nLab Fox derivative

Definitions

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

1. for $y \in X$,

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

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

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

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 original articles include:

• Ralph H. Fox, Free differential calculus I: Derivation in the free group ring, Annals Math. 57 3 (1953) 547–560 [doi:10.2307/1969736, jstor:1969736]

• Ralph H. Fox, Free differential calculus II: The isomorphism problem of groups, Annals Math. 59 2 (1954) 196-2-10 [doi:10.2307/1969686, jstor:1969686]

• Ralph H. Fox, Free differential calculus III: Subgroups, Annals Math. 64 3 (1956) 407–419 [doi:10.2307/1969592, jstor:1970044]

• Kuo Tsai Chen, Ralph H. Fox, R. C. Lyndon, Free differential calculus IV: The quotient groups of the lower central series, Ann. Math. 68 1 (1958) 81–95 [doi:10.2307/1970044, jstor:1970044]

• Ralph H. Fox, Free differential calculus V. The Alexander matrices re-examined, Ann. Math. 71 3 (1960) 408–422 [doi:10.2307/1969936, jstor:1969936]

• William M Goldman, The symplectic nature of fundamental groups of surfaces, Adv, Math. 54:2 (1984) 200–225 doi

with textbook treatments in

• Richard H. Crowell, Ralph H. Fox, Introduction to knot theory, Graduate Texts in Mathematics 57, Springer (1963) [doi:10.1007/978-1-4612-9935-6]

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

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

and more recently

• Valentino Zocca, Fox calculus, symplectic forms and moduli spaces, Trans. Amer. Math. Soc. 350, 4, (1998) 1429–1466, pdf

• en.wikipedia:Fox derivative

Connections to double Poisson structures/brackets are discussed in

• Gwénaël Massuyeau, Vladimir Turaev, Quasi-Poisson structures on representation spaces of surfaces, arXiv:1205.4898, irma pdf

See also

• Gwénaël Massuyeau, Vladimir Turaev, Fox pairings and generalized Dehn twists, Annales de l’Institut Fourier 63:5 (2013) 2403–2456 doi arXiv:1109.5248

For a pro-$l$-version of Fox calculus see

• Pro-$l$ Fox free differential calculus, section 8.3 of Masanori Morishita, Knots and primes: an introduction to arithmetic topology, Springer 2012