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 of $F$ .

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(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 x_i}{\partial x_i} = 1

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

\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_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 \mathbb{Z}F

be defined by

for $y \in X$ ,

$\frac{\partial y}{\partial x} = 1\,\,\, if\,\,\, x = y\,\,\, and \,\,\,= 0 \,\,\, y \neq x; .$
for any words, $w_1,w_2 \in F$ ,

$\frac{\partial}{\partial x}(w_1w_2) = \frac{\partial}{\partial x}w_1 + w_1\frac{\partial}{\partial x}w_2.$

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)

(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

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 groups.

Given any group $G$ with a 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 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 Jacobi matrix of the presentation is the matrix

J = \left(\phi(\frac{\partial r_i}{\partial x_j})\right)

and also the projected matrix $\bar{J}$ which is the image of $J$ as a matrix over $\mathbb{Z}\bar{G}$ . The 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 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

- R. H. Crowell and R. H. Fox, Introduction to Knot Theory, Springer, Graduate Texts 57, 1963.