Fox derivative

In mathematics, the Fox derivative is an algebraic construction in the theory of free groups which bears many similarities to the conventional derivative of calculus. The Fox derivative and related concepts are often referred to as the Fox calculus, or (Fox's original term) the free differential calculus. The Fox derivative was developed in a series of five papers by mathematician Ralph Fox, published in Annals of Mathematics beginning in 1953.

Definition

If "G" is a free group with identity element "e" and generators "g_i", then the Fox derivative with respect to "g_i" is a function from "G" into the integral group ring "ZG" which is denoted $frac\{partial\}\{partial\; g\_i\}$, and obeys the following axioms:
* $frac\{partial\}\{partial\; g\_i\}(g\_j)\; =\; delta\_\{ij\}$, where $delta\_\{ij\}$ is the Kronecker delta
* $frac\{partial\}\{partial\; g\_i\}(e)\; =\; 0$
* $frac\{partial\}\{partial\; g\_i\}(uv)\; =\; frac\{partial\}\{partial\; g\_i\}(u)\; +\; ufrac\{partial\}\{partial\; g\_i\}(v)$ for any elements "u" and "v" of "G".The first two axioms are identical to similar properties of the partial derivative of calculus, and the third is a modified version of the product rule.

Applications

The Fox derivative has applications in knot theory and covering space theory, among other areas of mathematics.

References

*cite journal |last=Fox |first=Ralph |authorlink=Ralph Fox |year=1953 |month=May |title=Free Differential Calculus, I: Derivation in the Free Group Ring |journal=Annals of Mathematics |volume=57 |issue=3 |pages=547–560 |doi=10.2307/1969736 MathSciNet|id=0053938
*cite journal |last=Fox |first=Ralph |year=1954 |month=March |title=Free Differential Calculus, II: The Isomorphism Problem of Groups |journal=Annals of Mathematics |volume=59 |issue=2 |pages=196–210 |doi=10.2307/1969686 MathSciNet|id=0062125
*cite journal |last=Fox |first=Ralph |year=1956 |month=November |title=Free Differential Calculus, III: Subgroups |journal=Annals of Mathematics |volume=64 |issue=2 |pages=407–419 |doi=10.2307/1969592 MathSciNet|id=0095876
*cite journal |last=Chen |first=Kuo-Tsai |authorlink=Kuo-Tsai Chen |coauthors=Ralph Fox, Roger Lyndon |year=1958 |month=July |title=Free Differential Calculus, IV: The Quotient Groups of the Lower Central Series |journal=Annals of Mathematics |volume=68 |issue=1 |pages=81–95 |doi=10.2307/1970044 MathSciNet|id=0102539
*cite journal |last=Fox |first=Ralph |year=1960 |month=May |title=Free Differential Calculus, V: The Alexander Matrices Re-Examined |journal=Annals of Mathematics |volume=71 |issue=3 |pages=408–422 |doi=10.2307/1969936 MathSciNet|id=0111781