The Ma-Qiu index and the Nakanishi index for a fibered knot are equal, and $\omega$-solvability

Abstract

For a knot $K$ in $S^3$, let $G(K)$ be the knot group of $K$, $a(K)$ the Ma-Qiu index (the MQ index, for short), which is the minimal number of normal generators of the commutator subgroup of $G(K)$, and $m(K)$ the Nakanishi index of $K$, which is the minimal number of generators of the Alexander module of $K$.We generalize the notions for a pair of a group $G$ and its normal sugroup $N$, and we denote them by $a(G, N)$ and $m(G, N)$ respectively.Then it is easy to see $m(G, N)\le a(G, N)$ in general.We also introduce a notion ``$\omega$-solvability" for a group that the intersection of all higher commutator subgroups is trivial.Our main theorem is that if $N$ is $\omega$-solvable, then we have $m(G, N)=a(G, N)$.As corollaries, for a fibered knot $K$, we have $m(K)=a(K)$, and we could determine the MQ indices of prime knots up to $9$ crossings completely.