The Ma–Qiu index and the Nakanishi index for a fibered knot are equal, and ω-solvability

Abstract

For a knot [Formula: see text] in [Formula: see text], let [Formula: see text] be the knot group of [Formula: see text], [Formula: see text] the Ma–Qiu (MQ) index, which is the minimal number of normal generators of the commutator subgroup of [Formula: see text], and [Formula: see text] the Nakanishi index of [Formula: see text], which is the minimal number of generators of the Alexander module of [Formula: see text]. We generalize the notions for a pair of a group [Formula: see text] and its normal subgroup [Formula: see text], and we denote them by [Formula: see text] and [Formula: see text] respectively. Then it is easy to see [Formula: see text] in general. We also introduce a notion “[Formula: see text]-solvability” for a group that the intersection of all higher commutator subgroups is trivial. Our main theorem is that if [Formula: see text] is [Formula: see text]-solvable, then we have [Formula: see text]. As corollaries, for a fibered knot [Formula: see text], we have [Formula: see text], and we could determine the MQ indices of prime knots up to nine crossings completely.