• Corpus ID: 252532120

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

@inproceedings{Kadokami2022TheMI,
  title={The Ma-Qiu index and the Nakanishi index for a fibered knot are equal, and \$\omega\$-solvability},
  author={Teruhisa Kadokami},
  year={2022}
}

• Teruhisa Kadokami
• Published 26 September 2022
• Mathematics

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 ) ≤ a ( G, N ) in general. We also introduce a notion “ ω solvability” for a group that the intersection of all higher commutator subgroups is trivial. Our main theorem is that if N is ω -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.