Abstract
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.
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Acknowledgements
I thank Prof. Degtyarev under whose supervision I completed this work. He introduced me to the subject and constantly encouraged me during the preparation of this paper.
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The author was partially supported by the TÜBİTAK Grant 118F413.
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Üçer, M. On the Alexander invariants of trigonal curves. Rev Mat Complut 35, 265–286 (2022). https://doi.org/10.1007/s13163-020-00381-9
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DOI: https://doi.org/10.1007/s13163-020-00381-9