{itemize}
We define $\A(X)$ to be the quotient of the $\QQ$-vector space generated by
$X$-diagram by the relations:
\begin{enumerate}
\item The (AS) relations for antisymmetry: $$\begin{array}{cccc}\put(-10,-10)
{\epsfbox{As+.eps}}&+& \put(-10,-10) {\epsfbox{As-.eps}}&=0\end{array}$$
\item The (IHX) relations for three diagrams which differ only in a
neighborhood of an edge:
  $$\begin{array}{ccccc}\put(-8,-5) {\epsfbox{IHXI.eps}}&=&
    \put(-8,-5) {\epsfbox{IHXH.eps}} &
    -&\put(-8,-5) {\epsfbox{IHXX.eps}}\end{array}$$
\end{enumerate}
These spaces are graduated, the degree of an $X$-diagram being given by half the
total number of vertices.\\ 
% %
Set $[n]=\{1,2,\ldots,n\}$ and define $F_n$ to be the subspace of $\A([n])$
generated by connected diagram with at least one trivalent vertex. The
permutation group $\S(X)$ acts on $\A(X)$ and we will denote by $B(*)$ the
coinvariants for this action:
$$B(*)=\bigoplus_{n\in\NN}\A([n])\otimes_{\S_n}\QQ$$
and by $\hat B(*)$ its completion for the grading.\\
% %
Finally $\L$ will denote Vogel's algebra formed by totally antisymmetric
elements of $F_3$ (for the action of $\S_3$).\\
% %
We recall (see \cite{Vo1}) that $\L$ acts on the modules $F_n$ and that for this
action, $F_0$ and $F_2$ are free $\L$-modules of rank one. Furthermore, the
following elements are in $\L$:
$$\begin{array}{cccccc}t=&\put(-8,-5) {\epsfbox{elt_t.eps}}&=\frac12&
\put(-8,-5) {\epsfbox{elt_2t.eps}} & \qquad x_n=&\put(-8,-10) {\epsfbox{elt_xn.eps}}
\end{array}$$

\begin{theo}(Vogel \cite{Vo2}) The element $t$ is a divisor of zero in $\Lambda$.
\end{theo}

\begin{coro}
There exists an element $r\in\L\setminus\{0\}$ such that $t.r=0$. So one has
$$\put(-90,-15){\input{t-killedF0.pstex_t}}\quad\hbox{but}
\quad\put(20,-15){\input{t-killedF3.pstex_t}}$$
\end{coro}
\begin{dem}
This is just a consequence of the fact that $F_0$ is a rank one free $\L$-module
generated by the diagram $\Theta$. The preceding element of $F_3$ is in fact the
product $r.\put(2,-7){\epsfbox{t-inF3.eps}}\hspace{25pt}=2tr=0\in\L$.
\end{dem}

\begin{Rq} Vogel's computation shows that $r$ can be chosen in degree
fifteen (in $\L$), living in the algebra generated by the $x_n$ and is killed by
all the weight systems coming from Lie algebras (but is not killed by the Lie
superalgebras $\mathfrak{D}_{2,1,\alpha}$).
\end{Rq}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Diagrams with beads}
Diagrams with beads have been introduced by Kricker and Garoufalidis (see
\cite{Kr}, \cite{GaKr}). A presentation of $\B$ which uses the first cohomology
classes of diagrams is already present in \cite{Roz}. Vogel explains me this
point of view for diagrams with beads.\\

Let $G$ be the multiplicative group $\{b^n,\,n\in\ZZ\}\simeq (\ZZ,+)$ and
consider its group algebra $R=\QQ G =\QQ[b,b^{-1}]$. We denote by $a\mapsto
\overline a$ the involution of the $\QQ$-algebra $R$ sending $b$ to $b^{-1}$.\\
% %
A diagram with beads in $R$ is an $\emptyset$-diagram with the following
supplementary data: The beads which consist of a map $f:E\go R$ from the set of
oriented edges of $K$ such that if $-e$ denotes the same edge than $e$ with
opposite orientation, one has $f(-e)=\overline{f(e)}$.\\
% %
We will represent the beads by some arrows on the edges with label in $R$. The
value of the bead $f$ on $e$ is given by the product of these labels and we
won't represent the beads with value $1$. So one has the notations:
$$\input{involution.pstex_t}\quad\hbox{and}\quad\input{Multilin1.pstex_t}$$
% %
The loop degree of a diagram with beads $K$ is the rank of the first homology
group of the underlying graph. \\ 
% %
Let $\A^R(\emptyset)$ denotes the quotient of the $\QQ$-vector space generated by
diagrams with beads in $R$ by the following relations:
\begin{enumerate}
\item (AS) 
\item The (IHX) relations are to be consider only near an edge with bead $1$.
\item PUSH:$$\put(-50,-10){\epsfbox{push.eps}}$$
\item Multilinearity: $$\input{Multilin2.pstex_t}$$
\end{enumerate}
$\A^R(\emptyset)$ is graded by the loop degree:
$$\A^R(\emptyset)=\bigoplus_{n\in\NN}\A^R_n(\emptyset)$$
We will prefer an other presentation of $\A^R(\emptyset)$:
\begin{itemize}
\item Remark that it is enough to consider diagrams with beads in $G$ and the
multilinear relation can be viewed as a notation.
\item Next remark that for a diagrams with beads in $G$, the map $f$ is the same
that a $1$-cochain $\tilde f$ with value in $\ZZ\simeq G$ on the underlying
simplicial set of $K$. The elements $\tilde f$ are in fact the $1$-cocycles
because of the condition $f(-e)=\overline{f(e)}$ which means $\tilde
f(-e)=-\tilde f(e)$.
\item The ``PUSH'' relation at a vertex $v$ say that $\tilde f$ is only given up
to the coboundary of the $0$-cochain with value $1$ on $v$ and $0$ on the other
vertices. Hence $\A^R(\emptyset)$ is also the $\QQ$-vector space generated by the
pairs ($3$-valent graph $D$, $x\in H^1(D,\ZZ)$) quotiented by the relations (AS)
and (IHX). With these notations one can describe the (IHX) relations in the
following way:\\
% %
Let $K_I$, $K_H$ and $K_X$ be three graphs which occur in a (IHX) relation on a
edge $e$. Let $K_\bullet$ be the graph obtained by collapsing the edge $e$.The map
$p_?: K_?\go K_\bullet$ induce three cohomology isomorphisms.\\
% %
If $x\in H^1(K_\bullet,\ZZ)$ then the (IHX) relation at $e$ says that
$$(K_I,p_I^*x)=(K_H,p_H^*x)-(K_X,p_X^*x)$$ is true in $\A^R(\emptyset)$.
\end{itemize}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{The hair map}
The hair map $H:\A^R(\emptyset)\go\hat\B(*)$ replaces beads by legs (or hair):
Just replace a bear $t^n$ by the exponential of $n$ times a leg.
$$\input{hairmap.pstex_t}$$
It is well defined (see \cite{GaKr}).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Grading on diagrams with beads}
Remark that for a $3$-valent graph $K$, $H^1(K,\ZZ)$ is a free $\ZZ$-module. The
beads $x\in H^1(K,\ZZ)$ which occur in an (AS) or (IHX) relation are the same up
to isomorphisms. We will call $p$ the bead degree of $(K,x)$ if $x$ is
$p$ times an indivisible element of $H^1(K,\ZZ)$.
\begin{theo}
The bead degree is well defined in $\A^R_n(\emptyset)$. Thus we have a grading
$$\A^R_n(\emptyset)=\bigoplus_{p\in\NN} \A^R_{n,p}(\emptyset)$$ where
$\A^R_{n,p}(\emptyset)$ is the subspace of $\A^R_n(\emptyset)$ generated by
diagrams with bead degree $p$.\\
% %
Furthermore, $\A^R_{n,0}(\emptyset)\simeq \A_n(\emptyset)$ and for $p>0$, $\A^R_{n,p}(\emptyset)\simeq\A^R_{n,1}(\emptyset)$.
\end{theo}
\begin{dem}
This degree is a consequence of the second presentation we give for
$\A^R_n(\emptyset)$. The map $\psi:R\go \QQ$ sending $b$ to $1$ induces the
isomorphism $\A^R_{n,0}(\emptyset)\simeq \A_n(\emptyset)$ and the group morphism
$\phi_p:G\go G$ sending $b$ to $b^p$ (or the multiplication by $p$ in
$H^1(.,\ZZ)$) induces the isomorphism
$\A^R_{n,1}(\emptyset)\simeq\A^R_{n,p}(\emptyset)$. These maps are isomorphisms
because their inverses are evident (for $\phi_p$ just remark that for a
$3$-valent graph $K$, $H^1(K,\ZZ)$ has no $p$-torsion).
\end{dem}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{A non trivial element in the kernel of $H$}

\begin{theo}
This non trivial element of $\A^R(\emptyset)$ is in the kernel of $H$:
$$\input{hairkilled.pstex_t}$$
Thus $H$ is not injective.
\end{theo}
\begin{dem}
This element is not zero because its bead degree zero part is just the
opposite of the first element of corollary 2. Next see that
$$\input{hairkilled2.pstex_t}$$ 
% %
but all these diagrams are zero because they contain the same diagram of the
corollary 2.
\end{dem}

\begin{Rq}
The element of theorem 4 has a loop degree seventeen.\\
The hair map is evidently injective on the space of diagrams with bead degree
zero. I don't know if it is still true in the other degree.\\
\end{Rq}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{thebibliography}{000}
\bibitem[1]{BN} D. Bar-Natan - {\sl On the Vassiliev knot invariants},
  Topologie 34 $n^0$ 2 1995, 423-472.
\bibitem[2]{BG}D. Bar-Natan, S. Garoufalidis, - {\it On the
Melvin-Morton-Rozansky conjecture}, Invent. Math. 125 (1996) 103-133.
\bibitem[3]{BGRT}D. Bar-Natan, S. Garoufalidis, L. Rozansky, D. P. Thurston, -
{\it Wheels, wheeling, and the Kontsevich integral of the unknot}, Hebrew
University, Harvard University, University of Illinois at Chicago and University
of California at Berkeley preprint, March 1997. q-alg/9703025.
\bibitem[4]{GaKr} S. Garoufalidis, A. Kricker - {\it A rational
noncommutative invariant of boundary links} preprint math.GT/0105028, May 2001.
\bibitem[5]{Kr} A. Kricker - {\sl The lines of the Kontsevich integral and
Rozansky's rationality conjecture}, Tokyo Institute of Technology preprint, May
2000, arXiv:math.GT/0005284
\bibitem[6]{Oth} T. Ohtsuki - {\it Problems on Invariants of Knots and
3-Manifold} to appear in Geometry and Topology Monographs.
\bibitem[7]{Roz} L. Rozansky - {\it Une structure rationnelle sur des
fonctions g{\'e}n{\'e}ratrices d'invariants de Vassiliev} Summer school of the
University of Grenoble preprint, July 1999.
\bibitem[8]{Roz2} L. Rozansky - {\it A rationality conjecture about Kontsevich
integral of knots and its implications to the structure of the colored Jones
polynomial}, math.GT/0106097 .
\bibitem[9]{Vo1} P. Vogel - {\it Algebraic structures on modules of
  diagrams}, preprint to appear Inventiones Mathematicae.
\bibitem[10]{Vo2} P. Vogel - {\it private communication}.
\end{thebibliography}
\end{document}



math0202065.extracted_bib
\begin{thebibliography}{000}
\bibitem[1]{BN} D. Bar-Natan - {\sl On the Vassiliev knot invariants},
  Topologie 34 $n^0$ 2 1995, 423-472.
\bibitem[2]{BG}D. Bar-Natan, S. Garoufalidis, - {\it On the
Melvin-Morton-Rozansky conjecture}, Invent. Math. 125 (1996) 103-133.
\bibitem[3]{BGRT}D. Bar-Natan, S. Garoufalidis, L. Rozansky, D. P. Thurston, -
{\it Wheels, wheeling, and the Kontsevich integral of the unknot}, Hebrew
University, Harvard University, University of Illinois at Chicago and University
of California at Berkeley preprint, March 1997. q-alg/9703025.
\bibitem[4]{GaKr} S. Garoufalidis, A. Kricker - {\it A rational
noncommutative invariant of boundary links} preprint math.GT/0105028, May 2001.
\bibitem[5]{Kr} A. Kricker - {\sl The lines of the Kontsevich integral and
Rozansky's rationality conjecture}, Tokyo Institute of Technology preprint, May
2000, arXiv:math.GT/0005284
\bibitem[6]{Oth} T. Ohtsuki - {\it Problems on Invariants of Knots and
3-Manifold} to appear in Geometry and Topology Monographs.
\bibitem[7]{Roz} L. Rozansky - {\it Une structure rationnelle sur des
fonctions g{\'e}n{\'e}ratrices d'invariants de Vassiliev} Summer school of the
University of Grenoble preprint, July 1999.
\bibitem[8]{Roz2} L. Rozansky - {\it A rationality conjecture about Kontsevich
integral of knots and its implications to the structure of the colored Jones
polynomial}, math.GT/0106097 .
\bibitem[9]{Vo1} P. Vogel - {\it Algebraic structures on modules of
  diagrams}, preprint to appear Inventiones Mathematicae.
\bibitem[10]{Vo2} P. Vogel - {\it private communication}.
\end{thebibliography}