Twisted Hurwitz numbers: Tropical and polynomial structures

TL;DR

This paper explores twisted Hurwitz numbers, providing a tropical geometric interpretation and analyzing their polynomial structure, thereby extending the understanding of enumerative invariants related to branched coverings.

Contribution

It introduces a tropical interpretation of twisted Hurwitz numbers and investigates their polynomial properties, linking combinatorial, geometric, and algebraic perspectives.

Findings

Tropical covers provide a new geometric interpretation of twisted Hurwitz numbers.

Twisted Hurwitz numbers exhibit polynomial structure in certain parameters.

Connections established between tropical geometry and symmetric group factorizations.

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.