nLab
Fox derivative

Let $F$ be a free group with basis $X=\{x_i{\}}_{i\in I}$ and $\mathbb{Z}F$ 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 $D:\mathbb{Z}F\to \mathbb{Z}F$ such that for all $u,v\in F$,

$$D(uv)=D(u)+uD(v).$$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

$$\frac{\partial x_i^{-1}}{\partial x_i}=-x_i^{-1}$$\frac{\partial x_i^{-1}}{\partial x_i} = -x_i^{-1}

$$\frac{\partial x_j^{\pm 1}}{\partial x_i}=0,i\ne j$$\frac{\partial x_j^{\pm 1}}{\partial x_i} = 0,\;\;i\neq j

and extended to the products $u=y_1\dots y_n$ where $y_i=x_k$ or $y_i=x_k^{-1}$ for some $k=k(i)$ 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}.

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

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

be defined by

1. for $y\in X$,

   $$\frac{\partial y}{\partial x}=1\mathrm{if}x=y\mathrm{and}=0y\ne x;.$$\frac{\partial y}{\partial x} = 1\,\,\, if\,\,\, x = y\,\,\, and \,\,\,=0 \,\,\, y \neq x; .

2. for any words, $w_1,w_2\in F$,

   $$\frac{\partial }{\partial x}(w_1{w}_2)=\frac{\partial }{\partial x}{w}_1+w_1\frac{\partial }{\partial x}{w}_2.$$\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

$$D(u)=\sum _{i\in I}\frac{\partial u}{\partial x_i}D(x_i)$$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 $ϵ:\mathbb{Z}F\to \mathbb{Z}$ is the augmentation map given by $ϵ:x_i\mapsto 1$, then the differentiation $u\mapsto u-ϵ(u)1_F$ satisfies

$$u-ϵ(u)1_F=\sum _i\frac{\partial u}{\partial x_i}(x_i-1)$$u - \epsilon(u) 1_F = \sum_i \frac{\partial u }{\partial x_i} (x_i -1)

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 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 $\overline{G}:=G/[G,G]$, 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 $\varphi :\mathbb{Z}F\to \mathbb{Z}G$ and $\overline{\varphi }:\mathbb{Z}F\to \mathbb{Z}\overline{G}$. The Jacobi matrix of the presentation is the matrix

$$J=(\varphi (\frac{\partial r_i}{\partial x_j}))$$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 $\mathbb{Z}\overline{G}$. The determinant ideal $D_i$ of order $i$ of the matrix $\overline{J}$ is the ideal of $\mathbb{Z}\overline{G}$ generated by all minors (= determinants of submatrices) of size $i\times i$ 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 $G=\pi (S)$ 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 $\mathbb{Z}\overline{G}$ 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.

• 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)

• en.wikipedia:Fox derivative

• B. Chandler, W. Magnus, The history of combinatorial group theory: a case study in the history of ideas, Springer 1982

• R. Lindon, P. Schupp, Combinatorial group theory, Ch. II.3, Springer 1977(Russian transl. Mir, Moskva 1980)