nLab
Fox derivative

Let FF be a free group with basis X={x i} iIX = \{ x_i\}_{i\in I} and F\mathbb{Z}F the integer group ring.

Differentiation or derivation, DD, in this context is defined using a sort of nonsymmetric analogue of the Leibniz rule: it is an additive map D:FFD:\mathbb{Z}F\to\mathbb{Z}F such that for all u,vFu,v\in F,

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

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

1x i=0 \frac{\partial 1}{\partial x_i} = 0
x ix i=1 \frac{\partial x_i}{\partial x_i} = 1

extended to the products u=y 1y nu = y_1\ldots y_n where y i=x ky_i = x_k or y i=x k 1y_i=x_k^{-1} for some k=k(i)k = k(i) by the formula

ux i= s=1 ny 1y s1y sx 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

x i 1x i=x i 1 \frac{\partial x_i^{-1}}{\partial x_i} = -x_i^{-1}
x j ±1x i=0,ij \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 xXx \in X, let

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

be defined by

  1. for yXy \in X,

    yx=1ifx=yand=0yx;.\frac{\partial y}{\partial x} = 1\,\,\, if\,\,\, x = y\,\,\, and \,\,\,= 0 \,\,\, y \neq x; .
  2. for any words, w 1,w 2Fw_1,w_2 \in F,

    x(w 1w 2)=xw 1+w 1xw 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 FF with respect to xx.

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

D(u)= iIux iD(x i) D(u)=\sum_{i\in I} \frac{\partial u }{\partial x_i} D(x_i)

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

In particular if ϵ:F\epsilon:\mathbb{Z}F\to\mathbb{Z} is the augmentation map given by ϵ:x i1\epsilon:x_i\mapsto 1, then the differentiation uuϵ(u)1 Fu\mapsto u-\epsilon(u) 1_F satisfies

uϵ(u)1 F= iux i(x i1) u - \epsilon(u) 1_F = \sum_i \frac{\partial u }{\partial x_i} (x_i -1)

hence it belongs to the left ideal in F\mathbb{Z}F which is generated by (x i1)(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 GG with a presentation X;R=F/N\langle X; R\rangle = F/N such that F=XF=\langle X\rangle is the free group on the set of letters XX and NN the normal closure of the set of relations RR, let G¯:=G/[G,G]\bar{G}:=G/[G,G], let ϕ:FG\phi:F\to G, ϕ¯:FG¯\bar\phi:F\to \bar{G} be the canonical projections; denote by the same letter their linearizations for group rings ϕ:FG\phi:\mathbb{Z}F\to \mathbb{Z}G and ϕ¯:FG¯\bar\phi:\mathbb{Z}F\to\mathbb{Z}\bar{G}. The Jacobi matrix of the presentation is the matrix

J=(ϕ(r ix j)) J = \left(\phi(\frac{\partial r_i}{\partial x_j})\right)

and also the projected matrix J¯\bar{J} which is the image of JJ as a matrix over G¯\mathbb{Z}\bar{G}. The determinant ideal D iD_i of order ii of the matrix J¯\bar{J} is the ideal of G¯\mathbb{Z}\bar{G} generated by all minors (= determinants of submatrices) of size i×ii\times i in J¯\bar{J}. The sequence D 1,D 2,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=π(S)G=\pi(S) where SS is the complement of a knot, G¯\bar{G} is an infinite cyclic group. Let tt be its generator; then the highest nonzero determinant ideal (of J¯\bar{J}) in G¯\mathbb{Z}\bar{G} is a principal ideal, hence it has a normalized (in the sense that the heighest coefficient is 11) generator, which is a polynomial in tt. 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

see also

  • 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

Connections to double Poisson structures/brackets are discussed in

See also

  • Gwenael Massuyeau, Vladimir Turaev, Fox pairings and generalized Dehn twists, arxiv/1109.5248

For a pro-ll-version of Fox calculus see

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

Revised on August 28, 2014 05:12:21 by Zoran Škoda (161.53.130.104)