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\title[Multitriangulations on ciliated surfaces]{Multitriangulations on ciliated surfaces}

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\author[Solotko, Tung, Yang, Zhang]{Saskia Solotko\thanks{\href{mailto:saskia.solotko@tufts.edu}{saskia.solotko@tufts.edu}. S. Solotko was supported by the Tufts Summer Scholars program. }\addressmark{1}, Katherine Tung \thanks{\href{mailto:katherinetung@college.harvard.edu}{katherinetung@college.harvard.edu}. K. Tung was supported by the Harvard College Research Program. }\addressmark{2}, Mengyuan Yang \thanks{\href{mailto:myang7@swarthmore.edu}{myang7@swarthmore.edu}.}\addressmark{3}, \and Yuchong Zhang \thanks{\href{mailto:yzhang@math.harvard.edu}{yzhang@math.harvard.edu}.\\
All authors were supported by the UMN Combinatorics and Algebra REU, NSF grant DMS-2053288, and the D.E.\ Shaw group.}\addressmark{4}}

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\address{\addressmark{1}Department of Mathematics, Tufts University, Medford, MA \\ \addressmark{2}Department of Mathematics, Harvard University, Cambridge, MA\\ \addressmark{3}Department of Mathematics, Swarthmore College, Swarthmore, PA \\ \addressmark{4}Department of Mathematics, Harvard University, Cambridge, MA}

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\abstract{We generalize the definition of $k$-triangulations to arbitrary ciliated surfaces, and prove that the simplicial complex $\Delta _{\mathcal{C}_n,2}$ is pure and a weak pseudomanifold of dimension $2(n-1)$, where $\Delta _{\mathcal{C}_n,2}$ is the simplicial complex associated with $2$-triangulations on the half-cylinder, which is an annulus with $n$ marked points on one boundary. This result generalizes the work of Vincent Pilaud and Francisco Santos for polygons and resolves a conjecture of Mathias Lepoutre and Vincent Pilaud for $k=2$. To achieve this, we show that $2$-triangulations on the half-cylinder decompose as complexes of star polygons, and that $2$-triangulations on the half-cylinder are in bijection with $2$-triangulations on the $4n$-gon invariant under rotation by $\pi/2$ radians. Building on work by Vincent Pilaud and Christian Stump, we also introduce chevron pipe dreams, a new combinatorial model that more naturally captures the symmetries of $k$-triangulations. 
In addition, we propose some conjectures on $k$-triangulations of general ciliated surfaces.}

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\keywords{multitriangulations, flip graphs, pipe dreams, subword complexes}

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\section{Introduction}\label{sec:intro}


A \emph{$k$-triangulation}, or \emph{multitriangulation of order $k$}, on the convex $n$-gon is a maximal set of edges (namely, internal diagonals) such that no $k+1$ of them mutually intersect inside the polygon. See \Cref{3periodic2tri} for an example of a $2$-triangulation on the $12$-gon. When $k=1$, a $1$-triangulation corresponds to the usual notion of an ideal triangulation.

 \begin{figure}
\centering
\begin{tikzpicture}[auto=center,scale=2.7]
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\caption{A $2$-triangulation on the $12$-gon invariant under rotation by $\pi/2$ radians.  \label{3periodic2tri}}
\end{figure}

Jakob Jonsson, Vincent Pilaud, Francisco Santos, Christian Stump, and many other authors have proved various structural results about $k$-triangulations on polygons, including: Each $k$-triangulation $T$ on the $n$-gon can be regarded as a simplicial complex $\delta^T_{n,k}$, where each facet is a $k$-star \cite{PilaudSantos2009}; All $k$-triangulations on the $n$-gon have the same number of edges, which is equal to $k(2n-2k-1)$ \cite{CP92, NAKAMIGAWA2000271, DKM00}; Every $k$-relevant edge $e$ of a $k$-triangulation $T$ can be ``flipped" to a unique edge $f$ such that $(T\setminus \{e\})\cup \{f\}$ is a $k$-triangulation \cite{NAKAMIGAWA2000271, PilaudSantos2009}; There is a bijection between $k$-triangulations on the $n$-gon and reduced pipe dreams for the permutation $\pi_{n,k}=[1,\dots, k,n-k,n-k-1,\dots, k+1]$ \cite{PilaudThesis, STUMP20111794, PP12}; The simplicial complex $\Delta_{n,k}$, whose vertices are the $k$-relevant edges of the $n$-gon and whose facets correspond to $k$-triangulations, is a vertex-decomposable sphere \cite{Jonssonfake}.

Furthermore, $k$-triangulations on polygons admit many natural interpretations, such as $k$-compatible split systems \cite{DKKM01}, pseudoline arrangements on the Möbius strip \cite{PilaudThesis,PP12}, multi-cluster complexes of type A \cite{CeballosLabbeStump}, and bases of the Pfaffian variety $\mathcal{P}f_k(n)$ of antisymmetric matrices of rank $\le 2k$ \cite{CS24}.

In this paper, we generalize the notion of $k$-tri\-angulations on polygons to $k$-tri\-angulations on \emph{ciliated surfaces}, or \emph{bordered surfaces with marked points}.

\begin{definition}[ciliated surface]
    Let $\calS$ be a connected $2$-dimensional Riemann surface with boundary. Fix a finite set of marked points, denoted $M$, in the closure of $\calS$. The points in $M$ do not need to be on the boundary of $\calS$. %(including the interior and the boundary of $\calS$). 
    The pair $(\calS, M)$ is called a \emph{ciliated surface}.
\end{definition}

We will use $\calS$ to denote the ciliated surface $(\calS,M)$ when there is no ambiguity.

\begin{definition}[$k$-triangulations]
    Let $\calS$ be a ciliated surface, let $\overline{\calS}$ be its universal cover, and let $\pi: \overline{\calS} \to 
    \calS$ be an associated covering map. A \emph{(weak) $k$-triangulation} on $\calS$ is a maximal subset of edges $e$ so that the set of preimages $\pi^{-1}(e) \in \overline{\calS}$ does not contain any \emph{$(k+1)$-crossings} (namely, no $k+1$ of them mutually intersect inside the polygon).
\end{definition} 


\Cref{fig:univ-cover} is an example of a $2$-triangulation on the half-cylinder with $3$ marked points. 


\begin{figure}
    \centering 
    \includegraphics[width=0.8\linewidth]{univ-cover.png}
    \caption{A $2$-triangulation on the half-cylinder with $3$ marked points, lifted to the universal cover.
    \label{fig:univ-cover}}
    
\end{figure}

Our main result is the following.
\begin{theorem}\label{thm:main}
    Let $\calCn$ denote the annulus with $n$ marked points on one of its boundary components. The flip graph of $2$-triangulations on $\calCn$ is regular. Equivalently, the associated simplicial complex $\Delta_{\calCn,2}$ is pure and a weak pseudomanifold. 
\end{theorem}





\subsection{Amending a previous definition of $k$-triangulations on surfaces.} 
For readers familiar with the $k$-triangulation literature, we summarize in the next six paragraphs some previous attempts to define $k$-triangulations on ciliated surfaces and explain the advantages of our definition. In 2019, Mathias Lepoutre and Vincent Pilaud proposed generalizing $k$-triangulations on polygons to arbitrary surfaces by studying $k$-triangulations on their universal covers \cite[Part III]{PHDThesis}. Their $k$-triangulations are ``the projections of \textbf{periodic maximal} $(k+1)$-crossing-free edge sets on the universal cover" \cite[Definition~1.1.4~and~Definition~1.2.2~of~Part~III]{PHDThesis}. We call such objects \emph{strong $k$-triangulations}.

In contrast, our $k$-triangulations are ``the projections of \textbf{maximal periodic} $(k+1)$-crossing-free edge sets on the universal cover." We call these objects \emph{weak $k$-triangulations}.\footnote{We conceived of this definition before discovering \cite{PHDThesis}.} 

In \cite[Remark~12~of~Part~III]{PHDThesis}, it is conjectured that on any surface, the notions of strong $k$-triangulation and weak $k$-triangulation are equivalent. However, this conjecture is incorrect. In fact, on many naturally arising surfaces, weak $k$-triangulations exist and strong $k$-triangulations do not. We decided to study weak $k$-triangulations because they have a richer structure than strong $k$-triangulations in the following two ways. 

First, %the strong $k$-triangulations form a proper subset of weak $k$-triangulations on some naturally arising surfaces, and 
weak $1$-triangulations %are the right objects that 
always coincide with \emph{ideal triangulations} (maximal collection of disjoint non-null-homotopic arcs that are pairwise non-homotopic) whereas strong $1$-triangulations do not. For example, $\calCn$ has ideal triangulations but no strong $1$-triangulations. %consider the half-cylinder with $n$ marked points on a single boundary, denoted $\calCn$. While it is clear that $\calCn$ admits ideal triangulations, there are no strong $1$-triangulations on it. %Thus, the weak $k$-triangulations is a more natural generalization of ideal triangulations.

\begin{remark}\label{remark:revisedstrong}
    %While \cite[Section~1.3~of~Part~III]{PHDThesis} discussed the strong $k$-triangulations on the half-cylinder, 
    There is a problem with the proof of \cite[Proposition~1.3.3~of~Part~III]{PHDThesis}, namely that there are no strong $k$-triangulations on the half-cylinder for any $k$. (Though \cite[Figure~1.11~of~Part~III]{PHDThesis} seems to suggest otherwise, the figure does not correctly depict a strong $k$-triangulation, as a single edge connecting $u_0$ and $u_4$ can be added without creating a $4$-crossing). 
\end{remark}

Second, weak $k$-triangulations lend themselves to an underexplored analog of \cite[Theorem~4.1]{PilaudSantos2009}, which we call the Star Decomposition Theorem for polygons.
While the Star Decomposition Theorem for the strong $k$-triangulations on any finite-type surface follows from the corresponding result for infinite tidy polygons (see \cite[Section~1.2 of Part~III]{PHDThesis}), the Star Decomposition Conjecture for the weak $k$-triangulations is far more subtle. 
In this paper, we prove the Star Decomposition Conjecture for weak $k$-triangulations on the half-cylinder when $k=2$ (\Cref{2starparty}), and we regard this result as the main technical contribution of our work. 

The definition of $k$-triangulations implicitly used by Pilaud is ``the projections of a periodic maximal [$(k+1)$-crossing-free sets on the infinite polygon that contain no edge which, when extended periodically, would create a $(k+1)$-self-crossing on the universal cover]" \cite{vppc}. We further prove that on the half-cylinder, the weak $k$-triangulations are precisely the $k$-triangulations they implicitly used when $k=2$ (\Cref{maximalityfromstars}).

\begin{convention}
    In the rest of the paper, $k$-triangulations refer to weak $k$-triangulations.
\end{convention}

This paper is organized as follows. In \Cref{sec:back}, we define necessary terms 
including $k$-triangulations, $k$-stars, angles, and $k$-relevant edges. In \Cref{sec:multi}, we prove a number of properties about multitriangulations on the half-cylinder. 
A main result is \Cref{2starparty}, which serves as a generalization of the Star Decomposition Theorem for $n$-gons (\cite[Theorem~4.1]{PilaudSantos2009}). In \Cref{sec:bij}, we give a bijection between $2$-triangulations on the half-cylinder with $n$ marked points and $n$-periodic $k$-triangulations on the $2kn$-gon when $k=2$. In \Cref{sec:chev}, we define \emph{chevron pipe dreams}, a symmetric analogue of reduced pipe dreams \cite{BB93}, which captures central symmetries of $k$-triangulations through mirror symmetry and periodicity conditions. Finally, in \Cref{sec:regular}, we prove our main theorem (\Cref{thm:main}) using the aforementioned results.

This is an extended abstract summarizing our work, and the reader is referred to \cite{whole} for a complete version.

\section{Background}\label{sec:back}


Generalizing the $k$-triangulations on a convex $n$-gon, we define $k$-triangulations on the half-cylinder with $n$ marked points. Let $\calCn$ denote the annulus with $n$ marked points $\alpha_{[0]},\dots,\alpha_{[n-1]}$ on one of its boundary components, where subscripts are congruence classes modulo $n.$ We refer to $\calCn$ as a \emph{half-cylinder} or an \emph{$(n,0)$-annulus}.


The universal cover of $\calCn,$ denoted $\ocalCn,$ has vertices $\{\alpha_i\mid i\in \ZZ\}.$ Taking any vertex $v=\alpha_{[i]}\in \calCn,$ we define $v+j\coloneq \alpha_{[i+j]}$ for any integer $j.$ An \emph{edge} $e$ on $\ocalCn$ is defined by the isotopy class of topological curves connecting $u,v$ and is denoted by $[u,v]$ or $[v,u].$ Similarly, we define $e+n\coloneq [u+n,v+n].$ The covering map $\pi:\ocalCn\rightarrow \calCn,$ applied to vertices and edges is given by $\pi(\alpha_{[i]})=\alpha_i$ and $\pi([\alpha_{[i]},\alpha_{[j]}])=[\alpha_i,\alpha_j].$ We say an edge $e$ on $\ocalCn$ is a \emph{translation} of an edge $f$ if $e=f+\ell n$ for some $\ell\in \ZZ,$ or equivalently if $\pi(e)=\pi(f).$ Typically, we do not refer to vertices of $\calCn$ or $\ocalCn$ using these subscripts determined by their position.

There is a natural order of edges on $\ocalCn$ where $u>v$ if and only if $u=v+\ell$ for some $\ell\in \ZZ^{+}$ induced from a counterclockwise orientation on $\ocalCn.$ When considering a subset of vertices of $\ocalCn$, we additionally have a \emph{cyclic order} $\prec$ given by the counterclockwise cyclic order on the first polygon, as defined in \cite{PHDThesis}. On $\ocalCn,$ two edges $e_1$ and $e_2$ are said to \emph{intersect} if all representatives of the isotopy classes of $e_1$ and $e_2$ intersect, or equivalently, $a_1\prec b_1 \prec a_2 \prec b_2$ for some labeling of vertices $e_1=[a_1,b_1],e_2=[a_2,b_2].$ A \emph{$k$-crossing} on $\ocalCn$ is a set of edges $E=\{e_1,e_2,\dots, e_k\}$ that pairwise intersect.



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\end{tikzpicture} \begin{tikzpicture}[scale=2]
    \draw[line width=0.5mm,spurple] (0.766, 0.643) -- (-0.940, -0.342);
    \draw[line width=0.5mm,spurple] (-0.940, -0.342) -- (1, 0);
    \draw[line width=0.5mm,spurple] (1, 0) -- (-0.940, 0.342);
    \draw[line width=0.5mm,spurple] (-0.940, 0.342) -- (0.766, -0.643);
    \draw[line width=0.5mm,spurple] (0.766, -0.643) -- (-0.5, 0.866);
    \draw[line width=0.5mm,spurple] (-0.5, 0.866) -- (0.174, -0.985);
    \draw[line width=0.5mm,spurple] (0.174, -0.985) -- (0.174, 0.985);
    \draw[line width=0.5mm,spurple] (0.174, 0.985) -- (-0.5, -0.866);
    \draw[line width=0.5mm,spurple] (-0.5, -0.866) -- (0.766, 0.643);
    
    \draw[thick] (0.766, 0.643) node  {$\bullet$};
    \draw[thick] (0.174, 0.985) node  {$\bullet$};
    \draw[thick] (-0.5, 0.866) node  {$\bullet$};
    \draw[thick] (-0.940, 0.342) node  {$\bullet$};
    \draw[thick] (-0.940, -0.342) node  {$\bullet$};
    \draw[thick] (-0.5, -0.866) node  {$\bullet$};
    \draw[thick] (0.174, -0.985) node  {$\bullet$};
    \draw[thick] (0.766, -0.643) node  {$\bullet$};
    \draw[thick] (1, 0) node  {$\bullet$};

    
   
\end{tikzpicture}
\caption{A $2$-star in a pentagon, a $3$-star in a heptagon, and a $4$-star in a nonagon.\label{fig:kstar}}
\end{figure}


Take a subset $\{z_0,\dots, z_{2k}\}$ of vertices on $\ocalCn$ ordered such that
\[
z_0\prec z_1 \prec\cdots\prec z_{2k}.
\]
A \emph{$k$-star polygon}, or a \emph{$k$-star} $S$ on this subset, as defined in \cite{PilaudSantos2009} and \cite{PHDThesis}, contains all vertices $z_i$ and edges $[z_i,z_{i+k}]$ on $\ocalCn$ with subscripts reduced modulo $n.$ Examples of $k$-stars are given in \Cref{fig:kstar}. We use these definitions of $k$-crossings and $k$-stars on $\ocalCn$ to define analogously $k$-crossings and $k$-stars on $\calCn.$

As a generalization of the $k$-stars and the $k$-triangulations on $n$-gons, we introduce the following definitions.

\begin{definition}
A \emph{$k$-star} on $\calCn$ is the projection $\pi(S)$ of a $k$-star $S$ on $\ocalCn.$
\end{definition}


\begin{definition}
A \emph{k-triangulation} $T$ on $\calCn$ is a maximal set of edges such that the lifting $\pi^{-1}(T)$ is $(k+1)$-crossing-free. 
\end{definition}


We additionally define a \emph{$k$-triangulation} $\oT$ on $\ocalCn$ to be a maximal set of \emph{$n$-periodic} edges on $\ocalCn,$ that is, $v\in \oT \Longleftrightarrow v+n\in \oT.$ Equivalently, $\pi(\oT)$ is a $k$-triangulation on $\calCn$.

For vertices $u,v\in \ocalCn,$ we say $v-u=j$ if $u+j=v.$ The \emph{length} of an edge $[u,v]\in \ocalCn$ is defined by $|v-u|.$ We also say $\pi([v,u])$ has \emph{length} $|v-u|.$ 

An edge $[u,v]$ of $\ocalCn$ can appear in a $(k+1)$-crossing only if $|v-u|>k.$ Thus, each $k$-triangulation $T$ (respectively, $\oT$) contains all edges of length $\leq k.$ For simplicity, we adopt the following convention.

\begin{convention}\label{convention:exclude_short_edge}
    In this paper, when discussing a $k$-triangulation $T$ on $\calCn$ (respectively, $\oT$ on $\ocalCn$), we exclude all the edges of length $<k$.
\end{convention}


As we will show in \Cref{length>kn} later, a $k$-triangulation $T$ (respectively, $\oT$) contains no edges of length $>kn.$  
These observations inform us to introduce the following definitions, analogous to those in \cite{PHDThesis} and \cite{PilaudSantos2009}. An edge $[u,v]\in \oT$ is a \emph{$k$-relevant} edge if $k<|v-u|\leq kn.$ An edge $[u,v]\in \oT$ is a \emph{$k$-boundary} edge if $|v-u|=k.$ An \emph{angle} of a $k$-triangulation $\oT$ on $\ocalCn$ is given by a pair of edges $[u,v],[v,w]\in \oT$ such that $u \prec v \prec w$ and there exists no edge $[v,w_0]$ in $\oT$ with $w_0 \prec u \prec v \prec w.$ Such an angle is denoted by $\angle(u,v,w).$ An angle $\angle(u,v,w)$ of $\oT$ is called \emph{$k$-relevant} if at least one of $[u,v]$ and $[v,w]$ is $k$-relevant with length $<kn$. Given an angle $\angle(u,v,w)$ of $\oT$, an edge $[v,w_0]$ with $w_0 \prec u \prec v \prec w$ is called an \emph{angle bisector} of $\angle(u,v,w)$; or equivalently, we say $[v,w_0]$ \emph{bisects} the angle $\angle(u,v,w)$. An edge $[v,w_0]$ might also be an angle bisector of more than one angle, or an angle bisector of an angle contained in a collection of edges; in these cases we abuse terminology and just call $[v,w_0]$ an angle bisector. 
    



\section{Multitriangulations on the half-cylinder}\label{sec:multi}

In this section, we prove a series of properties about multitriangulations on the half-cylinder. The main result is \Cref{2starparty}, which generalizes the Star Decomposition Theorem for $n$-gons (\cite[Theorem~4.1]{PilaudSantos2009}) to $2$-triangulations on the half-cylinder with $n$ marked points.

In the following lemmas, let $T$ denote a $k$-triangulation on $\calCn$, which lifts to $\pi^{-1}(T)=\oT$ on $\ocalCn$.

\begin{lemma}\label{length>kn}
    A $k$-triangulation $T$ of $\calCn$ contains no edge of length $\ell > kn$, and exactly one edge of length $kn$.
\end{lemma}


In the remainder of this section, we consider only $2$-triangulations on $\calCn.$

\begin{lemma}\label{lemma:philipslemma}
Let $E=\{[a_1,b_1],[a_2,b_2],[a_3,b_3]\}$ be a $3$-crossing labeled such that \[a_1<a_2<a_3<b_1<b_2<b_3.\] Suppose that for $i=1$ or $i=2,$ the edge $[a_{i+1},b_{i+1}]$ is a translation of $[a_{i},b_{i}].$ Then $\pi^{-1}(\pi(E))$ contains a $3$-crossing that includes exactly one translation of $[a_i,b_i].$

\end{lemma}


In the following lemmas, fix a $2$-triangulation $T$ of $\calCn$ which lifts to a $2$-triangulation $\pi^{-1}(T)=\oT$ on $\ocalCn$ and a $2$-relevant angle $\angle(u,v,w)$ of $\oT.$ 
 We say an edge \emph{intersects} $\angle(u,v,w)$ if it intersects both $[u,v]$ and $[v,w].$ 
 %One could note that there always exists a $2$-boundary edge intersecting $\angle (u,v,w).$ 
 The following definitions are consistent with \cite{PilaudSantos2009} and \cite{PHDThesis}. We say an edge $e=[a,b]$ is $v$-\emph{farther} than $f=[c,d]$ if $u\prec a \preceq c \prec v \prec d \preceq b \prec w$ and $e\neq f.$ We say $e$ is $v$-\emph{maximal} if there does not exist any edge intersecting $\angle(u,v,w)$ that is $v$-farther than $e.$
 Let $e=[a,b]$ be the $v$-maximal edge intersecting $\angle{(u,v,w)}$ with order $u \prec a \prec v \prec b \prec w$. We work in the next two lemmas towards showing that $[u,b]$ and $[a,w]$ are edges in $\oT.$

\begin{lemma}\label{lemma:lessthan2n}
The edges $[u,b]$ and $[a,w]$ have length at most $2n.$ 
\end{lemma}

\begin{lemma}\label{lemma:3crossing2crossing}
Assume that there exists a $2$-crossing $F = \{f_1, f_2\} \subset \oT\cup \pi^{-1}(\pi([u,b]))$ such that $F\cup\{[u,b]\}$ is a $3$-crossing and $F$ contains a translation of $[u,b]$. Then there exists a $2$-crossing $F' \subset \oT$ intersecting $[u,b]$ such that $F' \cup\{[u,b]\}$ is a $3$-crossing.
\end{lemma}

Using the previous lemmas, we could prove the Star Decomposition Theorem for the half-cylinder for $k=2$.


\begin{theorem}[Star Decomposition Theorem]\label{2starparty}
Let $T$ be a $2$-triangulation on $\calCn$ which lifts to $\oT$ on $\ocalCn$, then any $2$-relevant angle $\angle (u,v,w)$ in $\oT$ belongs to a unique $2$-star in $\oT.$
\end{theorem}

We also have the following corollary that relates the $k$-triangulations on the half-cylinder and the $k$-triangulations on its universal cover.

\begin{corollary}[Maximal Lifting Theorem]\label{maximalityfromstars}
Given a $2$-triangulation $T$ on $\calCn$ which lifts to $\oT$ on $\ocalCn,$
let $e$ be an edge on $\ocalCn$ of length $\le 2n$ that is not contained in $\oT$. Then $\oT\cup \{e\}$ contains a $3$-crossing.
\end{corollary}


We conjecture that \Cref{2starparty} and \Cref{maximalityfromstars} on the half-cylinder can be generalized for arbitrary $k$ \cite[Conjectures~3.8, 3.9]{whole}. 
Furthermore, we conjecture that analogous results hold for the $k$-triangulations on all finite-type ciliated surfaces as well. Following \cite[Section~1.2~of~Part~III]{PHDThesis}, we view any bordered surface $\calS$ as the quotient of the hyperbolic disc $\D$ by a Fuchsian group $\Gamma$, where $\D=\ocalS$ is the universal cover of $\calS$. In this setting, we likewise define the \emph{length} of an edge on $\D$ (note that this length may not always be finite). Consistent with \Cref{convention:exclude_short_edge}, we exclude all edges of length $<k$ when discussing a $k$-triangulation on $\ocalS$ (or $\calS$).

\begin{conjecture}[Star Decomposition Conjecture]\label{conj:othersurface_stardecompk}
    Given a finite-type ciliated surface $\calS$ and its universal cover $\pi:\ocalS\rightarrow\calS$. Let $T$ be a $k$-triangulation on $\calS$ which lifts to $\oT$ on $\ocalS.$ Let $\angle (u,v,w)$ be an angle in $\oT$. Suppose that $[u,v]$ and $[v,w]$ are not the preimage of the same edge $e\in T$ under $\pi$, then $\angle (u,v,w)$ belongs to a unique $k$-star contained in $\oT.$ 
\end{conjecture}

\begin{conjecture}[Maximal Lifting Conjecture]\label{conj:othersurface_maxliftingk}[cf.\ \cite[Remark~12~of~Part~III]{PHDThesis} and \Cref{remark:revisedstrong}]
    Given a finite-type ciliated surface $\calS$ and its universal cover $\pi:\ocalS\rightarrow\calS$. Let $T$ be a $k$-triangulation on $\calS$ which lifts to $\oT$ on $\ocalS.$ Let $e$ be an edge on $\ocalS$ such that $e$ is not contained in $\oT$ and $\pi^{-1}(\pi(e))$ does not contain a $(k+1)$-crossing. Then, $\oT\cup \{e\}$ contains a $(k+1)$-crossing.
\end{conjecture}

\section{Bijection: $2$-triangulations on the half-cylinder and $n$-periodic $2$-triangulations on the $4n$-gon}\label{sec:bij}

\begin{definition}
    An \emph{$n$-periodic} $k$-triangulation on a $2kn$-gon is a $k$-triangulation on the $2kn$-gon that is invariant under rotation by $\frac{2\pi}{2k} = \frac{\pi}{k}$ radians.
\end{definition}

An example of a $3$-periodic $2$-triangulation on the $12$-gon was given in \Cref{3periodic2tri}.


\begin{theorem}
\label{thm:annular-triangulations-as-periodic}
    There is a canonical bijection between $2$-triangulations on $\calCn$ and $n$-periodic $2$-triangulations on the $4n$-gon.
\end{theorem}

As shown in \cite{PilaudSantos2009}, every $k$-triangulation on the $2kn$-gon has exactly $2kn-2k$ $k$-stars, $k(2kn-2k-1)$ $k$-relevant edges, and $k(4kn-2k-1)$ edges. The following counts then follow almost directly from the bijection in \Cref{thm:annular-triangulations-as-periodic}.

\begin{corollary}\label{Cnpurity}
Any $2$-triangulation on $\calCn$ contains exactly $n-1$ $2$-stars, $2(n-1)$ $2$-relevant edges, and $2(2n-1)$ total edges.
\end{corollary}



 \section{Chevron pipe dreams}\label{sec:chev}

 In the 2010s, Vincent Pilaud and Christian Stump independently established a connection between pipe dreams and multitriangulations \cite[Section~4.1.4]{PilaudThesis}, \cite[Theorem~2.1]{STUMP20111794} \cite[Section~7]{PP12}. Starting from a $k$-triangulation on the $n$-gon, the following is the construction of the pipe dream in the $(n-1)\times (n-1)$  staircase polyomino shape given by them:
 in the NW staircase of an $(n-1)\times (n-1)$ grid as follows: 
 Label the rows $n,n-1,\dots, 2$ from top to bottom and the columns $1,2,\dots, n-1$ from left to right.
 In each box $(i,j)$ where there is an edge between vertices $i$ and $j$ in the $k$-triangulation, place a bump tile \begin{mosaic}{0.5}
     \tiJF \\
 \end{mosaic} \ . In all other boxes, 
 place a crossing tile \begin{mosaic}{0.5}
     \tiX \\
 \end{mosaic}. \ This process yields a natural bijection between $k$-triangulations on the $n$-gon and reduced pipe dreams for the permutation $\pi_{n,k} := [1, \dots, k, n - k, n - k - 1, \dots, k + 1] \in \mathfrak{S}_{n - k}$ \cite{STUMP20111794}. For clarity, we call them \emph{staircase pipe dreams}. Further, reduced staircase pipe dreams for the permutation $\pi_{n,k}$ are in natural bijection with subsets $P$ of the word $Q=c^k\text{w}_{\circ}(c)$ such that $Q\setminus P$ is a reduced expression for $w_0$ \cite{CeballosLabbeStump}. 

 In \cite{CeballosLabbeStump}, an analogous result is shown in type $B.$ 
 In particular, $k$-triangulations of the $2m$-gon invariant under rotation by $180^\circ$ are in bijection with subsets $P$ of the word $Q=c^m$ such that $Q\setminus P$ is a reduced expression for $w_0\in B_{m-k.}$ In \cite{whole}, we construct \emph{chevron pipe dreams} which arise naturally from the bijection between type $B$ $k$-triangulations and these subword complexes. In particular, chevron pipe dreams more naturally capture the possible symmetries of $k$-triangulations. 

 \begin{figure}\label{fig:regular-pd}
 \centering
 \begin{minipage}{0.45\textwidth}
 \centering
          \begin{mosaic}{0.5}
     \Sbrown\& \Sbrown\& \tiX\& \tiX\& \Spink\& \tiX\& \tiX\& \tiX\& \tiX\& \tiJ \\ 
     \Sbrown\& \Steal\& \tiX \& \tiX\& \Sorange\& \Spink\& \Syellow\& \Steal\& \tiJ\\
     \tiX\& \Syellow\& \tiX\& \tiX\& \tiX\& \tiX\& \tiX\& \tiJ \\
     \tiX\& \Spink\& \tiX\& \tiX\& \tiX\& \tiX\& \tiJ \\
     \tiX\& \Sorange\& \Spink\& \Syellow\& \Steal\& \tiJ \\
     \tiX\& \tiX\& \tiX\& \tiX\& \tiJ \\ 
     \tiX\& \tiX\& \tiX\& \tiJ \\ 
     \Syellow\& \Steal\& \tiJ \\
     \tiX\& \tiJ \\
     \tiJ \\
     \end{mosaic}
     \caption{The staircase pipe dream corresponding to the $3$-periodic $2$-triangulation on a $12$-gon in \Cref{3periodic2tri}.}
 \end{minipage}
\hfill
\begin{minipage}{0.45\textwidth}
\centering
    \begin{mosaic}{0.5}
     \& \& \& \& \& \tiF\& \& \& \& \\
     \& \& \& \& \tiF\& \Steal \& \& \& \& \\
     \& \& \& \tiF\& \tiX\& \Syellow\& \& \& \& \\
     \& \& \tiF\& \tiX\& \tiX\& \Spink\& \& \& \& \\
     \& \tiF\& \Steal\& \tiX\& \tiX\& \Sorange\& \Spink\& \Syellow\& \Steal\& \tiJ \\
     \tiF\& \tiX\& \Syellow\& \tiX\& \tiX\& \tiX\& \tiX\& \tiX\& \tiJ \\
     \tiX\& \tiX\& \Spink\& \tiX\& \tiX\& \tiX\& \tiX\& \tiJ \\
     \tiX\& \tiX\& \Sorange\& \Spink\& \Syellow\& \Steal\& \tiJ \\
     \tiX\& \tiX\& \tiX\& \tiX\& \tiX\& \tiJ \\
     \tiX\& \tiX\& \tiX\& \tiX\& \tiJ \\
 \end{mosaic}

     \caption{The chevron pipe dream which results from applying the cutting and gluing operations as outlined in \cite{whole} to the staircase pipe dream in \cref{fig:regular-pd}.}
     \label{fig:reattach2}
\end{minipage}

 \end{figure}

\begin{lemma}
 The chevron pipe dream $\Chev_{2m,k}$ corresponding to a $k$-triangulation of the $2m$-gon is reduced, and moreover, each pair of pipes crosses exactly once.
\end{lemma}

 With \cref{thm:annular-triangulations-as-periodic} as an intermediate step, we use chevron pipe dreams to show that every edge of a $2$-triangulation on $\calCn$ can be flipped uniquely, leading to results which are discussed further in \cref{sec:regular}.
 

\section{The flip graph of rotationally symmetric $2$-triangulations}\label{sec:regular}



Let $\tilde{E}_{2kn,n}$ denote the set of equivalence classes of edges of the $2kn$-gon of the convex $n$-gon after identification via rotation by $\frac{\pi}{k}.$ Each element of $\tilde{E}_{2kn,n}$ is an $n$-periodic edge, namely, a set $\tilde{e}$ containing an edge $e$ and its rotational copies by integer multiples of $\frac{\pi}{k}.$ We identify an $n$-periodic $k$-triangulation on the $2kn$-gon $T$ with a subset $\tilde{T}\subset \tilde{E}_{2kn,n}.$ 
When $k=2$, elements of $\tilde{E}_{4n,n}$ correspond to edges of $\calCn$ as described in \Cref{thm:annular-triangulations-as-periodic}, and $\tilde T$ corresponds to a $2$-triangulation on $\calCn.$

In the following, let $T$ be an $n$-periodic 2-triangulation on the $4n$-gon, and let $\tilde{e} \in \tilde{T}$ be a set of edges containing a 2-relevant edge $e$ and its rotational copies, which we call a \emph{$2$-relevant $n$-periodic edge}.



\begin{figure}
    \centering
   \begin{tikzpicture}[auto=center,scale=2]

   
    
    
    \draw[line width=0.5mm,steal] (0, 1) -- (0, -1);
    \draw[line width=0.5mm,steal] (1, 0) -- (-1, 0);

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    \draw[line width=0.5mm,steal] (0.866, 0.5) -- (0.5, -0.866);
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    \draw[line width=0.5mm,sgrey] (0.5, 0.866) -- (1, 0);
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    \draw[line width=0.5mm,sgrey] (0, -1) -- (0.866, -0.5);

    \draw[line width=0.5mm,sgrey] (0, 1) -- (0.866, 0.5);
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    \draw[line width=0.5mm,sgrey] (0.866, 0.5) -- (0.5, 0.866);
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    \draw[line width=0.5mm,sgrey] (0, 1) -- (-0.5, 0.866);
    \draw[line width=0.5mm,sgrey] (-0.5, 0.866) -- (-0.866, 0.5);
    \draw[line width=0.5mm,sgrey] (-0.866, 0.5) -- (-1, 0);
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\begin{tikzpicture}[auto=center,scale=2]

   
    
    
    \draw[line width=0.5mm,steal] (0, 1) -- (0, -1);
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    \draw[line width=0.5mm,sgrey] (0, -1) -- (0.866, -0.5);

    \draw[line width=0.5mm,sgrey] (0, 1) -- (0.866, 0.5);
    \draw[line width=0.5mm,sgrey] (-1, 0) -- (-0.5, 0.866);
    \draw[line width=0.5mm,sgrey] (0, -1) -- (-0.866, -0.5);
    \draw[line width=0.5mm,sgrey] (0.5, -0.866) -- (1, 0);

    \draw[line width=0.5mm,sgrey] (0.866, 0.5) -- (0.5, 0.866);
    \draw[line width=0.5mm,sgrey] (0.5, 0.866) -- (0, 1);
    \draw[line width=0.5mm,sgrey] (0, 1) -- (-0.5, 0.866);
    \draw[line width=0.5mm,sgrey] (-0.5, 0.866) -- (-0.866, 0.5);
    \draw[line width=0.5mm,sgrey] (-0.866, 0.5) -- (-1, 0);
    \draw[line width=0.5mm,sgrey] (-1, 0) -- (-0.866, -0.5);
    \draw[line width=0.5mm,sgrey] (-0.866, -0.5) -- (-0.5, -0.866);
    \draw[line width=0.5mm,sgrey] (-0.5, -0.866) -- (0, -1);
    \draw[line width=0.5mm,sgrey] (0, -1) -- (0.5, -0.866);
    \draw[line width=0.5mm,sgrey] (0.5, -0.866) -- (0.866, -0.5);
    \draw[line width=0.5mm,sgrey] (0.866, -0.5) -- (1, 0);
    \draw[line width=0.5mm,sgrey] (1, 0) -- (0.866, 0.5);

    \draw[line width=0.5mm,syellow] (-0.866, -0.5) -- (0, 1);
    \draw[line width=0.5mm,syellow] (0.866, 0.5) -- (0, -1);
    \draw[line width=0.5mm,syellow] (1, 0) -- (-0.5, 0.866);
    \draw[line width=0.5mm,syellow] (0.5, -0.866) -- (-1, 0);



    \draw[thick,red] (0.866, 0.5) node  {$\bullet$};
    \draw[thick,Green] (0.5, 0.866) node {$\bullet$};
    \draw[thick,blue] (0, 1) node {$\bullet$};
    \draw[thick,red] (-0.5, 0.866) node {$\bullet$};
    \draw[thick,Green] (-0.866, 0.5) node {$\bullet$};
    \draw[thick,blue] (-1, 0) node {$\bullet$};
    \draw[thick,red] (-0.866, -0.5) node {$\bullet$};
    \draw[thick,Green] (-0.5, -0.866) node {$\bullet$};
    \draw[thick,blue] (0, -1) node {$\bullet$};
    \draw[thick,red] (0.5, -0.866) node {$\bullet$};
    \draw[thick,Green] (0.866, -0.5) node {$\bullet$};
    \draw[thick,blue] (1, 0) node {$\bullet$};
\end{tikzpicture}
    \caption{(Left) The $3$-periodic $2$-triangulation $T$ of the $12$-gon. 
    (Right) The $2$-triangulation $(T\setminus \tilde e)\cup \tilde f$.}
    \label{fig:3-2triangulation}
\end{figure}


\begin{proposition}\label{flipexist}
For each $2$-relevant $n$-periodic edge $\tilde{e} \in \tilde{T}$, there exists a unique $2$-relevant $n$-periodic edge $\tilde{f} \notin \tilde{T}$ such that replacing $\tilde{e}$ with $\tilde{f}$ yields another $n$-periodic 2-triangulation $(\tilde{T} \setminus \tilde{e}) \cup \tilde{f}.$
\end{proposition}

We say that we obtain the $n$-periodic $2$-triangulation on the $4n$-gon $(T\setminus \tilde e)\cup \tilde f$ from $T$ by flipping the $n$-periodic edge $e.$ An example of a $3$-periodic $2$-triangulation of the $12$-gon $T$ and the result of flipping a $3$-periodic edge $\tilde{e}$ are illustrated in \cref{fig:3-2triangulation}. Let $G^n_{4n,2}$ denote the \emph{flip graph} of $n$-periodic $2$-triangulations on the $4n$-gon, the graph whose vertices are $n$-periodic $2$-triangulations on the $4n$-gon and whose edges are pairs related by a flip. Equivalently, $G^n_{4n,2}$ is the graph of flips of $2$-triangulations on the half-cylinder $\calCn.$


\begin{theorem}\label{flipgraphthm}
The graph $G^n_{4n,2}$ is regular of degree $2(n-1).$
\end{theorem}

\begin{proof}


The regularity of $G^n_{4n,2}$ follows from the count of $2$-relevant edges in \Cref{Cnpurity}, in addition to the fact that every $2$-relevant $n$-periodic edge can be flipped and this flip is unique, as shown in \Cref{flipexist}. 
\end{proof}

We additionally define a simplicial complex $\Delta_{\calCn,k}$ whose vertices are $k$-relevant edges of $\calCn$ and whose facets correspond to $k$-triangulations on $\calCn.$ \Cref{maximalityfromstars} shows that the faces of $\Delta _{\calCn,2}$ are sets of $k$-relevant edges of $\calCn$ such that the lifting to $\ocalCn$ is $(k+1)$-crossing-free. The results of this paper additionally imply the following:



\begin{corollary}\label{cor:weak pseudomanifold}
The simplicial complex $\Delta _{\calCn,2}$ is pure of dimension $2(n-1)$; moreover, it is a weak pseudomanifold (without boundary).
\end{corollary}


\begin{proof}
Purity of dimension $2(n-1)$ follows from the count of $2$-relevant edges in \Cref{Cnpurity}. As every $k$-relevant edge can be uniquely flipped as shown in \Cref{flipexist}, each codimension $1$ face lies in exactly $2$ distinct facets, and thus $\Delta _{\calCn,2}$ is a weak pseudomanifold without boundary. % the simplicial complex is non-branching.
\end{proof}

\begin{proof}[Proof of \Cref{thm:main}]
    This follows directly from \Cref{flipgraphthm} and \Cref{cor:weak pseudomanifold}.
\end{proof}

In \cite[Theorem~2.1]{STUMP20111794}, the simplicial complex $\Delta_{n,k},$ whose vertices are diagonals of the $n$-gon of length greater than $k$ and whose facets correspond to $k$-triangulations, is shown to be naturally isomorphic to a subword complex in type $A_{n-k-1}.$ 

In \cite[Theorem~2.10]{CeballosLabbeStump}, an analogous result is shown in type $B.$ In particular, the $B_{m-k}$ subword complex considered has vertices corresponding to a diagonal and its image under rotation by $180^\circ,$ and facets corresponding to $k$-triangulations of the $2m$-gon invariant under rotation by $180^\circ.$


We give an analogous result for the simplicial complex $\Delta_{\calCn,2}$ below. Let $Q$ be a word in the generators $S$ of a finite Coxeter group $W.$ For an element $\pi\in W,$ the subword complex $\mathcal{SC}(Q,\pi),$ which was introduced in \cite{KM05}, has faces given by subwords $P$ of $Q$ for which the complement $Q\setminus P$ contains a reduced expression of $\pi.$ 


\begin{proposition}\label{subwordcxunion}
    The simplicial complex $\Delta_{\calCn,2}$ is isomorphic to a union of subword complexes $\bigcup_{w^2=w_0} \mathcal{SC}(c^{n},w),$ where $w_0$ is the longest element of the Coxeter group $B_{2n-2}$ and $c$ is the word $s_1,s_2\cdots, s_{2n-2}.$ 
\end{proposition}


\begin{conjecture}
    The simplicial complex $\Delta_{\calCn,k}$ is isomorphic to a union of subword complexes $\bigcup_{w \in \mathcal A_{k}}\mathcal{SC}(c^n,w),$ where $w_0$ is the longest element of the Coxeter group $B_{kn-k},$ $c$ is the word $s_1,s_2,\dots ,s_{nk-k},$ and $\mathcal A_k \coloneqq \{w\mid w^k=w_0,\ell(w)=\ell(w_0)/k\}\subset B_{kn-k}.$ 
\end{conjecture}

Note that when $k\geq3,$ for $w$ satisfying $w^k=w_0$, it is not automatically true that $\ell(w)=\ell(w_0)/k.$

\begin{conjecture}
   The simplicial complex $\Delta_{\calCn,k}$ has $\binom{2(n-1)}{n-1}^k$ facets.
\end{conjecture}

In \cite[Chapter 6]{Jahn}, a specific class of unions of subword complexes, called \emph{glued subword complexes}, are shown to be vertex-decomposable spheres. As we have shown that $\Delta _{\calCn,2}$ is a weak pseudomanifold -- analogous to the result for $n$-gons \cite[Theorem~1]{Jonssonfake}, we make the following conjecture.%that $\Delta _{\calCn,2}$ is, moreover, a piecewise linear sphere. 

\begin{conjecture}\label{conj:spherical k}
The simplicial complex $\Delta _{\calCn,k}$ is a piecewise linear sphere.
\end{conjecture}


By \cite[Proposition~2.3]{MR2448067}, the only ciliated surfaces whose multitriangulation complexes are finite (that is, contain finitely many simplices) are polygons and half‑cylinders. Hence, an affirmative answer to \Cref{conj:spherical k}, together with \cite[Theorem~1]{Jonssonfake}, would imply that all finite multitriangulation complexes are piecewise linear spheres.

\acknowledgements{
    We thank the UMN Combinatorics and Algebra REU program; our mentor, Pavlo Pyl\-yavskyy; our TA, Joe McDonough; and Ayah Almousa, Ilani Axelrod-Freed, Kieran Favazza, Sergey Fomin, Carl Lian, Molly MacDonald, Vincent Pilaud, Victor Reiner, George Seelinger, Roberta Shapiro, Volkmar Welker, and Lauren Williams for their support.
}



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