We show that most of the genus-zero subgroups of the braid group \(\mathbb {B}_3\) (which are roughly the braid monodromy groups of the trigonal curves on the Hirzebruch surfaces) are irrelevant as far as the Alexander invariant is concerned: there is a very restricted class of “primitive” genus-zero subgroups such that these subgroups and their genus-zero intersections determine all the Alexander invariants. Then, we classify the primitive subgroups in a special subclass. This result implies the known classification of the dihedral covers of irreducible trigonal curves.

中文翻译：

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关于三角曲线的亚历山大不变量

我们表明，辫子群\（\ mathbb {B} _3 \）（大致是Hirzebruch曲面上的三角曲线的辫子单峰群）的零属零子群与亚历山大不变式无关。有关：“原始”零族子集的类别非常有限，因此这些子群及其零族交集决定了所有亚历山大不变量。然后，我们将原始子组分类为一个特殊的子类。该结果暗示了不可约三角曲线的二面覆盖的已知分类。