Twisted Hurwitz numbers: Tropical and polynomial structures

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 [4], a new enumerative invariant called <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>b</mml:mi> </mml:math> -Hurwitz number was introduced, which enumerates non-orientable branched coverings. For <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>b</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> , we obtain twisted Hurwitz numbers which were linked to surgery theory in [1] and admit a representation as factorisations in the symmetric group. In this paper, we derive a tropical interpretation of twisted Hurwitz numbers in terms of tropical covers and study their polynomial structure.