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\begin{document}

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\section*{Fox derivative}

Let $F$ be a [[free group]] with basis $X = \{ x_i\}_{i\in I}$ and $\mathbb{Z}F$ the integer [[group ring]] of $F$.

Differentiation or [[derivation on a group|derivation]], $D$, in this context is defined using a sort of nonsymmetric analogue of the Leibniz rule: it is an additive map $D:\mathbb{Z}F\to\mathbb{Z}F$ such that for all $u,v\in F$,

\begin{displaymath}
D(u v) = D(u) + u D(v).
\end{displaymath}
The \textbf{Fox partial derivatives} $\frac{\partial}{\partial x_i}$ are defined by the rules

\begin{displaymath}
\frac{\partial 1}{\partial x_i} = 0
\end{displaymath}
\begin{displaymath}
\frac{\partial x_i}{\partial x_i} = 1
\end{displaymath}
extended to the products $u = y_1\ldots y_n$ where $y_i = x_k$ or $y_i=x_k^{-1}$ for some $k = k(i)$ by the formula

\begin{displaymath}
\frac{\partial u }{\partial x_i} = \sum_{s=1}^n y_1\cdots y_{s-1} \frac{\partial y_s }{\partial x_i}.
\end{displaymath}
This then implies that

\begin{displaymath}
\frac{\partial x_i^{-1}}{\partial x_i} = -x_i^{-1}
\end{displaymath}
\begin{displaymath}
\frac{\partial x_j^{\pm 1}}{\partial x_i} = 0,\;\;i\neq j
\end{displaymath}
Notice that the summands on the right-hand side are ``of different length''.

The lemma given in [[derivation on a group]] allows the following alternative form of the above definition to be given:

\begin{udefn}
For each $x \in X$, let

\begin{displaymath}
\frac{\partial}{\partial x} : F \to \mathbb{Z}F
\end{displaymath}
be defined by

\begin{enumerate}%
\item for $y \in X$,

\begin{displaymath}
\frac{\partial y}{\partial x} = 1\,\,\, if\,\,\,  x = y\,\,\, and \,\,\,=
0  \,\,\, y \neq x;
.
\end{displaymath}

\item for any words, $w_1,w_2 \in F$,

\begin{displaymath}
\frac{\partial}{\partial x}(w_1w_2) = \frac{\partial}{\partial x}w_1 + w_1\frac{\partial}{\partial x}w_2.
\end{displaymath}


\end{enumerate}
Then these uniquely determine the Fox derivative of $F$ with respect to $x$.

\end{udefn}
The Fox derivatives give a way of expanding any derivation (differentiation) defined on $F$. For every differentiation

\begin{displaymath}
D(u)=\sum_{i\in I} \frac{\partial u }{\partial x_i} D(x_i)
\end{displaymath}
(This is a finite sum since $u$ will only involve finitely many of the generators.)

In particular if $\epsilon:\mathbb{Z}F\to\mathbb{Z}$ is the augmentation map given by $\epsilon:x_i\mapsto 1$, then the differentiation $u\mapsto u-\epsilon(u) 1_F$ satisfies

\begin{displaymath}
u - \epsilon(u) 1_F = \sum_i \frac{\partial u }{\partial x_i} (x_i -1)
\end{displaymath}
hence it belongs to the left [[ideal]] in $\mathbb{Z}F$ which is generated by $(x_i-1)$.

This construction is important in [[combinatorial group theory]], particularly in the study of [[free product]]s of groups and the study of [[metabelian group]]s.

Given any group $G$ with a [[group presentation|presentation]] $\langle X; R\rangle = F/N$ such that $F=\langle X\rangle$ is the [[free group]] on the set of letters $X$ and $N$ the [[normal subgroup|normal closure]] of the set of relations $R$, let $\bar{G}:=G/[G,G]$, let $\phi:F\to G$, $\bar\phi:F\to \bar{G}$ be the canonical projections; denote by the same letter their linearizations for group rings $\phi:\mathbb{Z}F\to \mathbb{Z}G$ and $\bar\phi:\mathbb{Z}F\to\mathbb{Z}\bar{G}$. The \textbf{Jacobi matrix} of the presentation is the matrix

\begin{displaymath}
J = \left(\phi(\frac{\partial r_i}{\partial x_j})\right)
\end{displaymath}
and also the projected matrix $\bar{J}$ which is the image of $J$ as a matrix over $\mathbb{Z}\bar{G}$. The \textbf{determinant ideal} $D_i$ of order $i$ of the matrix $\bar{J}$ is the ideal of $\mathbb{Z}\bar{G}$ generated by all minors (= determinants of submatrices) of size $i\times i$ in $\bar{J}$. The sequence $D_1,D_2,\ldots$ is invariant (up to some technical details), that is does not depend on the presentation. In the case when $G=\pi(S)$ where $S$ is the complement of a [[knot]], $\bar{G}$ is an infinite cyclic group. Let $t$ be its generator; then the highest nonzero determinant ideal (of $\bar{J}$) in $\mathbb{Z}\bar{G}$ is a principal ideal, hence it has a normalized (in the sense that the heighest coefficient is $1$) generator, which is a polynomial in $t$. This polynomial is an invariant of the knot, the \textbf{Alexander polynomial} of the knot.

\hypertarget{references}{}\subsubsection*{{References}}\label{references}

The orginal articles include:

\begin{itemize}%
\item R. H. Fox, \emph{Free differential calculus I: Derivation in the free group ring}, Annals Math. (2) \textbf{57}, 547--560 (1953) \href{http://dx.doi.org/10.2307/1969736}{doi:10.2307/1969736}


\item R. H. Fox, \emph{Free differential calculus II: The Isomorphism Problem of Groups}, Annals Math. (2) \textbf{59} 196--210 (1954); \emph{III:Subgroups}, Annals Math. (2) \textbf{64}, 407--419; IV: \textbf{71}, 408--422 (1960)



\end{itemize}
with a nice introduction in

\begin{itemize}%
\item \begin{itemize}%
\item [[R. H. Crowell]] and [[R. H. Fox]], Introduction to Knot Theory, Springer, Graduate Texts 57, 1963.

\end{itemize}


\end{itemize}
see also

\begin{itemize}%
\item B. Chandler, [[W. Magnus]], \emph{The history of combinatorial group theory: a case study in the history of ideas}, Springer 1982


\item [[R. Lyndon]], [[P. Schupp]], \emph{Combinatorial group theory}, Ch. II.3, Springer 1977(Russian transl. Mir, Moskva 1980)



\end{itemize}
and more recently

\begin{itemize}%
\item Valentino Zocca, \emph{Fox calculus, symplectic forms and moduli spaces}, Trans. AMS\_\_350\_\_, 4, (1998) 1429-1466, \href{http://www.ams.org/journals/tran/1998-350-04/S0002-9947-98-02052-2/S0002-9947-98-02052-2.pdf}{pdf}


\item \href{http://en.wikipedia.org/wiki/Fox_derivative}{en.wikipedia:Fox derivative}



\end{itemize}
Connections to double Poisson structures/brackets are discussed in

\begin{itemize}%
\item Gwenael Massuyeau, [[Vladimir Turaev]], \emph{Quasi-Poisson structures on representation spaces of surfaces}, \href{http://arxiv.org/abs/1205.4898}{arxiv/1205.4898}, \href{http://www-irma.u-strasbg.fr/~massuyea/papers/double_brackets.pdf}{irma pdf}

\end{itemize}
See also

\begin{itemize}%
\item Gwenael Massuyeau, Vladimir Turaev, \emph{Fox pairings and generalized Dehn twists}, \href{http://arxiv.org/abs/1109.5248}{arxiv/1109.5248}

\end{itemize}
For a pro-$l$-version of Fox calculus see

\begin{itemize}%
\item \emph{Pro-$l$ Fox free differential calculus}, section 8.3 of [[Masanori Morishita]], \emph{Knots and primes: an introduction to arithmetic topology}, Springer 2012

\end{itemize}
[[!redirects Fox derivatives]]



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