Knots and Primes by Masanori Morishita

Author:Masanori Morishita
Language: eng
Format: epub
Publisher: Springer London, London

We call α j the Fox free derivative of α with respect to x j and write α j =∂α/∂x j .

Proof

For (e j =±1), we set

and for α=∑a f f∈ℤ[F], we set

Noting , one has

and hence one has, for α∈ℤ[F],

Next, suppose

Applying the Magnus embedding, one has

and therefore

By Lemma 8.1, we have (1≤j≤n). □

Proposition 8.3

The Fox derivative ∂/∂x j :ℤ[F]→ℤ[F] satisfies the following properties: (1) .

(2) .

(3) .

(4) .

Proof

(1) follows from the definition. We leave the proof of (2) to readers. Let us prove (3) and (4).

(3) By Theorem 8.2, we have

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