论文标题
纤维结的Ma-QIU指数和中nakanishi指数相等，$ω$ -SOLVISIABLE
The Ma-Qiu index and the Nakanishi index for a fibered knot are equal, and $ω$-solvability
论文作者
论文摘要
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 $ k $的模块。我们分别以$ a（g，n）$和$ m（g，n）$表示的一组$ g $及其普通sugroup $ n $的概念概括它们，然后很容易看到$ m（g，n）\ le a（g，n）$。通勤子组是微不足道的。我们的主要定理是，如果$ n $是$ω$ -solvable，那么我们就有$ m（g，n）= a（g，n）$。作为固定的$ k $，我们有$ k $，我们有$ m（k）= a（k）$，我们可以确定$ 9 $ 9 $ 9 $ 9 $ 9 $ 9 $ 9 $ 9。
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 ``$ω$-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.
