Differentiation or derivation, , in this context is defined using a sort of nonsymmetric analogue of the Leibniz rule: it is an additive map such that for all ,
The Fox partial derivatives are defined by the rules
extended to the products where or for some by the formula
This then implies that
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:
For each , let
be defined by
for any words, ,
Then these uniquely determine the Fox derivative of with respect to .
The Fox derivatives give a way of expanding any derivation (differentiation) defined on . For every differentiation
(This is a finite sum since will only involve finitely many of the generators.)
In particular if is the augmentation map given by , then the differentiation satisfies
hence it belongs to the left ideal in which is generated by .
Given any group with a presentation such that is the free group on the set of letters and the normal closure of the set of relations , let , let , be the canonical projections; denote by the same letter their linearizations for group rings and . The Jacobi matrix of the presentation is the matrix
and also the projected matrix which is the image of as a matrix over . The determinant ideal of order of the matrix is the ideal of generated by all minors (= determinants of submatrices) of size in . The sequence is invariant (up to some technical details), that is does not depend on the presentation. In the case when where is the complement of a knot, is an infinite cyclic group. Let be its generator; then the highest nonzero determinant ideal (of ) in is a principal ideal, hence it has a normalized (in the sense that the heighest coefficient is ) generator, which is a polynomial in . This polynomial is an invariant of the knot, the Alexander polynomial of the knot.
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
B. Chandler, W. Magnus, The history of combinatorial group theory: a case study in the history of ideas, Springer 1982
and more recently
Valentino Zocca, Fox calculus, symplectic forms and moduli spaces, Trans. AMS__350__, 4, (1998) 1429-1466, pdf
Connections to double Poisson structures/brackets are discussed in