Number of Triangulations of a Möbius Strip

Véronique Bazier-Matte Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269-1009, United States of America [email protected] , Ruiyan Huang Lester B. Pearson College, BC, Canada [email protected] and Hanyi Luo The Affiliated High School of South China Normal University [email protected]

Abstract.

Consider a Möbius strip with $n$ chosen points on its edge. A triangulation is a maximal collection of arcs among these points and cuts the strip into triangles. In this paper, we proved the number of all triangulations that one can obtain from a Möbius strip with $n$ chosen points on its edge is given by $4^{n-1}+\binom{2n-2}{n-1}$ , then we made the connection with the number of clusters in the quasi-cluster algebra arising from the Möbius strip.

1. Introduction

We consider the Möbius strip as a marked surface, that is, a two-dimensional Riemann surface $\mathbf{S}$ with a set $\mathbf{M}$ of marked points on the boundary of $\mathbf{S}$ . We cut this surface into triangles with arcs, which are curves on the marked surface. A maximal collections of arcs without intersections is called a triangulation. Such an example of triangulation is given at Figure 1.

Refer to caption — Figure 1. A triangulation of Möbius strip with three marked points

We initiate a study about the number of triangulations of the Möbius strip, the only non-oriented surface with a finite number of triangulations.

In the theory of quasi-cluster algebras developed by Dupont and Palesi [DP15], the triangulations of a Möbius strip correspond naturally to the clusters in the quasi-cluster algebra from a Möbius strip, which is explained in section 4.

Our main result is the following theorem:

Theorem 1.1.

The number of triangulations of a Möbius strip with $n$ marked points is given by

4^{n-1}+\binom{2n-2}{n-1}.

To prove this result, we count the number of triangulations in an iterative way. Then we simplify the equation into a more succinct form.

2. Preliminaries

In this section, we start by reviewing some notions about non-orientable surfaces, and aim to define the triangulations of these surfaces. The review combines results from [FST08] and [DP15].

Let $\mathbf{S}$ be a compact 2-dimensional manifold with boundary and let $\mathbf{M}$ be a finite set of marked points of $\mathbf{S}$ such that each boundary component contains at least one marked point. We denote the surface $(\mathbf{S},\mathbf{M})$ . In the scope of this paper, we only consider the case when all the points of $\mathbf{M}$ are on the boundary. Moreover, to exclude pathological cases, we do not allow $\mathbf{S}$ to be a monogon or digon.

Intuitively, a surface is orientable if moving continuously any figure on this surface cannot result in the mirror image of the figure when it is back to its starting point. Else, it is non-orientable.

Definition 2.1.

Let $\mathbf{S}$ be a surface. A closed curve on $\mathbf{S}$ is defined as two-sided if it has a regular, orientable neighbourhood. If any neighbourhood is non-orientable, the curve is one-sided.

A surface is non-orientable if and only if there is a one-sided curve on it. A Möbius strip is such an example. This will be an important notion in the proof of Proposition 3.3.

Definition 2.2.

A cross-cap is obtained from an annulus by identifying antipodal points on the inner boundary. This is represented (as in Figure 2) by drawing a cross inside the inner circle.

Definition 2.3.

The isotopy class of a curve $\gamma$ is the set of embeddings of $\gamma$ that can be deformed into each other through a continuous path of homomorphisms such that:

•

the endpoints of each embedding are always the same;
•

except for its endpoints, each embedding is disjoint from $\mathbf{M}$ and the boundary.

We define two types of curves without self-intersections in $(\mathbf{S},\mathbf{M})$ . The first type of curves without self-intersections consists of closed one-sided curves in the interior of $S$ . The second type consists of curves with both endpoints in $M$ . We refer to the union of isotopy classes of these two types as arcs.

Two arcs are compatible if there exist representatives in their isotopy class not intersecting with each other.

Example 2.4.

In figure 4, even if they intersect each other, the curves $\alpha$ and $\beta$ in (a) are representatives of compatible arcs, because $\alpha^{\prime}$ and $\beta^{\prime}$ in (b) are representatives of the same respective isotopy classes and they do not intersect.

Note that what we refer to as arcs, Dupont and Palesi refer to as quasi-arcs, [DP15, Definition 1].

Let $\mathbf{B(\mathbf{S},\mathbf{M})}$ denote the set of boundary segments, that is, the set of connected components of the boundary of $\mathbf{S}$ with the points of $\mathbf{M}$ removed.

With the above definitions, we are now able to introduce triangulations of a surface.

Definition 2.5.

A triangulation of $(\mathbf{S},\mathbf{M})$ is a maximal collection of compatible arcs. Each area of $(\mathbf{S},\mathbf{M})$ cut by this maximal collection of compatible arcs is a triangle.

Example 2.6.

The Figure 5 is the same triangulation as in Figure 1.

Remark that the definition implies that if another arc can be added into the current collection of arcs, then this is not a triangulation.

Once again, our terminology does not totally coincide with that of Dupont and Palesi. What we call a triangulation, they would call a quasi-triangulation, [DP15, Defintion 6].

Let $M_{n}$ denote the Möbius strip with $n$ marked points on its boundary.

Example 2.7.

Figure 6 shows a triangulation of $M_{2}$ .

Remark 2.8.

In some cases, the triangles formed as a result of triangulation can have two edges identified; these triangles are called anti-self-folded triangles. The identified edges must go through the cross-cap and all the three vertices are also identified.

Example 2.9.

There is an anti-self-folded triangle in the Figure 6 formed by the arcs $a$ and $b$ . The three edges are represented with different colors, with the yellow one and the green one identified.

Definition 2.10.

A quasi-triangle consists in a monogon encircling only a cross-cap and a one-sided curve through the cross-cap in the interior of $\mathbf{S}$ , as in Figure 7.

Figure 7. A quasi-triangle

Proposition 2.11.

[DP15, Proposition 7] A triangulation cuts $M_{n}$ into a finite union of triangles and quasi-triangles.

Proposition 2.12.

[DP15, Example 13] The number of arcs in a triangulation of $M_{n}$ is equal to $n$ the number of marked points.

Theorem 2.13.

[DP15, Theorem 45] Let $(\mathbf{S},\mathbf{M})$ be a surface with finite number of triangulations and marked points on the boundary of $\mathbf{S}$ . If $(\mathbf{S},\mathbf{M})$ is orientable, it is an $n$ -gon ( $n\geq 3$ ); if $(\mathbf{S},\mathbf{M})$ is non-orientable, it is a Möbius strip.

3. Number of Triangulations of A Möbius Strip

Recall that the Catalan numbers are defined recursively as:

(3.1)

C_{n}=\sum_{i=0}^{n-1}C_{i}C_{n-1-i}=\frac{(4n-2)C_{n-1}}{n+1},\text{ for }n\geq 1,\;C_{0}=1

and is defined explicitly by

(3.2)

C_{n}=\frac{1}{n+1}\binom{2n}{n},

Lemma 3.1.

The number of triangulations of a $(n+2)$ -gon is given by $C_{n}$ , [Lam38].

We provide this simple proof because we will use a similar approach to the Möbius strip case.

Proof.

We prove by induction that $C_{n}$ is the number of triangulations of a $(n+2)$ -gon.

Label the vertices of the $(n+2)$ -gon from $1$ to $n+2$ . For each triangulation of the $(n+2)$ -gon, there is a single triangle containing the edge between the vertices $1$ and $n+2$ . Denote the third vertex of this triangle by $i$ . By the induction hypothesis, there are $C_{i-2}$ triangulations of the $i$ -gon on one side of this triangle and $C_{n+1-i}$ triangulations of the $(C_{n+3-i})$ -gon on the other side of this triangle, see Figure 8. Here $2\leq i\leq n+1$ . Therefore, the number of triangulations of a $(n+2)$ -gon is

\sum_{i=2}^{n+1}C_{i-2}C_{n+1-i}=\sum_{i=0}^{n-1}C_{i}C_{n-1-i}=C_{n}.

∎

Example 3.2.

The pentagon has 5 triangulations shown in the Figure 9.

Proposition 3.3.

The number of triangulations of a Möbius strip with $n$ marked points is given by $T_{n}$ , where

T_{n}=2\sum_{i=0}^{n-2}C_{i}T_{n-i-1}+nC_{n-1}\text{ for }n\geq 2,\text{ with }T_{1}=2.

Proof.

Let $(\mathbf{S},\mathbf{M})$ be a marked surface and let $\mathbf{T}$ be a triangulation of $(\mathbf{S},\mathbf{M})$ .

Recall that in any triangulation, each boundary edge is part of one and only one triangle. Therefore, there is a single triangle containing the edge between the marked points $n$ and $1$ ; denote this triangle $\Delta$ . It must be of one of the following three forms shown in Figure 10.

Indeed, a triangle cannot contain a cross-cap: a surface with one boundary component, three marked points, and a cross-cap is a Möbius strip with three marked points instead of a triangle.

Moreover, a triangle cannot have an edge going through a cross-cap. On the contrary, suppose that a triangle $\Delta$ has such an edge. Then, since we can add another arc in $\Delta$ , as depicted by the dotted arc in Figure 11, $T$ was not part of maximal collection of compatible arcs, i.e a triangulation, so $T$ was not a triangle.

Therefore, we proceed with our proof in three cases. Since Type 1 and Type 2 are mirror images, the number of triangulations occurring in each type are the same, and we discuss them in one case.

Case 1: Consider the triangle $\Delta$ of the form Type 1 or Type 2. Without loss of generality, we consider the case of Type 2.

We count the number of triangulations by considering the three areas cut out by the two arcs, see Figure 12.

Area 1 includes $i$ marked points numbered from 1 to $i$ . Since Area 1 is orientable, it is isotopic to an $i$ -gon. Thus by the Lemma 3.1, Area 1 has $C_{i-2}$ triangulations.

There is no more compatible arc in Area 2; alternatively, it is a triangle. Therefore, there is only one triangulation in this area.

Area 3 is a Möbius strip with $n-i+1$ marked points. Hence, there are $T_{n-i+1}$ triangulations in Area 3.

Thus, the number of triangulations in this case is the product of triangulations coming from these three areas. The number of triangulations containing a triangle $\Delta$ of Type 2 is

\sum_{i=2}^{n}T_{n-1+1}\times 1\times C_{i-2}=\sum_{i=0}^{n-2}T_{n-i-1}C_{i}.

Thus, the total number of triangulations containing triangle $\Delta$ of Type 1 and 2 in Case 1 is

2\sum_{i=0}^{n-2}T_{n-i-1}C_{i}.

Case 2: In this case, we consider $\Delta$ of Type 3, in which both arcs go through the cross-cap. The Möbius Strip is thus divided into two areas, see Figure 13.

Area 1 is a triangle, and is thus already a triangulation and no other arcs can be added.

Consider the number of triangulations in Area 2.

We first notice that the Area 2 is isotopic to an $(n+1)$ -gon, because:

•

$i-1$ edges between vertices $1$ and $i$ ,
•

one edge between vertices $i$ and $n$ going through the cross-cap,
•

$n-i$ edges between vertices $n$ and $i$ ,
•

one edge between vertices $i$ and $1$ going through the cross-cap.

This gives us a total of $n+1$ edges; since the interior of Area 2 contains no cross-cap, it is an $(n+1)$ -gon. Therefore, it has $C_{n-1}$ triangulations.

Since in this case, $1\leq i\leq n$ , the total number of triangulations can be found by

(3.3)

\sum_{i=1}^{n}C_{n-1}=nC_{n-1}

Therefore, taking all three cases into consideration, the total number of triangulations $T_{n}$ on a Möbius strip with $n$ marked points is the sum:

\displaystyle T_{n}=2\sum_{i=0}^{n-2}C_{i}T_{n-i-1}+nC_{n-1}\text{ for }n\geq 2,\;T_{1}=2

∎

We now want to simplify this formula and obtain a direct formula for the number of triangulations of a Möbius strip.

Proposition 3.4.

The number of triangulations of a Möbius strip with $n$ marked points is

T_{n}=4^{n-1}+\binom{2n-2}{n-1}.

Proof.

We proceed by induction to show that

(3.4)

2\sum_{i=0}^{n-2}C_{i}T_{n-i-1}=4^{n-1}.

Clearly, when $n=2$ , we have

2\sum_{i=0}^{n-2}C_{i}T_{n-i-1}=2\sum_{i=0}^{0}C_{i}T_{i+1}=2(C_{0}T_{1})=4=4^{1}=4^{n-1},

which verifies Equation 3.4. Now, suppose Equation 3.4 is true for all $k<n$ for some positive integer $n$ and prove that it is also verified for $k=n$ . Using both our induction hypothesis and Proposition 3.3, for $k<n$ , we know that

T_{k}=2\sum_{i=0}^{k-2}C_{i}T_{k-i-1}+kC_{k-1}=4^{k-1}+kC_{k-1}.

By Equation 3.1, we obtain, for $2\leq k<n$ ,

kC_{k-1}=(4k-6)C_{k-2}.

Using this equation, we compute:

	$\displaystyle T_{k}$	$\displaystyle=4^{k-1}+kC_{k-1}$
		$\displaystyle=4\left(4^{k-2}+(k-1)C_{k-2}\right)-4(k-1)C_{k-2}+kC_{k-1}$
		$\displaystyle=4T_{k-1}-4(k-1)C_{k-2}+(4k-6)C_{k-2}$
		$\displaystyle=4T_{k-1}-2C_{k-2}.$

Therefore,

	$\displaystyle 2\sum_{i=0}^{n-2}C_{i}T_{n-1-i}$	$\displaystyle=2\sum_{i=0}^{n-3}C_{i}T_{n-1-i}+2C_{n-2}T_{1}$
		$\displaystyle=2\sum_{i=0}^{n-3}C_{i}\left(4T_{n-i-2}-2C_{n-i-3}\right)+4C_{n-2}$
		$\displaystyle=4\left(2\sum_{i=0}^{n-3}C_{i}T_{n-1-i}\right)-4\sum_{i=0}^{n-3}C_{i}C_{n-i-3}+4C_{n-2}$
		$\displaystyle=4\cdot 4^{n-2}-4C_{n-2}+4C_{n-2}$
		$\displaystyle=4^{n-1}.$

Since we have now proven equation 3.4, we have now shown that

T_{n}=4^{n-1}+nC_{n-1}.

Let’s simplify this formula again: by Equation 3.2, we have

nC_{n-1}=\frac{n}{n+1-1}\binom{2n-2}{n-1}=\binom{2n-2}{n-1}.

Hence, by Proposition 3.3,

T_{n}=4^{n-1}+\binom{2n-2}{n-1}.

∎

4. Connection to Quasi-Cluster Algebras

In this section, we apply our previous results to quasi-cluster algebras, which are generalizations of cluster algebras.

Introduced in 2002 by Fomin and Zelevinsky [FZ02, FZ03, FZ07], cluster algebras create an algebraic framework for canonical bases and total positivity. They related to various areas of mathematics such as combinatorics, representation theory of algebras [BMRT07], mathematical physics [AHBHY18], Poisson geometry [GSV10], Lie theory [GLS13], and, of course, triangulation of surfaces [FST08], just to name a few. Indeed, in 2008, Fomin, Shapiro and Thurston introduced a particular class of cluster algebras, called cluster algebras from surfaces, [FST08]. Later work from Dupont and Palesi introduced the quasi-cluster algebras from non-orientable surfaces [DP15]. Then, Wilson studied the shellability and sphericity of the arc complex [Wil18], and proved the positivity theorem for quasi-cluster algebras [Wil19].

In this section, we will not define cluster algebras, but rather quasi-cluster algebras, which generalize them.

Quasi-cluster algebras are commutative $\mathbb{Z}$ -algebras of Laurent polynomials with a distinguished set of generators, called cluster variables, grouped into sets called clusters. The set of all cluster variables is constructed recursively from a set of initial cluster variables using an involutive operation called mutation. Let’s define these concepts formally.

The difference between classical cluster algebras from surfaces, as defined by Fomin and Zelevinsky [FZ02] and Fomin, Shapiro Thurston, [FST08] and quasi-cluster algebras, as defined by Dupont and Palesi, [DP15], is that cluster algebras arise only from orientable surfaces, while quasi-cluster algebras arise from both orientable and non-orientable surfaces.

Let $\mathcal{F}$ denote the field of rational functions in $n+k$ indeterminates, that is

\mathcal{F}=\left\{\left.\frac{p(a_{1},\dots,a_{n},b_{1},\dots b_{k})}{q(a_{1},\dots,a_{n},b_{1},\dots,b_{k})}\right|p,q\in\mathbb{Z}[a_{1},\dots,a_{n},b_{1},\dots,b_{k}]\right\}.

Definition 4.1.

Let $K$ be a field and $k$ a sub-field of $K$ . A set $X=\left\{x_{1},\dots,x_{m}\right\}\subseteq K$ is algebraically independent on $k$ if $(x_{1},\dots x_{m})$ is not a root for any non-zero polynomial on $k\left[t_{1},\dots,t_{m}\right]$ .

Example 4.2.

The singletons $\{\pi\}$ and $\{\sqrt{2\pi+1}\}\subseteq\mathbb{R}$ are algebraically independent on $\mathbb{Q}$ , but the set $\{\pi,\sqrt{2\pi+1}\}$ is not. Indeed, $\left(\sqrt{2\pi+1}\right)^{2}-2\pi-1=0$ , so the non-zero polynomial $f(t_{1},t_{2})=t_{1}^{2}-2t_{2}-1$ is null in $\left(\sqrt{2\pi+1},\pi\right)$ .

To each boundary segments $b$ , we associate an indeterminate $y_{b}$ such that $\mathbf{y}=\left\{y_{b}|b\in\mathbf{B(\mathbf{S},\mathbf{M})}\right\}$ is algebraically independent. The elements of $\mathbf{y}$ are called the coefficients. The ground ring is set as

\mathbb{Z}\mathbb{P}=\mathbb{Z}[y_{b}^{\pm 1}\mid b\in\mathbf{B}(\mathbf{S},\mathbf{M})].

Definition 4.3.

An initial seed is a triplet $(\mathbf{x},T)$ such that:

•

$T$ is a triangulation of $(\mathbf{S},\mathbf{M})$ ;
•

$\mathbf{x}=\{x_{t}|t\in T\}$ is a cluster and together with $\mathbf{y}$ , it forms a generating set of $\mathcal{F}$ .

In other words, a seed is triangulation with a variable associated to each arc.

From a given seed, we can obtain a new one by a process called mutation. First, let’s define the mutation only on triangulations.

Proposition 4.4.

[DP15, Proposition 9] Let $T$ be a triangulation of a marked surface $(\mathbf{S},\mathbf{M})$ and let $t$ be an arc of $T$ . There exists a unique arc $t^{\prime}\neq t$ such that $\left(T\setminus\{t\}\right)\cup t^{\prime}$ is still a triangulation of $(\mathbf{S},\mathbf{M})$ .

Example 4.5.

Figure 14 depicts the mutation in a Möbius strip with 3 marked points $M_{3}$ , where each line indicates a mutation.

Definition 4.6.

Let $(T,\mathbf{x})$ be an initial seed and $t\in T$ . The mutation of $(T,\mathbf{x})$ in the direction $t$ transforms $(T,\mathbf{x})$ in a seed $\mu_{t}(T,\mathbf{x})=(\mu_{t}(T),\mu_{t}(\mathbf{x}))$ where $\mathbf{x}=\mathbf{x}\setminus\{x_{t}\}\cup\{x_{t^{\prime}}\}$ with $x_{t^{\prime}}$ such that:

(1)

If the arc $t$ is the diagonal in a rectangle $abcd$ that creating two triangles as shown in Figure 15, then the relation is the following:

(4.1) $x_{t}x_{t}^{\prime}=x_{a}x_{c}+x_{b}x_{d}.$

This is called the Ptolemy relation because it is analogous to the relation that holds among the lengths of the sides and diagonals of a cyclic quadrilateral by Ptolemy’s theorem.

Figure 15. Mutation of type 1
(2)

Consider an anti-self-folded-triangle, as in Figure 16. Let $t$ be the arc corresponding to the two identified edges. Then, the relation is

(4.2) $x_{t}x_{t}^{\prime}=x_{a}.$

Figure 16. Mutation of type 2
(3)

If a one-sided closed curve $t$ is in the triangulation with boundary $y$ as in Figure 17, the relation is

(4.3) $x_{t}x_{t}^{\prime}=x_{a}.$

Figure 17. Mutation of type 3
(4)

Let $t$ be the diagonal in a rectangle with sides $a,b,c,d$ , as Figure 18 shows below, then the relation is

(4.4) $x_{t}x_{t}^{\prime}=(x_{b}+x_{c})^{2}+x_{a}x_{b}x_{c}.$

Figure 18. Mutation of type 4

In general, a seed is obtained by a sequence of mutations from the initial seed.

Example 4.7.

Figure 19 shows a mutation of a seed in a Möbius strip with 3 marked points $M_{3}$ . $x_{3}^{\prime}$ is the new cluster variable.

Definition 4.8.

Let $\mathscr{X}$ be the union of all clusters obtained by successive mutation from an initial seed $(T,\mathbf{x})$ . The quasi-cluster algebra is

\mathscr{A}(T,\mathbf{x})=\mathbb{Z}\mathbb{P}\left[\mathscr{X}\right].

That is to say, every element of $\mathcal{F}$ which can be expressed as a polynomial in the elements of $\mathscr{X}$ with coefficients in $\mathbb{Z}\mathbb{P}$ .

Example 4.9.

In figure 20, cluster variables from the quasi-cluster algebra from a Möbius Strip with 2 marked points ( $M_{2}$ ) are calculated. In the figure, $x_{1}^{\prime}$ and $x_{2}^{\prime}$ are derived from the Ptolemy relation 4.1:

x_{1}^{\prime}=\frac{{x_{2}}^{2}+y_{1}y_{2}}{x_{1}},x_{2}^{\prime}=\frac{x_{1}^{\prime}(y_{1}+y_{2})}{x_{2}}=\frac{x_{1}({y_{1}}^{2}y_{2}+y_{1}{y_{2}}^{2}+{x_{2}}^{2}y_{1}+{x_{2}}^{2}y_{2})}{x_{2}}.

$x_{1}^{\prime\prime}$ is mutated from an anti-self-folded triangle according to equation 4.3, and can be calculated as

x_{1}^{\prime\prime}=\frac{x_{2}^{\prime}}{x_{1}^{\prime}}=\frac{{x_{1}}^{2}(y_{1}+y_{2})}{x_{2}}.

$x_{2}^{\prime\prime}$ is calculated according to formula 4.4, which yields

x_{2}^{\prime\prime}=\frac{(y_{1}+y_{2})^{2}+{x_{1}^{\prime\prime}}^{2}y_{1}y_{2}}{x_{2}^{\prime}}=\frac{(y_{1}+y_{2})({x_{1}}^{4}y_{1}y_{2}+{x_{2}}^{2})}{x_{1}x_{2}(y_{1}y_{2}+{x_{2}}^{2})}.

The cluster algebra consists of all polynomials in $x_{1},x_{2},x_{1}^{\prime},x_{2}^{\prime},x_{1}^{\prime\prime}$ and $x_{2}^{\prime\prime}$ with coefficients in $\mathbb{Z}[y_{1},y_{1}^{-1},y_{2},y_{2}^{-1},y_{3},y_{3}^{-1}]$ .

Proposition 4.10.

There is a bijection between the set of triangulations of a marked surface and the set of seeds of the quasi-cluster algebra arising from this surface.

Proof.

Clearly, to any seed $(T,\mathbf{x})$ , we can associate the triangulation $T$ .

Moreover, to any arc, we can associate a unique cluster variable, [DP15]. Wilson also gave a formula to express the cluster variable associated to a given arc in term of the cluster variables in any cluster, [Wil19]. Therefore, to any triangulation $T$ , we can associate a unique cluster $(T,\mathbf{x})$ .

∎

Corollary 4.11.

There are $4^{n}+\binom{2n-2}{n-1}$ seeds in a quasi-cluster algebra arising from a Möbius strip. Any other quasi-cluster algebra arising from a non-orientable surface has infinitely many seeds.

Proof.

It follows directly from Proposition 4.10 and Theorem 2.13. ∎

Acknowledgements

We would like to thank Jon Wilson for fruitful discussions. We would also like to thank Hugh Thomas for providing editorial advice of this paper and helping us. Finally, we would like to thank Canada/USA Mathcamp for giving us the opportunity to meet together and spark discussions on this topic.

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