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arXiv · 2210.00595

Twisted Hurwitz numbers: Tropical and polynomial structures

Marvin Anas Hahn·Hannah Markwig
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Abstract

Hurwitz numbers count covers of curves satisfying fixed ramification data. Via monodromy representation, this counting problem can be transformed to a problem of counting factorizations in the symmetric group. This and other beautiful connections make Hurwitz numbers a longstanding active research topic. In recent work Chapuy and Dol\k{e}ga, a new enumerative invariant called b-Hurwitz number was introduced, which enumerates non-orientable branched coverings. For b=1, we obtain twisted Hurwitz numbers which were linked to surgery theory in work of Burman and Fesler and admit a representation as factorisations in the symmetric group. In this paper, we derive a tropical interperetation of twisted Hurwitz numbers in terms of tropical covers and study their polynomial structure.

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Marvin Anas Hahn, Hannah Markwig. 2022-10-02. Twisted Hurwitz numbers: Tropical and polynomial structures. https://arxiv.org/abs/2210.00595

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