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\title
[Tropical twisted Hurwitz numbers]
{Twisted Hurwitz numbers: Tropical and polynomial structures}
\author[\initial{M.} \middlename{A.} Hahn]{\firstname{Marvin} \middlename{Anas} \lastname{Hahn}}
\address{Trinity College Dublin\\
School of Mathematics\\
17 Westland Row\\
Dublin\\
Ireland}
\email{hahnma@tcd.ie}
\urladdr{https://www.marvinanashahn.com/}
\author[\initial{H.} Markwig]{\firstname{Hannah} \lastname{Markwig}}
\address{Eberhard-Karls-University Tuebingen\\
Fachbereich Mathematik\\
Auf der Morgenstelle 10\\
72076 T\"{u}bingen\\
Germany}
\keywords{Tropical geometry, Hurwitz numbers}
\subjclass{14T15, 14N10, 57M12, 05C30}
\crefname{theo}{Theorem}{Theorems}
\crefname{prop}{Proposition}{Propositions}
\crefname{conj}{Conjecture}{Conjectures}
\crefname{ques}{Question}{Questions}
\crefname{coro}{Corollary}{Corollaries}
\crefname{lemm}{Lemma}{Lemmas}
\crefname{defi}{Definition}{Definitions}
\crefname{nota}{Notation}{Notations}
\crefname{rema}{Remark}{Remarks}
\crefname{exam}{Example}{Examples}
\crefname{section}{Section}{Sections}
\crefname{subsection}{Subsection}{Subsections}
\crefname{figure}{Figure}{Figures}
\crefname{construction}{Construction}{Constructions}
\datepublished{2024-09-03}
\begin{document}
\newtheorem{construction}[cdrthm]{Construction}
\begin{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~\cite{chapuy2020non}, a new enumerative invariant called \textit{$b$-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~\cite{BF21} 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.
\end{abstract}
\maketitle
\section{Introduction}
Hurwitz numbers are enumerations of branched morphisms between Riemann surfaces with fixed numerical data. They go back to work by Adolf Hurwitz in the 1890s~\cite{hurwitz1892algebraische} and are now important invariants in enumerative geometry.
They admit various equivalent descriptions in the language of different areas of mathematics, e.g., as shown by Hurwitz in his above-mentioned work, they can be computed by an enumeration of transitive factorisations in the symmetric group. This equivalence gives rise to a deep connection between Hurwitz theory and the representation theory of the symmetric group; it will also play a key role in the present work. Moreover, Hurwitz numbers are closely related to the algebraic topology underlying Riemann surfaces, since they turn out to be \textit{topological invariants}. While the theory of Hurwitz numbers has been dormant for most of the 20th century, the close relationship between Hurwitz and Gromov--Witten theory discovered in the 1990s has rekindled interest in these enumerative invariants and led to several exciting developments.\vspace{\baselineskip}
\subsection{Hurwitz numbers, Gromov--Witten theory and variants}
When studying relations between Hurwitz numbers and Gromov--Witten theory, certain classes of Hurwitz numbers with particularly well--behaved structures take center stage. Among these classes are so-called \textit{double Hurwitz numbers}, which are defined as follows.
\begin{defi}[Double Hurwitz numbers]\label{def-hur}
Let $g\ge0$ be a non-negative integer, $n>0$ a positive integer and $\mu,\nu$ partitions of $n$. Moreover, we fix $p_1,\dots,p_b\in\mathbb{P}^1$, where $b=2g-2+\ell(\mu)+\ell(\nu)$. Then, we define a cover of type $(g,\mu,\nu)$ to be a map $f\colon S\to\mathbb{P}^1$, such that
\begin{itemize}
\item $S$ is a connected Riemann surface of genus $g$;
\item the ramification profile of $0$ is $\mu$;
\item the ramification profile of $\infty$ is $\nu$;
\item the ramification profile of $p_1,\dots,p_b$ is $(2,1\dots,1)$.
\end{itemize}
Two covers $f\colon S\to\mathbb{P}^1$ and $f'\colon S'\to\mathbb{P}^1$ are called equivalent if there exists a homeomorphism $g\colon S\to S'$, such that $f= f'\circ g$.
Then, we define \textbf{double Hurwitz numbers} as
\begin{equation*}
h_g(\mu,\nu)=\sum_{[f]}\frac{1}{|\mathrm{Aut}(f)|},
\end{equation*}
where the sum runs over all equivalence classes of covers of type $(g,\mu,\nu)$.
When $\nu=(1,\dots,1)$, we call $h_g(\mu,\nu)$ a \textbf{single Hurwitz number} and denote it by~$h_g(\mu)$.
\end{defi}
At the core of the relationship between double Hurwitz numbers and Gromov--Witten theory is a polynomial structure in the prescribed ramification data of these enumerative invariants. First discovered in the seminal work of Goulden, Jackson and Vakil in~\cite{GJV05}, double Hurwitz numbers exhibit a \textit{piecewise polynomial} behaviour. More precisely, we consider the space \begin{equation*}
\mathcal{H}_{m.n}\coloneqq\{(\mu,\nu)\in\mathbb{N}^m\times\mathbb{N}^n\mid \sum \mu_i=\sum \nu_j\}
\end{equation*}
of partitions $(\mu,\nu)$ of fixed lengths $m,n$ and of the same size. For fixed $g$, we consider the map
\begin{align*}
h_g\colon \mathcal{H}_{m,n}&\to\mathbb{Q}\\
(\mu,\nu)&\mapsto h_g(\mu,\nu)
\end{align*}
which parametrises double Hurwitz numbers.
The authors of~\cite{GJV05} showed that there exists a hyperplane arrangement $\mathcal{R}_{m,n}$ in $\mathcal{H}_{m,n}$ (called the resonance arrangement), such that the map $h_g$ restricted to each connected component (called \textit{chamber}) of $\mathcal{H}_{m,n}\backslash\mathcal{R}_{m,n}$ may be represented as a polynomial in the entries of $\mu$ and $\nu$. In~\cite{SSV08, CJM11, Joh15}, the natural question of how the polynomials differ from chamber to chamber was studied. It was observed that there is a recursive structure in the sense that this difference can be expressed by double Hurwitz numbers with smaller input data. This is called a \textit{wall-crossing formula}.
We want to highlight the work in~\cite{CJM11}, in which a graph theoretic approach towards the polynomiality of double Hurwitz numbers was established. The key technique in this paper revolves around the field of \textit{tropical geometry}. Tropical geometry is a relatively new field of mathematics, which may be described as a combinatorial shadow of algebraic geometry. The tropical geometry perspective allows to degenerate algebraic curves to certain metric graphs that are called \textit{tropical curves}. In this manner, branched morphisms between Riemann surfaces are \textit{tropicalised} to maps between tropical curves that are called \textit{tropical covers}. Motivated by this point of view, a combinatorial interpretation of double Hurwitz numbers in terms of tropical covers was derived in~\cite{CJM10}, which laid the groundwork for the analysis of the polynomial behaviour of double Hurwitz numbers undertaken in~\cite{CJM11}. In particular, by proceeding along an intricate combinatorial analysis of tropical covers in different chambers, the authors of~\cite{CJM11} were able to derive the desired wall-crossing structure.
In the past years, several variants of Hurwitz numbers have appeared in the literature in a plethora of different contexts. Among the most prominent ones are so-called \textit{pruned Hurwitz numbers}~\cite{do2018pruned,hahn2020bi}, \textit{monotone Hurwitz numbers}~\cite{goulden2014monotone}, \textit{strictly monotone Hurwitz numbers}~\cite{kazarian2015virasoro}, \textit{completed cycles Hurwitz numbers}~\cite{okounkov2006gromov} and many more. For all of these variants the piecewise polynomiality of the double Hurwitz numbers analogue was established and for the majority a wall-crossing structure as well (see e.g.~\cite{zbMATH06791415, hahn2018wall,hahn2020wall,shadrin2012double}).
While classical Hurwitz theory deals with the enumeration of branched morphisms between \textit{orientable surfaces}, it is very natural to ask for an analogous theory for non-orientable surfaces. Such a new and exciting theory for so-called \textit{$b$-Hurwitz numbers} was introduced in~\cite{chapuy2020non}.
\subsection{Twisted Hurwitz numbers}\label{sec-twistedsingle}
The construction of $b$-Hurwitz numbers is based on the following idea: Let $\overline{\mathbb{H}}$ be the compactified complex upper half-plane of $\mathbb{P}^1$ and $\mathcal{J}$ the corresponding natural involution on $\mathbb{P}^1$. A generalised branched covering is a covering $f\colon S\to\overline{\mathbb{H}}$ where $S$ is a not-necessarily connected compact orientable surface with orientation double cover $\hat{S}$, such that $f$ may be ``lifted'' to a branched covering $\hat{S}\to\mathbb{P}^1$. We give a precise formulation in~\cref{sec:twistdoub}. Via these generalised coverings the authors of~\cite{chapuy2020non} introduce a new one-parameter deformation of classical Hurwitz numbers called $b$-Hurwitz numbers in reference to the $b$-conjecture by Goulden and Jackson in the context of Jack polynomials~\cite{goulden1996connection}. In order to obtain $b$-Hurwitz numbers, the authors of~\cite{chapuy2020non} associate a non-negative integer $\nu_p(f)$ to any generalised branched covering $f$ which ``measures'' the non-orientability of the the surface $\hat{S}$. This non-negative integer is zero if and only if $\hat{S}$ is orientable. Based on this idea $b$-Hurwitz numbers are defined -- depending on a \textit{measure of non-orientability} $p$ -- as a sum over generalised branched coverings weighted by $b^{\nu_p(f)}$. Thus, one obtains a Hurwitz-type enumeration for any value of $b$. For example, under the convention that $0^0=1$, one recovers classical Hurwitz numbers for $b=0$. For $b=1$, one obtains enumerations of generalised branched coverings which are called \textit{twisted Hurwitz numbers}. This is the case we study in the present paper.
The term twisted Hurwitz numbers was coined in~\cite{BF21} in the context of \textit{surgery theory}. Surgery theory studies the construction of new manifolds from given ones via cutting and gluing, such that key properties are preserved. In~\cite{BF21}, the enumeration of decompositions of a given surface with boundary and marked points is studied. The term \textit{twisted} is motivated by the fact in~\cite{BF21} gluings are performed with respect to a twist of the natural boundary orientations. It was proved in~\cite{BF21} that the enumeration of certain decompositions with respect to such a twist may be computed in terms of factorisations in the symmetric group, reminiscent of Hurwitz' result in his original work~\cite{hurwitz1892algebraische}. More precisely, we fix the involution $$\tau= (1\;\; n+1) (2 \;\;n+2) \ldots (n \;\;2n)\in \mathbb{S}_{2n},$$ and use the notation
\begin{equation*}
B_n=C(\tau)=\{\sigma \in \mathbb{S}_{2n} \mid \sigma \tau \sigma^{-1}=\tau\}, \; C^\sim(\tau)=\{\sigma \in \mathbb{S}_{2n} \mid \tau \sigma \tau^{-1} = \tau \sigma \tau = \sigma^{-1}\},
\end{equation*}
where $B_n$ is the hyperoctahedral group. We further define the subset $B^\sim_n \subset C^\sim(\tau)$ consisting of those permutations that have no self-symmetric cycles (see \cite[Lemma~2.1]{BF21}). We then set, for a partition $\lambda$ of $n$, $B^\sim_\lambda\subset B^\sim_n$ to consist of those permutations that have $2\ell(\lambda)$ cycles, two of length $\lambda_i$ for each i, that pair up under conjugation with~$\tau$. We are now ready to define \textbf{twisted single Hurwitz numbers} in terms of the symmetric group.
\begin{defi}[Twisted single Hurwitz numbers,~\cite{BF21}]
\label{def:twisthur}
Fix a partition $\lambda$ of $n$ and a number $b$ (the number of transpositions).
Then define
\begin{align*}
\tilde{h}_{b}(\lambda)= \frac{1}{n!}\sharp \Big\{(\sigma_1,\ldots,\sigma_b) \mid \sigma_s=(i_s\;\;j_s),\, j_s\neq \tau(i_s),\, \sigma_1\ldots\sigma_b (\tau\sigma_b\tau)\ldots (\tau\sigma_1\tau) \in B^{\sim}_\lambda\Big\}.
\end{align*}
\end{defi}
Maybe surprisingly, it was then proved in \cite[Theorem 3.2]{BF21} that these numbers coincide with $b$-Hurwitz numbers for $b=1$ by showing that the generating series of both invariants satisfy the same PDE with equal initial data.
\subsection{Tropical geometry of twisted Hurwitz numbers}
The present paper develops a tropical theory of twisted Hurwitz numbers and demonstrates some first applications. In~\cref{sec:twistdoub}, we define a generalisation of~\cref{def:twisthur} to twisted double Hurwitz numbers $\tilde{h}_g(\mu,\nu)$ which by the same arguments as in \cite[Theorem 3.2]{BF21} coincides with $b$-Hurwitz numbers for $b=1$. This generalisation arises naturally from the symmetric group expression for twisted single Hurwitz numbers. Moreover, we define in~\cref{sec:troptwist} a tropical analogue of twisted double Hurwitz numbers in terms of tropical covers. We prove in~\cref{sec:corr} that twisted double Hurwitz numbers coincide with their tropical counterpart, thus giving a tropical correspondence theorem for these enumerative invariants. This allows us to derive a purely graph-theoretic interpretation of twisted double Hurwitz numbers in~\cref{sec:monodr} by reinterpreting the tropical covers as directed graphs.
Finally, we employ this expression of twisted double Hurwitz numbers as a weighted enumeration of directed graphs to study the polynomiality of twisted Hurwitz numbers in~\cref{sec:poly}. Finally, we discuss the wall-crossing behaviour of twisted Hurwitz numbers.
\section{Twisted double Hurwitz numbers}
\label{sec:twistdoub}
In this section, we define twisted double Hurwitz numbers as a factorization problem in the symmetric group, generalizing the case of single twisted Hurwitz numbers discussed in \cref{sec-twistedsingle}. We use the notation of \cref{sec-twistedsingle}. To begin with, we recall that
\begin{equation*}
C^\sim(\tau)=\{\sigma\in \mathbb{S}_{2n} \mid \tau \sigma \tau^{-1} = \tau \sigma \tau = \sigma^{-1}\}.
\end{equation*}
It was proved in \cite[Lemma 2.1]{BF21} for $\sigma \in C^\sim(\tau)$ with the decomposition in cycles $\sigma=c_1\cdots c_m$, we have for any $i$ that either
\begin{itemize}
\item there exists $j\neq i$ with $\tau c_i\tau=c_j^{-1}$ or
\item we have $\tau c_i\tau=c_i^{-1}$ and $c_i$ has even length.
\end{itemize}
In the first case, $c_i$ and $c_j$ are called \textit{$\tau$-symmetric}, while in the second case $c_i$ is called \textit{self-symmetric}. As mentioned above, we denote by $B_n^\sim\subset C^\sim(\tau)$ the set of permutations without self-symmetric cycles and by $B^\sim_\lambda\subset B^\sim_n$ the set of permutations in $B^\sim_n$ with $2\ell(\lambda)$ cycles and for each $i$ two cycles of length $\lambda_i$ that are $\tau$-symmetric.
\begin{defi}[Cycle type]
Let $\sigma\in B^{\sim}_n$. We denote its cycle type by $C(\sigma)$, which is a partition of $2n$ recording the lengths of the cycles of $\sigma$.
\end{defi}
For a partition $\mu$ of $n$ we denote by $2\mu$ the partition of $2n$ with twice as many parts, where each part is repeated once.
We may now define twisted double Hurwitz numbers generalising~\cref{def:twisthur}.
\begin{defi}[Twisted double Hurwitz numbers]
\label{def:twisthurdou}
Let $g\ge0,n>0$ and $\mu,\nu$ partitions of $n$. We define $C_g(\mu,\nu)$ as the set of tuples $(\sigma_1,\eta_1,\dots,\eta_b,\sigma_2)$,
such that we have:
\begin{enumerate}
\item $b=\frac{2g-2+2\ell(\mu)+2\ell(\nu)}{2}>0$,
\item $C(\sigma_1)=2\mu$, $C(\sigma_2)=2\nu$, $\eta_i$ are transpositions satisfying $\eta_i\neq \tau \eta_i \tau$,
\item $\sigma_1\in B^\sim_\mu$,
\item $\eta_b\cdots\eta_1\sigma_1(\tau\eta_1\tau)\cdots(\tau\eta_b\tau)=\sigma_2$,
\item the subgroup
$$\langle \sigma_1,\eta_1,\ldots,\eta_{b},\tau \eta_1\tau, \ldots,\tau\eta_b\tau,\sigma_2\rangle$$ acts transitively on the set $\{1,\ldots,2d\}$.
\end{enumerate}
Then, we define the associated \textit{twisted double Hurwitz number} as
\begin{equation*}
\tilde{h}_{g}(\mu,\nu)=\frac{1}{(2n)!!}|C_g(\mu,\nu)|.
\end{equation*}
When we drop the transitivity condition, we obtain possibly disconnected twisted double Hurwitz numbers which we denote $\tilde{h}_{g}^\bullet(\mu,\nu)$.
\end{defi}
\begin{rema}[Conventional differences]
Compared with the definition of twisted single Hurwitz numbers in \cref{def:twisthur}, there are two conventional differences:
\begin{enumerate}
\item Rather than the number of transpositions, we use the genus (of the source of a twisted tropical cover, see \cref{def-twistedtropcover}) as subscript in the notation. In the usual case of Hurwitz numbers (without twisting) counting covers, this corresponds to the genus of the source curves. By the Riemann-Hurwitz formula, the genus $g$ and the number $b$ of transpositions (i.e.\ simple branch points) are related via
$$b=\frac{2g-2+2\ell(\mu)+2\ell(\nu)}{2}.$$
\item We choose to normalize with the factor $\frac{1}{(2n)!!}$ rather than $\frac{1}{n!}$. This leads to nicer formulae and structural results.
\end{enumerate}
\end{rema}
\begin{rema}[Connectedness and transitivity]
As noted above, one can also drop the transitivity condition (5) in \cref{def:twisthurdou} to obtain $\tilde{h}_g^\bullet(\mu,\nu)$. On the tropical side, this amounts to allowing disconnected tropical curves as source of a twisted tropical cover, see \cref{def-twistedtropcover}. In this setting, it can happen that a disconnected twisted tropical cover contains two twisted components which both just correspond to a single edge without any interior vertex. This also happens in the connected case if we allow $b=0$. For this case, one has to adapt the tropical multiplicity we set in \cref{def:twisttrophn}: a twisted tropical cover which consists of a pair of twisted single edges of weight $\mu$ each has multiplicity $\frac{1}{\mu}$. With this adaption, one can easily generalize our results to the disconnected case.
\end{rema}
As mentioned in the introduction, twisted Hurwitz numbers first appeared in the Hurwitz theory of non-orientable surfaces. More precisely, we denote by $\mathcal{J}\colon\mathbb{P}^1\to\mathbb{P}^1$ the complex conjugation, by $\mathbb{H}\coloneqq\{z\in\mathbb{C}\mid\mathrm{Im}(z)\ge0\}$ the complex upper half-plane and by $\overline{\mathbb{H}}\coloneqq\mathbb{H}\cup\{\infty\}$ the compactified complex upper half-plane. Moreover, we denote by $\pi\colon\mathbb{P}^1\to\overline{\mathbb{H}}$ the quotient map.
We call continuous maps $f\colon S\to\overline{\mathbb{H}}$ \textit{generalised branched coverings}, if $S$ is a not-necessarily orientable surface and there exists a further map $\hat{f}\colon \hat{S}\to\mathbb{P}^1$ with
\begin{enumerate}
\item $p\colon\hat{S}\to S$ is the orientation double cover,
\item $\pi\circ\hat{f}=f\circ p$,
\item all the branch points of $\hat{f}$ are real.
\end{enumerate}
Let $\mathcal{T}\colon \hat{S}\to\hat{S}$ be an orientation reversing involution without fix points, such that $p\circ \mathcal{T}=p$. Then, the second condition of $\hat{f}$ may be reformulated as $\hat{f}\circ\mathcal{T}=\mathcal{J}\circ \hat{f}$. As $\mathcal{T}$ has no fixed points, for any branch points $c\in\mathbb{P}^1_{\mathbb{R}}\subset\mathbb{P}^1$ of $\hat{f}$, the points in its pre-image come in pairs $(a,\mathcal{T}(a))$ with the same ramification index. Thus, the degree of $\hat{f}$ is even and the ramification profile of $c$ repeats any entry twice, e.g. $(\lambda_1,\lambda_1,\dots,\lambda_s,\lambda_s)$. We then say that $f$ has ramification profile $(\lambda_1,\dots,\lambda_s)$ at $\pi(s)\in\partial\overline{\mathbb{H}}$. We further call two generalised branched coverings $f_1$ and $f_2$ equivalent if their lifts $\hat{f_1}$ and $\hat{f_2}$ are, and denote the equivalence class of $f$ by $[f]$.
\begin{defi}\label{def-nonoriented}
Let $g\ge0$, $n>0$, $\mu,\nu$ partitions of $n$. Let $b=\frac{2g-2+2\ell(\mu)+2\ell(\nu)}{2}$ and fix $p_1,\dots,p_b$ pairwise distinct real points on $\overline{\mathbb{H}}$. We define $G_g(\mu,\nu)$ as the set of equivalence classes $[f]$ of generalised branched coverings $f\colon S\to\overline{\mathbb{H}}$, such that
\begin{itemize}
\item $f$ is of degree $n$,
\item $f$ has ramification profile $\mu$ over $0$ and $\nu$ over $\infty$,
\item $f$ has ramification profile $(2,1,\dots,1)$ over $p_i$.
\end{itemize}
Then, we define $1$-Hurwitz numbers as
\begin{equation*}
h^1_g(\mu,\nu)=\sum_{[f]\in G_g(\mu,\nu)}\frac{1}{|\mathrm{Aut}(f)|}.
\end{equation*}
\end{defi}
The parameter $g$ used here is not equal to the genus of the surface $S$, this is $g'=\frac{g+1}{2}.$
The following was proved in \cite[Theorem 3.2]{BF21} for $\nu=(1,\dots,1)$. However, the same approach works for arbitrary $\nu$. In particular, the idea is that in \cite[Theorem 6.5]{chapuy2020non} a recursion of $b$--Hurwitz numbers was derived. Moreover, twisted Hurwitz numbers were proved to satisfy a recursion in \cite[Theorem 2.12]{BF21}. It turns out that these recursions only differ by a factor of $2$. The same argument as in the proof of \cite[Theorem 3.2]{BF21} proves the following result.
\begin{theo}
Let $g\ge0$, $n>0$ and $\mu$, $\nu$ partitions of $d$. Then, we have
\begin{equation*}
h^1_g(\mu,\nu)=2^{-b}\tilde{h}_g^{\bullet}(\mu,\nu).
\end{equation*}
\end{theo}
\section{Tropical twisted covers and twisted Hurwitz numbers}
\label{sec:troptwist}
In~\cite{CJM10}, tropical Hurwitz numbers have been introduced as a count of tropical covers, parallel to the count of covers of algebraic curves from \cref{def-hur}. We start by recalling the basic notions of tropical curves and covers. Then we introduce twisted tropical covers, which can roughly be viewed as tropical covers with an involution. By fixing branch points, we produce a finite count of twisted tropical covers for which we show in the following section that it coincides with the corresponding twisted double Hurwitz number. Readers with a background in the theory of tropical curves are pointed to the fact that we only consider explicit tropical curves in the following, i.e.\ there is no genus hidden at vertices.
\begin{defi}[Abstract tropical curves]
An abstract tropical curve is a connected graph $\Gamma$ with the following data:
\begin{enumerate}
\item The vertex set of $\Gamma$ is denoted by $V(\Gamma)$ and the edge set of $\Gamma$ is denoted by $E(\Gamma)$.
\item The $1$-valent vertices of $\Gamma$ are called \textit{leaves} and the edges adjacent to leaves are called \textit{ends}.
\item The set of edges $E(\Gamma)$ is partitioned into the set of ends $E^\infty(\Gamma)$ and the set of \textit{internal edges} $E^0(\Gamma)$.
\item There is a length function
\begin{equation*}
\ell\colon E(\Gamma)\to\mathbb{R}\cup\{\infty\},
\end{equation*}
such that $\ell^{-1}(\infty)=E^\infty(\Gamma)$.
\end{enumerate}
The genus of an abstract tropical curve $\Gamma$ is defined as the first Betti number of the underlying graph, i.e.\ $g=1+\# E^0(\Gamma)-\# V(\Gamma)$. An isomorphism of abstract tropical curves is an isomorphism of the underlying graphs that respects the length function. The combinatorial type of an abstract tropical curve is the underlying graph without the length function.
\end{defi}
We are now ready to define the notion of a tropical cover. We restrict to the case where the target $\Gamma_2$ is a subdivided version of $\mathbb{R}$, i.e.\ a line with some $2$-valent vertices.
\begin{defi}[Tropical covers]
A tropical cover of a subdivided version of $\mathbb{R}$, $\Gamma_2$, is a surjective harmonic map between abstract tropical curves $\pi\colon \Gamma_1\to\Gamma_2$, i.e.:
\begin{enumerate}
\item We have $\pi(V(\Gamma_1))\subset V(\Gamma_2)$.
\item Let $e\in E(\Gamma_1)$. Then, we interpret $e$ and $\pi(e)$ as intervals $[0,\ell(e)]$ and $[0,\ell(\pi(e))]$ respectively. We require $\pi$ restricted to $e$ to be a bijective integer linear function $[0,\ell(e)]\to[0,\ell(\pi(e))]$ given by $t\mapsto \omega(e)\cdot t$, with $\omega(e) \in \mathbb{Z}$. If $\pi(e)\in V(\Gamma_2)$, we define $\omega(e)=0$. We call $\omega(e)$ the \textit{weight} of $e$.
\item For a vertex $v\in V(\Gamma_1)$, we denote by $\mathrm{Inc}(v)$ the set of incoming edges at $v$ (edges adjacent to $v$ mapping to the left of $\pi(v)$) and by $\mathrm{Out}(v)$ the set of outgoing edges at $v$ (edges adjacent to $v$ mapping to the right of $\pi(v)$). We then require
\begin{equation*}
\sum_{e\in\mathrm{Inc}(v)}\omega(e)=\sum_{e\in\mathrm{Out}(v)}\omega(e).
\end{equation*}
This number is called the local degree of $\pi$ at $v$.
We call this equality the \textit{harmonicity} or \textit{balancing condition}.
For a point $v$ in the interior of an edge $e$ of $\Gamma_1$, the local degree of $\pi$ at $v$ is defined to be the weight $\omega(e)$.
\end{enumerate}
Moreover, we define the \textit{degree} of $\pi$ as the sum of local degrees of all preimages in $\Gamma_1$ of a given point of $\Gamma_2$. The degree is independent of the choice of point of $\Gamma_2$. This follows from the harmonicity condition.
For any end $e$ of $\Gamma_2$, we define a partition $\mu_e$ as the partition of weights of ends of~$\Gamma_1$ mapping to $e$. We call $\mu_e$ the \textit{ramification profile} of $e$.
We call two tropical covers $\pi_1\colon\Gamma_1\to\Gamma_2$ and $\pi_2\colon\Gamma_1'\to\Gamma_2$ equivalent if there exists an isomorphism $g\colon\Gamma_1\to\Gamma_1'$ of metric graphs, such that $\pi_2\circ g=\pi_1$.
\end{defi}
\begin{defi}[Twisted tropical covers]\label{def-twistedtropcover}
We define a twisted tropical cover of type $(g,\mu,\nu)$ to be a tropical cover $\pi\colon\Gamma_1\to\Gamma_2$ with an involution $\iota\colon\Gamma_1\to\Gamma_1$ which respects the cover $\pi$, such that:
\begin{itemize}
\item The target $\Gamma_2$ is a subdivided version of $\mathbb{R}$ with vertices $\{p_1,\dots,p_b\}=V(\Gamma_2)$, where $p_i0$ the given graph is counted with the weight
\begin{equation*}
2^3\cdot (\delta-1)\delta^2.
\end{equation*}
However, in the chamber $\delta<0$, the graph contributes a weight of
\begin{equation*}
2^3\cdot (-\delta-1)\delta^2.
\end{equation*}
Therefore, factoring out the edge weights is not possible in this situation, which represents the first obstacle for the tropical approach of~\cite{CJM11}.
The second obstacle arises from the possible number of orientations after gluing together the cut-graphs. This is illustrated in~\cref{fig:badexcut}. In particular, we observe that the $2$-valent vertex imposes the condition that it lies between the upper and the lower vertex of the respective cut-graphs. The enumeration of possible orderings is therefore not a simple binomial coefficient and depends on the graph at hand.
\end{exam}
\begin{figure}
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\draw [line width=1.5] (47.48,371) -- (623,371) ;
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\draw [color={rgb, 255:red, 0; green, 0; blue, 0 } ,draw opacity=1 ][fill={rgb, 255:red, 0; green, 0; blue, 0 } ,fill opacity=1 ][line width=1.5] (305.88,330) .. controls (305.88,326.96) and (308.34,324.5) .. (311.38,324.5) .. controls (314.42,324.5) and (316.88,326.96) .. (316.88,330) .. controls (316.88,333.04) and (314.42,335.5) .. (311.38,335.5) .. controls (308.34,335.5) and (305.88,333.04) .. (305.88,330) -- cycle ;
\draw (26.49,78.4) node [anchor=north west][inner sep=0.75pt] {$\mu $};
\draw (26.49,119.4) node [anchor=north west][inner sep=0.75pt] {$\mu $};
\draw (27.99,159.4) node [anchor=north west][inner sep=0.75pt] {$\nu $};
\draw (27.99,39.4) node [anchor=north west][inner sep=0.75pt] {$\nu $};
\draw (630.76,10.4) node [anchor=north west][inner sep=0.75pt] {$\lambda $};
\draw (629.26,193.4) node [anchor=north west][inner sep=0.75pt] {$\lambda $};
\draw (630.51,70.4) node [anchor=north west][inner sep=0.75pt] {$\kappa $};
\draw (633.51,129.4) node [anchor=north west][inner sep=0.75pt] {$\kappa $};
\draw (233,84.4) node [anchor=north west][inner sep=0.75pt] {$\delta $};
\draw (234,118.4) node [anchor=north west][inner sep=0.75pt] {$\delta $};
\draw (306,74.4) node [anchor=north west][inner sep=0.75pt] {$\delta $};
\draw (302,126.4) node [anchor=north west][inner sep=0.75pt] {$\delta $};
\draw (27.49,277.4) node [anchor=north west][inner sep=0.75pt] {$\mu $};
\draw (632,277.4) node [anchor=north west][inner sep=0.75pt] {$\lambda $};
\draw (30.99,362.4) node [anchor=north west][inner sep=0.75pt] {$\nu $};
\draw (629,361.4) node [anchor=north west][inner sep=0.75pt] {$\kappa $};
\draw (271,295.4) node [anchor=north west][inner sep=0.75pt] {$\delta $};
\draw (370,327.4) node [anchor=north west][inner sep=0.75pt] {$\delta $};
\end{tikzpicture}
\caption{The graph on the top contributes to $\tilde{h}_{0}((\mu,\nu),(\lambda,\kappa))$. The graph on the bottom is the quotient by the involution.}
\label{fig:badex}
\end{figure}
\begin{figure}
\tikzset{every picture/.style={line width=0.75pt}}
\begin{tikzpicture}[scale=0.75,x=0.75pt,y=0.75pt,yscale=-1,xscale=1]
\draw [line width=1.5] (30.99,59) -- (153.89,59) ;
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\draw [line width=1.5] (153.89,59) -- (604.51,59) ;
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\draw [line width=1.5] (299.68,132.11) -- (435.89,172) ;
\draw (10,47.4) node [anchor=north west][inner sep=0.75pt] {$\mu $};
\draw (614.51,47.4) node [anchor=north west][inner sep=0.75pt] {$\lambda $};
\draw (253.51,65.4) node [anchor=north west][inner sep=0.75pt] {$\delta $};
\draw (15.5,163.4) node [anchor=north west][inner sep=0.75pt] {$\nu $};
\draw (613.51,162.4) node [anchor=north west][inner sep=0.75pt] {$\kappa $};
\draw (354.51,128.4) node [anchor=north west][inner sep=0.75pt] {$\delta $};
\end{tikzpicture}
\caption{The graph at the bottom of~\cref{fig:badex} cut along the edge $\delta$.}
\label{fig:badexcut}
\end{figure}
\longthanks{We would like to thank an anonymous referee for their thorough work and for numerous helpful suggestions on how to improve the first versions of this paper. We thank Rapha\"el Fesler and Veronika K\"orber for useful discussions and comments.
The second author acknowledges support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), Project-ID 286237555, TRR 195. Computations have been made using the Computer Algebra System \textsc{GAP} and \textsc{OSCAR}~\cite{GAP4, Oscar}.}
\bibliographystyle{mersenne-plain}
\bibliography{ALCO_Hahn_945}
\end{document}