# nLab Fox derivative

Let $F$ be a free group with basis $X=\left\{{x}_{i}{\right\}}_{i\in I}$ and $ℤF$ the integer group ring.

Differentiation or derivation, $D$, in this context is defined using a sort of nonsymmetric analogue of the Leibniz rule: it is an additive map $D:ℤF\to ℤF$ such that for all $u,v\in F$,

$D\left(uv\right)=D\left(u\right)+uD\left(v\right).$D(u v) = D(u) + u D(v).

The Fox partial derivatives $\frac{\partial }{\partial {x}_{i}}$ are defined by the rules

$\frac{\partial 1}{\partial {x}_{i}}=0$\frac{\partial 1}{\partial x_i} = 0
$\frac{\partial {x}_{i}}{\partial {x}_{i}}=1$\frac{\partial x_i}{\partial x_i} = 1

extended to the products $u={y}_{1}\dots {y}_{n}$ where ${y}_{i}={x}_{k}$ or ${y}_{i}={x}_{k}^{-1}$ for some $k=k\left(i\right)$ by the formula

$\frac{\partial u}{\partial {x}_{i}}=\sum _{s=1}^{n}{y}_{1}\cdots {y}_{s-1}\frac{\partial {y}_{s}}{\partial {x}_{i}}.$\frac{\partial u }{\partial x_i} = \sum_{s=1}^n y_1\cdots y_{s-1} \frac{\partial y_s }{\partial x_i}.

This then implies that

$\frac{\partial {x}_{i}^{-1}}{\partial {x}_{i}}=-{x}_{i}^{-1}$\frac{\partial x_i^{-1}}{\partial x_i} = -x_i^{-1}
$\frac{\partial {x}_{j}^{±1}}{\partial {x}_{i}}=0,\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}i\ne j$\frac{\partial x_j^{\pm 1}}{\partial x_i} = 0,\;\;i\neq j

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:

###### Definition

For each $x\in X$, let

$\frac{\partial }{\partial x}:F\to ℤF$\frac{\partial}{\partial x} : F \to \mathbb{Z}F

be defined by

1. for $y\in X$,

$\frac{\partial y}{\partial x}=1\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\mathrm{if}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}x=y\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\mathrm{and}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}=0\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}y\ne x;.$\frac{\partial y}{\partial x} = 1\,\,\, if\,\,\, x = y\,\,\, and \,\,\,= 0 \,\,\, y \neq x; .
2. for any words, ${w}_{1},{w}_{2}\in F$,

$\frac{\partial }{\partial x}\left({w}_{1}{w}_{2}\right)=\frac{\partial }{\partial x}{w}_{1}+{w}_{1}\frac{\partial }{\partial x}{w}_{2}.$\frac{\partial}{\partial x}(w_1w_2) = \frac{\partial}{\partial x}w_1 + w_1\frac{\partial}{\partial x}w_2.

Then these uniquely determine the Fox derivative of $F$ with respect to $x$.

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

$D\left(u\right)=\sum _{i\in I}\frac{\partial u}{\partial {x}_{i}}D\left({x}_{i}\right)$D(u)=\sum_{i\in I} \frac{\partial u }{\partial x_i} D(x_i)

(This is a finite sum since $u$ will only involve finitely many of the generators.)

In particular if $ϵ:ℤF\to ℤ$ is the augmentation map given by $ϵ:{x}_{i}↦1$, then the differentiation $u↦u-ϵ\left(u\right){1}_{F}$ satisfies

$u-ϵ\left(u\right){1}_{F}=\sum _{i}\frac{\partial u}{\partial {x}_{i}}\left({x}_{i}-1\right)$u - \epsilon(u) 1_F = \sum_i \frac{\partial u }{\partial x_i} (x_i -1)

hence it belongs to the left ideal in $ℤF$ which is generated by $\left({x}_{i}-1\right)$.

This construction is important in combinatorial group theory, particularly in the study of free products of groups and the study of metabelian group?s.

Given any group $G$ with a presentation $⟨X;R⟩=F/N$ such that $F=⟨X⟩$ is the free group on the set of letters $X$ and $N$ the normal closure of the set of relations $R$, let $\overline{G}:=G/\left[G,G\right]$, let $\varphi :F\to G$, $\overline{\varphi }:F\to \overline{G}$ be the canonical projections; denote by the same letter their linearizations for group rings $\varphi :ℤF\to ℤG$ and $\overline{\varphi }:ℤF\to ℤ\overline{G}$. The Jacobi matrix of the presentation is the matrix

$J=\left(\varphi \left(\frac{\partial {r}_{i}}{\partial {x}_{j}}\right)\right)$J = \left(\phi(\frac{\partial r_i}{\partial x_j})\right)

and also the projected matrix $\overline{J}$ which is the image of $J$ as a matrix over $ℤ\overline{G}$. The determinant ideal ${D}_{i}$ of order $i$ of the matrix $\overline{J}$ is the ideal of $ℤ\overline{G}$ generated by all minors (= determinants of submatrices) of size $i×i$ in $\overline{J}$. The sequence ${D}_{1},{D}_{2},\dots$ is invariant (up to some technical details), that is does not depend on the presentation. In the case when $G=\pi \left(S\right)$ where $S$ is the complement of a knot, $\overline{G}$ is an infinite cyclic group. Let $t$ be its generator; then the highest nonzero determinant ideal (of $\overline{J}$) in $ℤ\overline{G}$ 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 Alexander polynomial of the knot.

### References

The orginal articles include:

• R. H. Fox, Free differential calculus I: Derivation in the free group ring, Annals Math. (2) 57, 547–560 (1953) doi:10.2307/1969736

• R. H. Fox, Free differential calculus II: The Isomorphism Problem of Groups, Annals Math. (2) 59 196–210 (1954); III:Subgroups, Annals Math. (2) 64, 407–419; IV: 71, 408–422 (1960)

with a nice introduction in