Spirals, tic-tac-toe partition, and deep diagonal maps

Spirals, tic-tac-toe partition, and deep diagonal maps

1 Introduction

2 Background

3 The Spirals and $T_{k}$ -Orbit Invariance

4 Coordinate Representation of 3-Spirals

5 A Birational Formula for $T_{3}$

6 The Precompactness of $T_{3}$ Orbits

7 Type- $\beta$ 2-Spirals and Precompact $T_{2}$ Orbits

8 Appendix

References

1 Introduction

2 Background

3 The Spirals and Tk-Orbit Invariance

4 Coordinate Representation of 3-Spirals

3 The Spirals and $T_{k}$ -Orbit Invariance

5 A Birational Formula for $T_{3}$

6 The Precompactness of $T_{3}$ Orbits

7 Type- $\beta$ 2-Spirals and Precompact $T_{2}$ Orbits

1.1 Context and Motivation

1.2 The $k$ -Spirals under the Map $T_{k}$

1.3 Tic-Tac-Toe Partition and Precompact $T_{3}$ Orbits

1.4 The Type- $\beta$ 2-Spirals and Precompact $T_{2}$ Orbits

1.5 Obstacles for $k>3$ and Future Directions

1.6 Accompanying Program

1.7 Acknowledgements

2.1 Projective Geometry

2.2 Orientation of Affine Triangles

2.3 The Cross-Ratio

2.4 Twisted Polygons, Corner Invariants

4.1 The Tic-Tac-Toe Grids

4.2 The Correspondence of $\mathcal{S}_{3,n}^{\alpha}$ and $S_{n}(J,I)$

4.3 The Correspondence of $\mathcal{S}_{3,n}^{\beta}$ and $S_{n}(K,J)$

8.1 Conjectures for Invariants

1.1 Context and Motivation

1.2 The k-Spirals under the Map Tk

1.3 Tic-Tac-Toe Partition and Precompact T3 Orbits

1.4 The Type-β 2-Spirals and Precompact T2 Orbits

1.5 Obstacles for k>3 and Future Directions

1.6 Accompanying Program

1.7 Acknowledgements

2.1 Projective Geometry

2.2 Orientation of Affine Triangles

2.3 The Cross-Ratio

2.4 Twisted Polygons, Corner Invariants

3.1 The Geometry of k-Spirals

3.2 Transversals of the Spirals

3.3 Invariance of Forward Orbit

3.4 Invariance of Backward Orbit

4.1 The Tic-Tac-Toe Grids

4.2 The Correspondence of 𝒮3,nα and Sn⁢(J,I)

1.2 The $k$ -Spirals under the Map $T_{k}$

1.3 Tic-Tac-Toe Partition and Precompact $T_{3}$ Orbits

1.4 The Type- $\beta$ 2-Spirals and Precompact $T_{2}$ Orbits

1.5 Obstacles for $k>3$ and Future Directions

3.1 The Geometry of $k$ -Spirals

4.2 The Correspondence of $\mathcal{S}_{3,n}^{\alpha}$ and $S_{n}(J,I)$

4.3 The Correspondence of $\mathcal{S}_{3,n}^{\beta}$ and $S_{n}(K,J)$

5.2 Conjugated Corner Invariants and Its $T_{3}$ Formula

5.3 Relation to $Y$ -Variables

7.1 The Corner Invariants of Type- $\beta$ 2-Spirals

7.2 The Precompactness of $T_{2}$ Orbits

Abstact.

Mathematics Subject Classification:

Key words and phrases:

Theorem 1.1.

Conjecture 1.2.

Theorem 1.3.

Theorem 2.1.

Definition 4.1.

Proposition 4.2.

Proposition 4.3.

Proof.

Remark 4.4.

Corollary 4.5.

Proof.

Theorem 4.6.

Corollary 4.7.

Proof.

Lemma 4.8.

Proof.

Proposition 4.9.

Proof.

Proposition 4.10.

Proof.

Lemma 4.11.

Proof.

Proposition 4.12.

Proof.

Lemma 4.13.

Proof.

Proposition 4.14.

Proof.

Proposition 8.1.

Proof.

Remark 8.2.

Conjecture 8.3.

Abstact.

Mathematics Subject Classification:

Key words and phrases:

Theorem 1.1.

Conjecture 1.2.

Theorem 1.3.

Theorem 1.4.

Theorem 2.1.

Proposition 2.2.

Definition 3.1.

Remark 3.2.

Definition 3.3.

Proposition 3.4.

Proof.

Corollary 3.5.

Proof.

Remark 3.6.

Proposition 3.7.

Proof.

Proposition 3.8.

Proof.

Proposition 3.9.

Proof.

Proposition 3.10.

Proof.

Problem 3.11.

Problem 3.12.

Problem 3.13.

Lemma 3.14.

Proof.

Proposition 3.15.

Proof.

Remark 3.16.

Lemma 3.17.

Proof.

Proposition 3.18.

Proof.

Proposition 3.19.

Proof.

Proposition 3.20.

Proof.

Proposition 3.21.

Proof.

Definition 4.1.

Proposition 4.2.

Proposition 4.3.

Proof.

Remark 4.4.

Corollary 4.5.

Proof.

Theorem 4.6.

Corollary 4.7.

Proof.

Lemma 4.8.

Proof.

Proposition 4.9.

Proof.

Proposition 4.10.

Arnold Mathematical Journal
Volume 11, Issue 4, 2025

Research Contribution DOI:10.56994/ARMJ.011.004.006
Received: 25 Dec 2024; Accepted: 9 Aug 2025

Zhengyu Zou

Given a polygon $P$ in the real projective plane, let $T_{k}$ be the map that connects its $k$ -th diagonals and intersects them successively to form another polygon $P^{\prime}$ whose vertices are given by the following formula:

P^{\prime}_{i}=P_{i}P_{i+k}\cap P_{i+1}P_{i+k+1}.

(1)

Figure 1 demonstrates an example of the action of $T_{2}$ on a convex heptagon. The map $T_{2}$ is called the pentagram map, a well-studied discrete dynamical system (see [20, 21, 23, 18]). A well-known result is that $T_{2}$ preserves convexity.¹¹1A projective polygon is convex if some projective transformation maps it to a planar convex polygon in the affine patch The $T_{2}$ -orbit of a convex polygon sits on a flat torus in the moduli space of projective equivalent convex polygons. On the other hand, the geometry of the map $T_{k}$ is less well-behaved. For $k\geq 3$ , the $T_{k}$ images of convex polygons may not even be embedded. See Figure 1 for an example of $T_{3}$ taking a convex heptagon to a polygon that is not even embedded.

Refer to caption — Figure 1: Left: The iterative images of a convex heptagon under the action of $T_{2}$ . Right: A convex heptagon whose image under $T_{3}$ is not even embedded.

Previous results of $T_{k}$ often had an algebraic and combinatorial flavor, motivated by two branches of studies. The first one was a sequence of works [23, 18, 28, 19] that established that the $T_{2}$ action on the moduli space of projective convex polygons is a discrete completely integrable system; the second one was M. Glick’s discovery in [5] of the connection between $T_{2}$ and cluster algebras. In [4], M. Gekhtman, M. Shapiro, S. Tabachnikov, and A. Vainshtein generalized the cluster transformations in [5] to the map $T_{k}$ acting on so-called “corrugated polygons,” which are polygonal curves in $\mathbb{R}\mathbb{P}^{k}$ satisfying certain coplanarity conditions. [4] showed that $T_{k}$ is a discrete integrable system. There are numerous integrability results for these higher-dimensional analogs. See [13, 15, 16, 14, 11]. These led to many applications and connections of $T_{k}$ to other fields, such as octahedral recurrence [23, 3], the condensation method of computing determinants [23, 6], cluster algebras [5, 4, 7, 3], Poisson Lie groups [2, 10], $T$ -systems [12, 3], Grassmannians [1], algebraically closed fields [29], Poncelet polygons [22, 25, 9, 26], and integrable partial differential equations [23, 18, 17].

The geometric aspects of $T_{k}$ and other deep diagonal maps on planar polygons remain underexplored. There are only a few studies on the geometries of $T_{k}$ that focused on small $k$ or polygons with many symmetries. See [25, 26]. There is no established general framework on the type of geometric properties preserved under $T_{k}$ for $k\geq 3$ that is analogous to convexity under $T_{2}$ . Even less is known for geometric objects that have precompact orbits under $T_{k}$ .

The most relevant result to this endeavor is the discovery of $k$ -birds under the map $\Delta_{k}$ in [27]. A $k$ -bird $P$ is a planar $n$ -gon with $n>3k$ , such that there exists a continuous path of polygons $P^{(t)}$ connecting $P$ to the regular $n$ -gon where the four lines

P^{(t)}_{i}P^{(t)}_{i-k-1},\hskip 10.0ptP^{(t)}_{i}P^{(t)}_{i-k},\hskip 10.0% ptP^{(t)}_{i}P^{(t)}_{i+k},\hskip 10.0ptP^{(t)}_{i}P^{(t)}_{i+k+1}

are distinct for all $i=1,\ldots,n$ and $t\in I$ . The map $\Delta_{k}$ connects the $(k+1)$ -th diagonal of a polygon and intersects the diagonals that are $k$ clicks apart. See Figure 2 for the action of $\Delta_{2}$ on 2-birds. In [27], R. Schwartz showed that the $k$ -birds are invariant under both $\Delta_{k}$ and $\Delta_{k}^{-1}$ . Experimentally, the $k$ -birds seem to have toroidal orbits under $\Delta_{k}$ , which highly resembles the orbit of convex $n$ -gons under $T_{2}$ . Schwartz also showed that the $k$ -birds have precompact forward $\Delta_{k}$ -orbits modulo affine transformations—a property satisfied by convex $n$ -gons under $T_{2}$ .

This paper has two main results. The first one is the discovery of two classes of geometric objects called type- $\alpha$ and type- $\beta$ $k$ -spirals that are preserved under $T_{k}$ for all $k\geq 2$ . These two classes of objects are subsets of twisted polygons: bi-infinite sequences $P:\mathbb{Z}\rightarrow\mathbb{R}\mathbb{P}^{2}$ such that no three consecutive points are collinear, and $P_{i+n}=\phi(P_{i})$ for some fixed projective transformation $\phi$ called the monodromy. The moduli space of projective equivalent twisted $n$ -gons is conventionally denoted by $\mathcal{P}_{n}$ . The type- $\alpha$ and type- $\beta$ $k$ -spirals are the first discovered classes of geometric constructions of $T_{k}$ that generalize the pentagram map, which provides crucial evidence for a more general understanding of geometrically preserved classes under $T_{k}$ .

The second result is the precompactness of both forward and backward $T_{k}$ -orbits of type- $\alpha$ and type- $\beta$ $k$ -spirals modulo projective transformations for $k=2$ and $3$ , a key property satisfied by convex polygons under the pentagram map discovered by Schwartz in [20]. We first examine the action of $T_{3}$ on type- $\alpha$ and type- $\beta$ 3-spirals. We show that one can characterize type- $\alpha$ and type- $\beta$ 3-spirals via linear constraints on the corner invariants. We also derive a birational formula of $T_{3}$ for the corner invariants, which is a generalization of the combinatorial formulas developed by [7]. Then, we present four global invariants under $T_{3}$ , which we use to prove the precompactness of $T_{3}$ -orbits modulo projective transformations. For the case $k=2$ , we show that there exists no type- $\alpha$ 2-spirals and that the type- $\beta$ 2-spirals are distinct from closed convex polygons. We use the Casimir functions of the $T_{2}$ -invariant Poisson structure developed in [23] and [18] to show that type- $\beta$ 2-spirals have precompact $T_{2}$ -orbits modulo projective transformations.

Here we describe the geometric picture of a $k$ -spiral. For the formal definition, see $\S$ 3.1. Geometrically, $[P]\in\mathcal{P}_{n}$ is a $k$ -spiral if for all $N\in\mathbb{Z}$ , we can find a representative $P$ such that $\{P_{i}\}_{i\geq N}$ lies on the affine patch, and the triangles $(P_{i},P_{i+1},P_{i+2})$ and $(P_{i},P_{i+1},P_{i+k})$ have positive orientation for all $i\geq N$ . We call $P$ an $N$ -representative of $[P]$ .

We are mainly interested in two types of $k$ -spirals, which we term type- $\alpha$ and type- $\beta$ (although there certainly exist many more types of spirals, we only consider these two types here). They are $k$ -spirals with additional constraints on the arrangement of the four points $P_{i}$ , $P_{i+1}$ , $P_{i+k}$ , $P_{i+k+1}$ . For type- $\alpha$ spirals, we require $P_{i+k}$ to be contained in the interior of the triangle $(P_{i},P_{i+1},P_{i+k+1})$ . For type- $\beta$ spirals, $P_{i+k+1}$ needs to be contained in the interior of $(P_{i},P_{i+1},P_{i+k})$ . We say $P$ is a type- $\alpha$ or type- $\beta$ $N$ -representative. A class of twisted polygons $[P]$ is a type- $\alpha$ $k$ -spiral (resp. $\beta$ ) if and only if it admits a type- $\alpha$ (resp. $\beta$ ) $N$ -representative for all $N\in\mathbb{Z}$ . Let $\mathcal{S}_{k,n}^{\alpha}$ and $\mathcal{S}_{k,n}^{\beta}$ denote the space of type- $\alpha$ and type- $\beta$ $k$ -spirals modulo projective equivalence. We will see in $\S$ 3.1 that they are both open in $\mathcal{P}_{n}$ and hence have dimension $2n$ . Figure 3 illustrates three examples of representatives of $\mathcal{S}_{5,n}^{\alpha}$ for $n=3$ , and $20$ .

It turns out that $\mathcal{S}_{k,n}^{\alpha}$ and $\mathcal{S}_{k,n}^{\beta}$ are invariant under both $T_{k}$ and $T_{k}^{-1}$ . Figure 4 shows the inward half of a representative $P$ of $[P]\in\mathcal{S}_{5,3}^{\beta}$ , with the red arc representing $P^{\prime}=T_{5}(P)$ . On the right we have five polygonal arcs by joining vertices of $P$ that are 5 clicks apart. We call them the transversals of $P$ . One way to distinguish type- $\alpha$ and type- $\beta$ spirals is by looking at the orientations of transversals. The transversals of type- $\alpha$ spirals are counterclockwise, whereas those of type- $\beta$ are clockwise (See Figure 11). In $\S$ 3, we use the orientations of these transversals to prove the following main theorem.

For all $n\geq 2$ and $k\geq 2$ , we have $T_{k}(\mathcal{S}_{k,n}^{\alpha})=\mathcal{S}_{k,n}^{\alpha}$ . The same is true for type- $\beta$ .

A key property satisfied by convex polygons under the pentagram map is that the forward and backward orbits of any convex polygon under the pentagram map are precompact modulo projective tranformations. See [20, Lemma 3.2]. Experimental results suggest that the $k$ -birds also have precompact $\Delta_{k}$ -orbits. In [27, Conjecture 8.2] Schwartz conjectured that the $k$ -birds have precompact forward $\Delta_{k}$ -orbits modulo affine transformations. We observed experimentally that $\mathcal{S}_{k,n}^{\alpha}$ and $\mathcal{S}_{k,n}^{\beta}$ behave analogously under $T_{k}$ .

For $n\geq 2$ and $k\geq 2$ , the forward and backward $T_{k}$ -orbit of any $[P]\in\mathcal{S}_{k,n}^{\alpha}$ is precompact in $\mathcal{P}_{n}$ . The same holds for type- $\beta$ .

In $\S$ 6 and $\S$ 7, we prove Conjecture 1.2 for $k=2$ and $k=3$ .

Our main focus will be the case $k=3$ , which we prove in $\S$ 6.2.

For $n\geq 2$ , the forward and backward $T_{3}$ -orbit of any $[P]\in\mathcal{S}_{3,n}^{\alpha}$ is precompact in $\mathcal{P}_{n}$ . The same holds for type- $\beta$ .

We discovered several interesting properties of the two types of $k$ -spirals and the map $T_{3}$ along our way to prove Theorem 1.3. One major discovery is that the sets $\mathcal{S}_{3,n}^{\alpha}$ and $\mathcal{S}_{3,n}^{\beta}$ fit well with a local parameterization of $\mathcal{P}_{n}\rightarrow\mathbb{R}^{2n}$ introduced by [20] called corner invariants (See $\S$ 2.4). The invariant sets of $\mathcal{P}_{n}$ under $T_{3}$ are partitioned by linear boundaries in the parameter space. The boundary lines give a grid pattern that resembles the board of the game “tic-tac-toe.” Each of the four “side-squares” is invariant under $T_{3}$ .

To construct the tic-tac-toe board, consider the three intervals $I, J, K$ of $\mathbb{R}$ given by $I=(-\infty,0)$ , $J=(0,1)$ , $K=(1,\infty)$ . The squares are of the form $I\times I$ , $I\times J$ , $I\times K$ , $J\times I$ , etc.. We mark the four side-squares $S_{n}(I,J)$ , $S_{n}(J,I)$ , $S_{n}(K,J)$ , $S_{n}(J,K)$ . See Figure 5 for a visualization of the tic-tac-toe grid. Given $[P]\in\mathcal{P}_{n}$ , we say $[P]\in S_{n}(I,J)$ if all even corner invariants of $[P]$ are in $I$ , and all odd ones are in $J$ . This means if we plot all $n$ pairs of corner invariants $(x_{2i},x_{2i+1})$ onto $\mathbb{R}^{2}$ , we would see $n$ points lying in $I\times J$ . The other three side squares are defined analogously.

Figure 6 shows vertices of a representative $P$ of $[P]\in S_{4}(K,J)$ and the image $P^{\prime}=T_{3}(P)$ . On the right, we have the projection of the first $2^{11}$ iterations of the orbit of $P$ under $T_{3}$ . Each point corresponds to $P^{(m)}_{3}$ after normalizing $(P_{-2}^{(m)},P_{-1}^{(m)},P_{0}^{(m)},P_{1}^{(m)})$ to the unit square (here $P^{(m)}=T_{3}^{m}(P)$ ). We speculate that the orbit lies on a flat torus, where the map $T_{3}$ acts as a translation on the flat metric.

Twisted polygons that are assigned to these squares have geometric properties. For example, the closed convex polygons always lie in the center square; two of the side-squares are $\mathcal{S}_{3,n}^{\alpha}$ and $\mathcal{S}_{3,n}^{\beta}$ ; the other two side-squares are obtained by reverting the indexing of vertices of these two types of $k$ -spirals. These facts will be proved in $\S$ 4.

The proof of Theorem 1.3 is algebraic. In $\S$ 5 I show that $T_{3}$ is a birational map on the corner invariants, which generalizes a direct application of [7, Theorem 1.6]. For the explicit formulas, see Equation (19). In $\S$ 6, I derive four algebraic invariants of $T_{3}$ , which allow me to show boundedness of the corner invariants of the $T_{3}$ -orbits, thereby proving Theorem 1.3. This approach is reminiscent of Schwartz’s second proof of precompactness of $T_{2}$ -orbits of convex polygons in [21, Section 3B & 3C].

The real projective plane $\mathbb{R}\mathbb{P}^{2}$ is the space of 1-dimensional subspaces of $\mathbb{R}^{3}$ . Points of $\mathbb{R}\mathbb{P}^{2}$ are lines in $\mathbb{R}^{3}$ that go through the origin. We say that $[x:y:z]$ is a homogeneous coordinate of $V\in\mathbb{R}\mathbb{P}^{2}$ if the vector $\tilde{V}=(x,y,z)$ is a representative of $V$ . Given two distinct points $V_{1},V_{2}\in\mathbb{R}\mathbb{P}^{2}$ , the line $l=V_{1}V_{2}$ connecting $V_{1}$ and $V_{2}$ is the 2-dimensional hyperplane spanned by the two 1-dimensional subspaces. Let $l_{1},l_{2}$ be two lines in $\mathbb{R}\mathbb{P}^{2}$ . The point of intersection $l_{1}\cap l_{2}$ is the 1-dimensional line given by the intersection of the two 2-dimensional subspaces. In $\mathbb{R}\mathbb{P}^{2}$ , there exists a unique line connecting each pair of distinct points and a unique point of intersection given two distinct lines. We call a collection of points $V_{1},V_{2},\ldots,V_{n}\in\mathbb{R}\mathbb{P}^{2}$ in general position if no three of them are collinear.

The affine patch $\mathbb{A}^{2}$ consists of points in $\mathbb{R}\mathbb{P}^{2}$ with homogeneous coordinate $[x:y:1]$ . We call this canonical choice of coordinate $(x,y,1)$ the affine coordinate of a point $V\in\mathbb{A}^{2}$ . There is a diffeomorphism $\Phi:\mathbb{R}^{2}\rightarrow\mathbb{A}^{2}$ given by $\Phi(x,y)=[x:y:1]$ . We often identify $\mathbb{A}^{2}$ as a copy of $\mathbb{R}^{2}$ in $\mathbb{R}\mathbb{P}^{2}$ . The line $\mathbb{R}\mathbb{P}^{2}-\mathbb{A}^{2}$ is called the line at infinity.

A map $\phi:\mathbb{R}\mathbb{P}^{2}\rightarrow\mathbb{R}\mathbb{P}^{2}$ is a projective transformation if it maps points to points and lines to lines and is bijective. Algebraically, the group of projective transformations is $\operatorname{PGL}_{3}(\mathbb{R})=\operatorname{GL}_{3}(\mathbb{R})/\mathbb{R% }^{*}I$ , where we are modding by the subgroup $\mathbb{R}^{*}I=\{\lambda I:\lambda\in\mathbb{R}^{*}\}$ and $I$ is the $3\times 3$ identity matrix. We state a classical result regarding projective transformations below with its proof omitted.

Given two 4-tuples of points $(V_{1},V_{2},V_{3},V_{4})$ and $(W_{1},W_{2},W_{3},W_{4})$ in $\mathbb{R}\mathbb{P}^{2}$ , both in general position, there exists a unique $\phi\in\operatorname{PGL}_{3}(\mathbb{R})$ such that $\phi(V_{i})=W_{i}$ .

The group of affine transformations $\operatorname{Aff}_{2}(\mathbb{R})$ on $\mathbb{A}^{2}$ is the subgroup of projective transformations that fixes the line at infinity. It is isomorphic to a semidirect product of $\operatorname{GL}_{2}(\mathbb{R})$ and $\mathbb{R}^{2}$ . Elements of $\operatorname{Aff}_{2}(\mathbb{R})$ can be uniquely expressed as a tuple $(M^{\prime},v)$ where $M^{\prime}\in\operatorname{GL}_{2}(\mathbb{R})$ and $v\in\mathbb{R}^{2}$ . Let $\operatorname{Aff}_{2}^{+}(\mathbb{R})$ denote the subgroup of $\operatorname{Aff}_{2}^{+}(\mathbb{R})$ where $(M^{\prime},v)\in\operatorname{Aff}_{2}^{+}(\mathbb{R})$ iff $\det(M^{\prime})>0$ . These are orientation-preserving affine transformations.

Introduced in [23], a twisted $n$ -gon is a bi-infinite sequence $P:\mathbb{Z}\rightarrow\mathbb{R}\mathbb{P}^{2}$ , along with a projective transformation $M\in\operatorname{PGL}_{3}(\mathbb{R})$ called the monodromy, such that every three consecutive points of $P$ are in general position, and $P_{i+n}=M(P_{i})$ for all $i\in\mathbb{Z}$ . When $M$ is the identity, we get an ordinary closed $n$ -gon. Two twisted $n$ -gons $P, Q$ are equivalent if there exists $\phi\in\operatorname{PGL}_{3}(\mathbb{R})$ such that $\phi(P_{i})=Q_{i}$ for all $i\in\mathbb{Z}$ . The two monodromies $M_{p}$ and $M_{q}$ satisfy $M_{q}=\phi M_{p}\phi^{-1}$ . Let $\mathcal{P}_{n}$ denote the space of twisted $n$ -gons modulo projective equivalence.

The cross-ratio allows us to parameterize $\mathcal{P}_{n}$ with coordinates in $\mathbb{R}^{2n}$ . Given a twisted $n$ -gon $P$ , the corner invariants of $P$ is a coordinate system $x_{0}(P),\ldots,x_{2n-1}(P)$ given by

\left\{\begin{aligned} x_{2i}(P)=\chi(P_{i-2},\,P_{i-1},\,P_{i-2}P_{i-1}\cap P% _{i}P_{i+1},\,P_{i-2}P_{i-1}\cap P_{i+1}P_{i+2});\\ x_{2i+1}(P)=\chi(P_{i+2},\,P_{i+1},\,P_{i+2}P_{i+1}\cap P_{i}P_{i-1},\,P_{i+2}% P_{i+1}\cap P_{i-1}P_{i-2}).\end{aligned}\right.

(7)

See the left side of Figure 9 for a geometric interpretation of the corner invariants. Let $l_{a,b}=P_{i+a}P_{i+b}$ . By Equation (5), the corner invariants can be computed by

\left\{\begin{aligned} x_{2i}(P)&=\chi(l_{1,-2},l_{1,-1},l_{1,0},l_{1,2});\\ x_{2i+1}(P)&=\chi(l_{-1,2},l_{-1,1},l_{-1,0},l_{-1,-2}).\end{aligned}\right.

(8)

See the right side of Figure 9 for the line configurations.

Since $\chi$ is invariant under projective transformations, for all $j$ we have $x_{j}(P)=x_{j+2n}(P)$ , so a $2n$ -tuple of corner invariants is enough to fully determine the projective equivalence class of a twisted $n$ -gon. We use $x_{j}(P)$ to denote the corner invariants of $[P]\in\mathcal{P}_{n}$ without adding square brackets around $P$ . To obtain the corner invariants of $[P]\in\mathcal{P}_{n}$ , one can simply choose an arbitrary representative $P$ and compute its corner invariants. [23, Equation (19) & (20)] showed that one can also revert the process and obtain a representative twisted polygon of the equivalence class given its corner invariants.

Recall the intervals $I=(-\infty,0)$ , $J=(0,1)$ , $K=(1,\infty)$ from $\S$ 1.3. One can partition $\mathbb{R}^{2}$ into a $3\times 3$ grid. See Figure 5. We make the following definition:

For $n\geq 2$ , let $S_{n}(I,J)$ be the subset of $\mathcal{P}_{n}$ that satisfies the following: given $[P]\in S_{n}(I,J)$ , for all $i\in\{0,\ldots,n-1\}$ , $(x_{2i},x_{2i+1})\in I\times J$ . We similarly define $S_{n}(K,J)$ , $S_{n}(J,I)$ , and $S_{n}(J,K)$ .

The following symmetries of the four grids follow directly from Definition 4.1.

For $i\in\mathbb{Z}$ , define the map $\sigma_{i}:\mathbb{Z}\rightarrow\mathbb{Z}$ by $\sigma_{i}(x)=x+i$ . Define the map $\iota:\mathbb{Z}\rightarrow\mathbb{Z}$ by $\iota(x)=-x$ . Given $[P]\in\mathcal{P}_{n}$ , the following are true:

•

If $[P]\in S_{n}(I,J)$ , then $[P\circ\sigma_{i}]\in S_{n}(I,J)$ for all $i\in\mathbb{Z}$ . This also holds for $S_{n}(K,J)$ , $S_{n}(J,I)$ , and $S_{n}(J,K)$ .
•

$[P]\in S_{n}(I,J)$ if and only if $[P\circ\iota]\in S_{n}(J,I)$ .
•

$[P]\in S_{n}(K,J)$ if and only if $[P\circ\iota]\in S_{n}(J,K)$ .

To understand the geometry implied by the corner invariants, we need to examine what happens when the corner invariants take value from $0,1,\infty$ .

For all $[P]\in\mathcal{P}_{n}$ with corner invariants $x_{j}=x_{j}(P)$ and $i\in\mathbb{Z}$ , we have the following correspondence between the position of $P_{i+2}$ and the values of $x_{2i}$ and $x_{2i+1}$ :

Configuration	Coordinates	Configuration	Coordinates
$P_{i+2}\in P_{i+1}P_{i}$	$x_{2i}=0$	$P_{i+2}\in P_{i-1}P_{i+1}$	$x_{2i+1}=0$
$P_{i+2}\in P_{i+1}P_{i-2}$	$x_{2i}=1$	$P_{i+2}\in P_{i-1}P_{i-2}$	$x_{2i+1}=1$
$P_{i+2}\in P_{i+1}P_{i-1}$	$x_{2i}=\infty$	$P_{i+2}\in P_{i-1}P_{i}$	$x_{2i+1}=\infty$

Consider the following lines:

	$\displaystyle l_{1}$	$\displaystyle=P_{i+1}P_{i-2};$	$\displaystyle l_{2}$	$\displaystyle=P_{i+1}P_{i-1};$	$\displaystyle l_{3}$	$\displaystyle=P_{i+1}P_{i};$	$\displaystyle l_{4}$	$\displaystyle=P_{i+1}P_{i+2};$
	$\displaystyle m_{1}$	$\displaystyle=P_{i-1}P_{i+2};$	$\displaystyle m_{2}$	$\displaystyle=P_{i-1}P_{i+1};$	$\displaystyle m_{3}$	$\displaystyle=P_{i-1}P_{i};$	$\displaystyle m_{4}$	$\displaystyle=P_{i-1}P_{i-2}.$

See Figure 17 for a visualization of the configurations of points and lines. Equation (8) implies $x_{2i}=\chi(l_{1},l_{2},l_{3},l_{4})$ and $x_{2i+1}=\chi(m_{1},m_{2},m_{3},m_{4})$ . This yields

Configuration	Lines	Coordinates	Configuration	Lines	Coordinates
$P_{i+2}\in P_{i+1}P_{i}$	$l_{4}=l_{3}$	$x_{2i}=0$	$P_{i+2}\in P_{i-1}P_{i+1}$	$m_{1}=m_{2}$	$x_{2i+1}=0$
$P_{i+2}\in P_{i+1}P_{i-2}$	$l_{4}=l_{1}$	$x_{2i}=1$	$P_{i+2}\in P_{i-1}P_{i-2}$	$m_{1}=m_{4}$	$x_{2i+1}=1$
$P_{i+2}\in P_{i+1}P_{i-1}$	$l_{4}=l_{2}$	$x_{2i}=\infty$	$P_{i+2}\in P_{i-1}P_{i}$	$m_{1}=m_{3}$	$x_{2i+1}=\infty$

which is precisely the relationship described in the proposition. ∎

Proposition 4.3 also gives us a way to determine the position of $P_{i+2}$ when neither $x_{2i}$ nor $x_{2i+1}$ takes value in $0,1,\infty$ . Suppose the four points $P_{i-2},P_{i-1},P_{i},P_{i+1}$ are in general position. For $i,j,k\in\{1,2,3\}$ distinct, we define $U_{i,j}$ to be the connected component of $\mathbb{R}\mathbb{P}^{2}-(l_{i}\cup l_{j})$ that does not intersect $l_{k}$ . For $i,j,k\in\{2,3,4\}$ distinct, we define $V_{i,j}$ to be the connected component of $\mathbb{R}\mathbb{P}^{2}-(m_{i}\cup m_{j})$ that does not intersect $m_{k}$ . See Figure 18 for a visualization of the $U_{i,j}$ ’s and $V_{i,j}$ ’s using the point configurations given in Figure 17. By Proposition 4.3 and continuity of $\chi$ , we have the following:

Configuration	Coordinates	Configuration	Coordinates
$P_{i+2}\in U_{2,3}$	$x_{2i}=I$	$P_{i+2}\in V_{2,3}$	$x_{2i+1}=I$
$P_{i+2}\in U_{1,3}$	$x_{2i}=J$	$P_{i+2}\in V_{2,4}$	$x_{2i+1}=J$
$P_{i+2}\in U_{1,2}$	$x_{2i}=K$	$P_{i+2}\in V_{3,4}$	$x_{2i+1}=K$

Given $[P]\in\mathcal{P}_{n}$ with corner invariants $x_{j}=x_{j}(P)$ , if $x_{j}\not\in\{0,1,\infty\}$ for all $j$ , then $P$ is $3$ -nice. Moreover, every four consecutive points of $P$ are in general position.

Using Proposition 4.3 we may check that

Collinearity	Coordinates	Collinearity	Coordinates
$P_{i-2}$ , $P_{i-1}$ , $P_{i+1}$	$x_{2i-1}=\infty$	$P_{i-1}$ , $P_{i}$ , $P_{i+2}$	$x_{2i+1}=\infty$
$P_{i-2}$ , $P_{i-1}$ , $P_{i+2}$	$x_{2i+1}=1$	$P_{i-1}$ , $P_{i+1}$ , $P_{i+2}$	$x_{2i+1}=0$
$P_{i-2}$ , $P_{i+1}$ , $P_{i+2}$	$x_{2i}=1$	$P_{i}$ , $P_{i+1}$ , $P_{i+2}$	$x_{2i}=0$
$P_{i-1}$ , $P_{i}$ , $P_{i+1}$	$x_{2i-2}=0$

All seven cases contradict the assumption in the corollary. Therefore, the four points $P_{i-2}$ , $P_{i-1}$ , $P_{i+1}$ , $P_{i+2}$ are in general position, and the four consecutive points $P_{i-1}$ , $P_{i}$ , $P_{i+1}$ , $P_{i+2}$ are in general position for all $i\in\mathbb{Z}$ . This shows $P$ is 3-nice, and every four consecutive points of $P$ are in general position. ∎

Our goal of this section is to prove the following correspondence theorem:

For all $n\geq 2$ , $\mathcal{S}_{3,n}^{\alpha}=S_{n}(J,I)$ , $\mathcal{S}_{3,n}^{\beta}=S_{n}(K,J)$ .

This theorem immediately produces the following important corollary.

For all $n\geq 2$ , the four cells $S_{n}(I,J)$ , $S_{n}(K,J)$ , $S_{n}(J,I)$ , $S_{n}(J,K)$ are both forward and backward invariant under $T_{3}$ .

The case $S_{n}(J,I)$ and $S_{n}(K,J)$ follows immediately from Theorem 1.1 and 4.6. We will prove the case $S_{n}(I,J)$ . The case $S_{n}(J,K)$ is completely analogous, so we will omit.

Fix $[P]\in S_{n}(I,J)$ . Recall the maps $\sigma_{i}$ and $\iota$ from Proposition 4.2. Equation (1) implies $T_{3}(P\circ\iota)=T_{3}(P)\circ\iota\circ\sigma_{4}$ . Then, Proposition 4.2 implies $[P\circ\iota]\in S_{n}(J,I)$ , so $[T_{3}(P\circ\iota)]\in S_{n}(J,I)$ . Finally, observe that

T_{3}(P)=(T_{3}(P)\circ\iota\circ\sigma_{4})\circ(\sigma_{-4}\circ\iota)=T_{3}% (P\circ\iota)\circ(\sigma_{-4}\circ\iota).

It follows that $[T_{3}(P)]\in S_{n}(I,J)$ . We omit the proof of $[T_{3}^{-1}(P)]\in S_{n}(I,J)$ . ∎

Here we show that $\mathcal{S}_{3,n}^{\alpha}$ is equivalent to $S_{n}(J,I)$ . We will first show that the corner invariants of a $0$ -representative $P$ of some $[P]\in\mathcal{S}_{3,n}^{\alpha}$ satisfies $S_{n}(J,I)$ . Then, we will show that we can find type- $\alpha$ $N$ -representatives for all $N\in\mathbb{Z}$ given any $[P]\in S_{n}(J,I)$ .

If $P$ is an $N$ -representative of $[P]\in\mathcal{S}_{3,n}^{\alpha}$ , then $P_{i+2}\in\operatorname{int}(P_{i-1},P_{i},P_{i+1})$ for all $i>N+1$ .

Since $(P_{i-1},P_{i},P_{i+1})$ is positive, we may normalize with $\operatorname{Aff}_{2}^{+}(\mathbb{R})$ so that $P_{i-1}=(-1,0)$ , $P_{i}=(0,0)$ , and $P_{i+1}=(0,1)$ . Let $P_{i+2}=(x,y)$ . It suffices to show that $x<0$ , $y>0$ , and $y-x<1$ . We get $x<0$ from positivity of $(P_{i},P_{i+1},P_{i+2})$ , and we get $y>0$ from positivity of $(P_{i-1},P_{i},P_{i+2})$ . Finally, since $(P_{i-2},P_{i-1},P_{i+2})$ is positive and $P_{i+1}\in\operatorname{int}(P_{i-2},P_{i-1},P_{i+2})$ , Proposition 2.2 implies $(P_{i+1},P_{i-1},P_{i+2})$ is positive, which gives us $y-x<1$ as desired. ∎

For all $n\geq 2$ , $\mathcal{S}_{3,n}^{\alpha}\subset S_{n}(J,I)$ .

Fix $i\in\mathbb{Z}$ . Let $P$ be an $(i-3)$ -representative of $[P]\in\mathcal{S}_{3,n}^{\alpha}$ with corner invariants $x_{j}=x_{j}(P)$ . Normalize with $\operatorname{Aff}_{2}^{+}(\mathbb{R})$ so that $P_{i-1}=(-1,0)$ , $P_{i}=(0,0)$ , and $P_{i+1}=(0,1)$ . Let $s_{a,b}$ denote the slope of the line $P_{i+a}P_{i+b}$ . See Figure 19 for the configuration of points.

We want to show that $(x_{2i},x_{2i+1})\in I\times J$ . By Lemma 4.8, $P_{i+1}\in\operatorname{int}(P_{i-2},P_{i-1},P_{i})$ . This implies $s_{1,-2}>s_{-1,-2}>1$ . On the other hand, since $P_{i+1}\in\operatorname{int}(P_{i-2},P_{i-1},P_{i+2})$ , we have $s_{1,2}>s_{1,-2}>1$ , and $s_{-1,2}\in(0,1)$ . This gives us

	$\displaystyle x_{2i}$	$\displaystyle=\frac{(s_{1,-2}-s_{1,-1})(s_{1,0}-s_{1,2})}{(s_{1,-2}-s_{1,0})(s% _{1,-1}-s_{1,2})}=\frac{s_{1,-2}-1}{s_{1,2}-1}\in J\hskip 20.0pt\text{and}$
	$\displaystyle x_{2i+1}$	$\displaystyle=\frac{(s_{-1,2}-s_{-1,1})(s_{-1,0}-s_{-1,-2})}{(s_{-1,2}-s_{-1,0% })(s_{-1,1}-s_{-1,-2})}=\frac{s_{-1,-2}(s_{-1,2}-1)}{s_{-1,2}(s_{-1,-2}-1)}\in I.$

This concludes the proof. ∎

For all $n\geq 2$ , $\mathcal{S}_{3,n}^{\alpha}=S_{n}(I,J)$ .

Proposition 4.9 implies we only need to show $\mathcal{S}_{3,n}^{\alpha}\supset S_{n}(I,J)$ . Given $[P]\in S_{n}(I,J)$ , let $P$ be a representative that satisfies $P_{-1}=(1,4)$ , $P_{0}=(-1,0)$ , $P_{1}=(0,0)$ , $P_{2}=(0,1)$ . Say that $P$ satisfies condition $(*)_{i}$ if the three triangles $(P_{i-1},P_{i},P_{i+1})$ , $(P_{i},P_{i+1},P_{i+2})$ , $(P_{i-1},P_{i},P_{i+2})$ are all positive, $P_{i+2}\in\operatorname{int}(P_{i-1},P_{i},P_{i+1})$ , and $P_{i+2}\in\operatorname{int}(P_{i-2},P_{i-1},P_{i+1})$ .

We show that for all $i>0$ , if $P$ satisfies $(*)_{i-1}$ , then $P$ satisfies $(*)_{i}$ . Since $(P_{i-1},P_{i},P_{i+1})$ is positive, we can normalize with $\operatorname{Aff}_{2}^{+}(\mathbb{R})$ so that $P_{i-1}=(-1,0)$ , $P_{i}=(0,0)$ , and $P_{i+1}=(0,1)$ . Let $s_{a,b}$ denote the slope of $P_{i+a}P_{i+b}$ . Since $P_{i+1}\in\operatorname{int}(P_{i-2},P_{i-1},P_{i})$ , we know that $s_{1,-2}>s_{-1,-2}>1$ . Then, $x_{2i}\in J$ implies $0<\frac{s_{1,-2}-1}{s_{1,2}-1}<1$ . This gives us $s_{1,2}>s_{1,-2}>1$ . On the other hand, $x_{2i+1}\in I$ implies $\frac{s_{-1,-2}(s_{-1,2}-1)}{s_{-1,2}(s_{-1,-2}-1)}<0$ . Since $s_{-1,-2}>1$ , this is equivalent to $1-\frac{1}{s_{-1,2}}<0$ , which implies $s_{-1,2}\in(0,1)$ . Thus, the two lines $P_{i-1}P_{i+2}$ and $P_{i+1}P_{i+2}$ must meet in the shaded triangle in Figure 19, which implies $(P_{i},P_{i+1},P_{i+2})$ , $(P_{i-1},P_{i},P_{i+2})$ are positive, $P_{i+2}\in\operatorname{int}(P_{i-1},P_{i},P_{i+1})$ , and $P_{i+2}\in\operatorname{int}(P_{i-2},P_{i-1},P_{i+1})$ , so $P$ satisfies $(*)_{i}$ . Finally, since $P$ clearly satisfies $(*)_{0}$ , by induction $P$ satisfies $(*)_{i}$ for all $i\geq 0$ , so $P$ is a type- $\alpha$ $0$ -representative of a 3-spiral. We conclude that $[P]\in\mathcal{S}_{3,n}^{\alpha}$ . ∎

Here we show that $\mathcal{S}_{3,n}^{\beta}$ is equivalent to $S_{n}(K,J)$ . The ideas behind the proofs are essentially the same as the ones in $\S$ 4.2. We will focus on explaining how to modify the details of the proofs in $\S$ 4.2 for type- $\beta$ 3-spirals and $S_{n}(K,J)$ .

If $P$ is an $N$ -representative of $[P]\in\mathcal{S}_{3,n}^{\beta}$ , then the quadrilateral joined by vertices $(P_{i},P_{i+1},P_{i+2},P_{i+3})$ is convex for all $i>N$ .

Normalize with $\operatorname{Aff}_{2}^{+}(\mathbb{R})$ so that $P_{i}=(-1,0)$ , $P_{i+1}=(0,0)$ , $P_{i+2}=(0,1)$ , and $P_{i+3}=(x,y)$ . Positivity of $(P_{i+1},P_{i+2},P_{i+3})$ and $(P_{i},P_{i+1},P_{i+3})$ implies $x<0$ and $y>0$ . Positivity of $(P_{i-1},P_{i},P_{i+2})$ , $P_{i+3}\in\operatorname{int}(P_{i-1},P_{i},P_{i+2})$ , and Proposition 2.2 shows $y-x>1$ . ∎

For all $n\geq 2$ , $\mathcal{S}_{3,n}^{\beta}\subset S_{n}(K,J)$ .

Let $P$ be a $(-3)$ -representative of $[P]\in\mathcal{S}_{3,n}^{\beta}$ with corner invariants $x_{j}=x_{j}(P)$ . Lemma 4.11 implies the quadrilateral $(P_{i-2},P_{i-1},P_{i},P_{i+1})$ is convex. Next, since $P$ is a type- $\beta$ $(-3)$ -representative, $P_{i}\in\operatorname{int}(P_{i-2},P_{i-1},P_{i+1})$ for all $i\geq 0$ (See Figure 20). Referring back to Remark 4.4, convexity of $(P_{i-2},P_{i-1},P_{i},P_{i+1})$ implies $P_{i}P_{i+1}$ doesn’t go through $P_{i-2},P_{i-1},P_{i+1}$ , so $(x_{2i},x_{2i+1})\in K\times J$ whenever $P_{i+2}\in\operatorname{int}(P_{i-2},P_{i-1},P_{i+1})$ . ∎

Given a $3$ -nice sequence $P:\mathbb{Z}\rightarrow\mathbb{R}\mathbb{P}^{2}$ and an integer $i\in\mathbb{Z}$ , let $x_{2i}=x_{2i}(P)$ and $x_{2i+1}=x_{2i+1}(P)$ be the corner invariants of $P$ . If the following conditions are true:

•

$(P_{i-2},P_{i-1},P_{i})$ and $(P_{i-1},P_{i},P_{i+1})$ are both positive;
•

The quadrilateral $(P_{i-2},P_{i-1},P_{i},P_{i+1})$ is convex;
•

$(x_{2i},x_{2i+1})\in K\times J$ .

Then, the following hold:

•

$P_{i+2}\in\operatorname{int}(P_{i-2},P_{i-1},P_{i+1})$ ;
•

The quadrilateral $(P_{i-1},P_{i},P_{i+1},P_{i+2})$ is convex;
•

$(P_{i},P_{i+1},P_{i+2})$ and $(P_{i-1},P_{i},P_{i+2})$ are both positive.

Recall that from the proof of Proposition 4.12, we claimed that if the quadrilateral $(P_{i-2},P_{i-1},P_{i},P_{i+1})$ is convex, then the line $P_{i}P_{i+1}$ doesn’t go through $(P_{i-2},P_{i-1},P_{i+1})$ . Since $(x_{2i},x_{2i+1})\in K\times J$ , Remark 4.4 implies $P_{i+2}\in\operatorname{int}(P_{i-2},P_{i-1},P_{i+1})$ , in which case all conclusions of this lemma will hold. See Figure 20 for a visualization of the five points. ∎

$\mathcal{S}_{3,n}^{\beta}=S_{n}(K,J)$ .

Proposition 4.12 gives us $\mathcal{S}_{3,n}^{\beta}\subset S_{n}(K,J)$ , so we show the other containment. Given $[P]\in S_{n}(K,J)$ , we can find a representative $P$ that satisfies $P_{N}=(0,0)$ , $P_{N+1}=(1,0)$ , $P_{N+2}=(1,1)$ , $P_{N+3}=(0,1)$ . Corollary 4.5 shows that $P$ is 3-nice. To see that $(P_{i},P_{i+1},P_{i+2})$ , $(P_{i},P_{i+1},P_{i+3})$ are positive, and $P_{i+4}\in\operatorname{int}(P_{i},P_{i+1},P_{i+3})$ , we may inductively apply Lemma 4.13. This implies $[P]\in\mathcal{S}_{3,n}^{\beta}$ . ∎

Given $[P]\in\mathcal{P}_{k,n}$ , we may consider the following quantity:

y_{i}^{(k)}(P)=-\chi(P_{i},P_{i}P_{i+k}\cap P_{i-1}P_{i+k-1},P_{i}P_{i+k}\cap P% _{i+1}P_{i+k+1},P_{i+k}).

(36)

When $T_{k}^{j}(P)$ is well-defined, we write $y_{i,j}^{(k)}=y_{i}^{(k)}(T_{k}^{j}(P))$ , or simply $y_{i,j}$ if the value of $k$ is clear from the context. Let $Y_{j}^{(k)}$ (or simply $Y_{j}$ ) denote the product $\prod_{i=0}^{n-1}y_{i,j}^{(k)}$ .

For all $k,n\geq 2$ , given $[P]\in\mathcal{S}_{k,n}^{\alpha}$ , there exists $C\in\mathbb{R}$ , $C\neq 0$ , such that

Y_{j+1}^{(k)}\left(Y_{j}^{(k)}\right)^{-1}=C

(37)

for all $j\in\mathbb{Z}$ . The same holds for $[P]\in\mathcal{S}_{k,n}^{\beta}$ .

The grid $(\hat{P}_{i,j})_{(i,j)\in\mathbb{Z}^{2}}$ where $\hat{P}_{i,j}$ is the $i$ -th vertex of $T_{k}^{j}(P)$ is a $Y$ -mesh of type $S=\{(0,0),(k,0),(-1,1),(0,1)\}$ with $y_{r}=y_{r}(\hat{P})$ for $r\in\mathbb{Z}^{2}$ . The proof is essentially the same as the one for Proposition 5.11, so we will omit it. Then, from [7, Theorem 1.6], we have

y_{i+k,j}y_{i-1,j+2}=\frac{(1+y_{i-1,j+1})(1+y_{i+k,j+1})}{(1+y_{i,j+1}^{-1})(% 1+y_{i+k-1,j+1}^{-1})}.

(38)

It follows that

	$\displaystyle Y_{j}Y_{j+2}$	$\displaystyle=\prod_{i=0}^{n-1}(y_{i+k,j}y_{i-1,j+2})=\prod_{i=0}^{n-1}\frac{(% 1+y_{i-1,j+1})(1+y_{i+k,j+1})}{(1+y_{i,j+1}^{-1})(1+y_{i+k-1,j+1}^{-1})}$		(39)
		$\displaystyle=\left(\prod_{i=0}^{n-1}(y_{i,j+1}\,y_{i+k-1,j+1})\right)\,\left(% \prod_{i=0}^{n-1}\frac{(1+y_{i-1,j+1})(1+y_{i+k,j+1})}{(1+y_{i,j+1})(1+y_{i+k-% 1,j+1})}\right)=Y_{j+1}^{2}.$		(39)

This implies $Y_{j+2}/Y_{j+1}=Y_{j+1}/Y_{j}$ for all $j\in\mathbb{Z}$ . Taking $C=Y_{1}/Y_{0}$ completes the proof. ∎

Combining the results of [7] and [4], we see that Proposition 8.1 is equivalent to [4, Theorem 2.1]. Specifically, the quantity in Equation (37) is shown to be a Casimir function with respect to a Poisson structure that is invariant under the $y$ -variable transformation of a quiver $Q_{k}$ , which we will define below.

Consider the infinite directed graph $Q_{k}$ whose vertices are indexed by $\mathbb{Z}\times\{0,1\}$ , with directed edges $(i,0)\rightarrow(i-1,1)$ , $(i,0)\rightarrow(i-k,1)$ , $(i,1)\rightarrow(i,0)$ , and $(i-k-1,1)\rightarrow(i,0)$ for all $i\in\mathbb{Z}$ . See Figure 25 for a visual representation of this quiver. We refer the readers to [7, $\S$ 9] for the construction of this quiver and the proof that the $y$ -variable transformations satisfy (38).

For all $n\geq 2$ , the $y$ -variables corresponding to $[P]\in\mathcal{P}_{k,n}$ are periodic modulo $n$ . We may then identify vertices of $Q_{k}$ via $(i,j)\sim(i+n,j)$ , and similarly identify the corresponding edges. The resulting directed graph $Q_{k,n}$ is isomorphic to the quiver $\mathcal{Q}_{k,n}$ from [4] by applying a translation to the first entry of the vertices $(i,1)$ . Moreover, the $y$ -variables $(y_{i,0})_{i\in[n]}$ of the quiver in [7] transforms in the same way as the $p$ -variables $(p_{i})_{i\in[n]}$ of $\mathcal{Q}_{k,n}$ under the map $\overline{T_{k}}$ (see [4, $\S$ 2]), and the $q$ -variables $(q_{i})_{i\in[n]}$ of $\mathcal{Q}_{k,n}$ correspond to the multiplicative inverse of $y_{i,-1}$ . As a result, $Y_{0}^{(k)}/Y_{-1}^{(k)}=\prod_{i=1}^{n}p_{i}q_{i}$ , which by [4, Theorem 2.1] is invariant under $\overline{T_{k}}$ and forms a Casimir function with respect to a Poisson structure that is invariant under $\overline{T_{k}}$ .

Both [7] and [4] demonstrate that the quiver $Q_{k,n}$ is a bipartite graph that can be embedded into a torus. For further details, see [7, $\S$ 9] and [4, $\S$ 3]. This connection links $Q_{k,n}$ to the Goncharov-Kenyon Dimer Integrable Systems in [8], where a more general definition of Casimir functions is provided.

The constant $C$ in Proposition 8.1 equals 1 for all $k\geq 2$ .

We prove Conjecture 8.3 for $k=2$ and $k=3$ . Let $x_{j}=x_{j}(P)$ be the corner invariants of $[P]$ . The case $k=2$ follows from $y_{i}^{(2)}(P)=-x_{2i+1}x_{2i+2}$ (see Equation (32)), so

\prod_{i=1}^{n}y_{i}^{(2)}(P)=(-1)^{n}\prod_{i=1}^{n}x_{2i+1}x_{2i+2}=(-1)^{n}% \,O_{n}(P)\,E_{n}(P),

which is $T_{2}$ -invariant by Lemma 7.3.

For the case $k=3$ , Equation (31) implies

\prod_{i=1}^{n}y_{i}^{(3)}(P)=(-1)^{n}\prod_{i=1}^{n}\frac{x_{2i}x_{2i+3}}{(x_% {2i}-1)(x_{2i+3}-1)}=(-1)^{n}\,\mathcal{F}_{1}(P)\,\mathcal{F}_{2}(P),

which is $T_{3}$ -invariant by Proposition 6.1.

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× R. Felipe and G. Marí Beffa. The pentagram map on Grassmannians. Annales de l’Institut Fourier, 69(1):421–456, 2019.
× V. V. Fock and A. Marshakov. Loop groups, clusters, dimers and integrable systems, pages 1–65. Springer International Publishing, Cham, 2016.
× P. Di Francesco and R. Kedem. T-systems with boundaries from network solutions, 2012.
× M. Gekhtman, M. Shapiro, S. Tabachnikov, and A. Vainshtein. Higher pentagram maps, weighted directed networks, and cluster dynamics. Electronic Research Announcements, 19(0):1–17, 2012.
× M. Glick. The pentagram map and y-patterns. Advances in Mathematics, 227(2):1019–1045, 2011.
× M. Glick. The limit point of the pentagram map. International Mathematics Research Notices, 2020(9):2818–2831, 2018.
× M. Glick and P. Pylyavskyy. Y-meshes and generalized pentagram maps. Proceedings of the London Mathematical Society, 112(4):753–797, 2016.
× A. B. Goncharov and R. Kenyon. Dimers and cluster integrable systems. Annales scientifiques de l’École Normale Supérieure, Ser. 4, 46(5):747–813, 2013.
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	$\displaystyle\mathcal{O}(V_{1},V_{2},V_{3})$	$\displaystyle=\det(V_{1},V_{2})+\det(V_{2},V_{3})+\det(V_{3},V_{1})$		(3)
		$\displaystyle=\det(V_{i}-V_{i-1},V_{i+1}-V_{i})\ \text{ for $i=1,2,3$}$		(3)

$\displaystyle P^{\prime}_{i}=(1-s_{1})P_{i+1}+s_{1}P_{i+k+1};$	$\displaystyle P^{\prime}_{i+1}=(1-t_{1})P_{i+1}+t_{1}P_{i+k+1};$	(9)
$\displaystyle P^{\prime}_{i+1}=(1-s_{2})P_{i+2}+s_{2}P_{i+k+2};$	$\displaystyle P^{\prime}_{i+2}=(1-t_{2})P_{i+2}+t_{2}P_{i+k+2};$
$\displaystyle P^{\prime}_{i+k}=(1-s_{3})P_{i+k+1}+s_{3}P_{i+2k+1};$	$\displaystyle P^{\prime}_{i+k+1}=(1-t_{3})P_{i+k+1}+t_{3}P_{i+2k+1},$

$\displaystyle\mathcal{O}(P^{\prime}_{i},P^{\prime}_{i+1},P^{\prime}_{i+2})$	$\displaystyle=\det(P^{\prime}_{i+1}-P^{\prime}_{i},P^{\prime}_{i+2}-P^{\prime}% _{i+1})$	(10)
	$\displaystyle=\det((s_{1}-t_{1})P_{i+1}+(t_{1}-s_{1})P_{i+k+1},(s_{2}-t_{2})P_% {i+2}+(t_{2}-s_{2})P_{i+k+2})$
	$\displaystyle=(t_{1}-s_{1})(t_{2}-s_{2})\det(P_{i+k+2}-P_{i+2},P_{i+1}-P_{i+k+% 1}).$

	$\displaystyle P^{\prime}_{i+k}$	$\displaystyle=\frac{t_{3}-s_{3}}{t_{3}(s_{3}-1)}\left(\frac{(1-s_{3})(t_{1}-1)% }{t_{1}-s_{1}}P^{\prime}_{i}+\frac{(1-s_{3})(1-s_{1})}{t_{1}-s_{1}}P^{\prime}_% {i+1}+\frac{s_{3}(1-s_{3})}{t_{3}-s_{3}}P^{\prime}_{i+k+1}\right)$
		$\displaystyle=\frac{(t_{3}-s_{3})(1-t_{1})}{t_{3}(t_{1}-s_{1})}P^{\prime}_{i}+% \frac{(t_{3}-s_{3})(s_{1}-1)}{t_{3}(t_{1}-s_{1})}P^{\prime}_{i+1}+\frac{s_{3}}% {t_{3}}P^{\prime}_{i+k+1}$
		$\displaystyle=(1-r_{2})(1-r_{1})P^{\prime}_{i}+(1-r_{2})r_{1}P^{\prime}_{i+1}+% r_{2}P^{\prime}_{i+k+1}.$

$\displaystyle\det(P^{\prime}_{i+1}-P^{\prime}_{i},P^{\prime}_{i+k+1}-P^{\prime% }_{i+k})$	$\displaystyle=\det((t_{1}-s_{1})(P_{i+k+1}-P_{i+1}),(t_{3}-s_{3})(P_{i+2k+1}-P% _{i+k+1}))$	(11)
	$\displaystyle=(t_{1}-s_{1})(t_{3}-s_{3})\det(P_{i+k+1}-P_{i+1},P_{i+2k+1}-P_{i% +k+1})$
	$\displaystyle=(t_{1}-s_{1})(t_{3}-s_{3})\mathcal{O}(P_{i+1},P_{i+k+1},P_{i+2k+% 1}).$

	$\displaystyle\mathcal{O}(B,O,C)=\det(-B,C)=-x_{b}y_{c}+x_{c}y_{b};$
	$\displaystyle\mathcal{O}(B,O,D)=\det(-B,D)=-x_{b}y_{d}+x_{d}y_{b};$
	$\displaystyle\mathcal{O}(C,O,D)=\det(-C,D)=-x_{c}y_{d}+x_{d}y_{c}.$

	$\displaystyle P^{\prime}_{i+k}$	$\displaystyle=(1-s_{3})P_{i+k+1}+s_{3}P_{i+2k+1}$
		$\displaystyle=\frac{(1-s_{3})(t_{1}-1)}{t_{1}-s_{1}}P^{\prime}_{i}+\frac{(1-s_% {3})(1-s_{1})}{t_{1}-s_{1}}P^{\prime}_{i+1}+\frac{s_{3}(t_{3}-1)}{t_{3}-s_{3}}% P^{\prime}_{i+k}+\frac{s_{3}(1-s_{3})}{t_{3}-s_{3}}P^{\prime}_{i+k+1}.$

$\displaystyle P^{\prime}_{i}=(1-t_{1})P_{i+1}+t_{1}P_{i+k+1};$	$\displaystyle P^{\prime}_{i+1}=(1-s_{1})P_{i+1}+s_{1}P_{i+k+1};$	(12)
$\displaystyle P^{\prime}_{i+1}=(1-t_{2})P_{i+2}+t_{2}P_{i+k+2};$	$\displaystyle P^{\prime}_{i+2}=(1-s_{2})P_{i+2}+s_{2}P_{i+k+2};$
$\displaystyle P^{\prime}_{i+k}=(1-t_{3})P_{i+k+1}+t_{3}P_{i+2k+1};$	$\displaystyle P^{\prime}_{i+k+1}=(1-s_{3})P_{i+k+1}+s_{3}P_{i+2k+1},$

		$\displaystyle\mathcal{O}(A,O,D)=s_{1}s_{2}\mathcal{O}(P^{\prime}_{i-k-1},P^{% \prime}_{i-k},P^{\prime}_{i-k+1});$		(16)
		$\displaystyle\mathcal{O}(C,O,D)=(1-s_{3})s_{2}\mathcal{O}(P^{\prime}_{i-k},P^{% \prime}_{i-k+1},P^{\prime}_{i-k+2});$
		$\displaystyle\mathcal{O}(B,O,D)=s_{2}(1-s_{5})\mathcal{O}(P^{\prime}_{i-k},P^{% \prime}_{i-k+1},P^{\prime}_{i+1})+s_{2}s_{5}\mathcal{O}(P^{\prime}_{i-k},P^{% \prime}_{i-k+1},P^{\prime}_{i});$

$\displaystyle x_{-4}$	$\displaystyle=\chi(l_{-1,-4},\,l_{-1,-3},\,l_{-1,-2},\,l_{-1,0})\overset{l_{-3% ,0}}{=}\chi(A,E,G,D);$	(22)
$\displaystyle x_{-2}$	$\displaystyle=\chi(l_{0,-3},\,l_{0,-2},\,l_{0,-1},\,l_{0,1})\overset{l_{-3,0}}% {=}\chi(E,B,H,D);$
$\displaystyle x_{-1}$	$\displaystyle=\chi(l_{-2,1},\,l_{-2,0},\,l_{-2,-1},\,l_{-2,-3})\overset{l_{-3,% 0}}{=}\chi(B,D,G,E);$
$\displaystyle x_{1}$	$\displaystyle=\chi(l_{-1,2},\,l_{-1,1},\,l_{-1,0},\,l_{-1,-2})\overset{l_{-3,0% }}{=}\chi(C,F,D,G).$

	$\displaystyle x_{-2}(x_{-4}+x_{-1}-1)$	$\displaystyle=\chi(G,B,F,D)\left(\chi(A,E,G,D)+\chi(B,D,G,E)-1\right)$
		$\displaystyle=\,\frac{(g-b)(f-d)}{(g-f)(b-d)}\left(\frac{(a-e)(g-d)}{(a-g)(e-d% )}+\frac{(b-e)(g-d)}{(b-g)(e-d)}\right)$
		$\displaystyle=\,\frac{(a-b)(g-d)(e-g)(d-f)}{(a-g)(b-d)(e-d)(g-f)}.$

	$\displaystyle\varphi^{2}(s,t)$	$\displaystyle=\varphi((-1)^{s+1}s,\,(-1)^{s}(2s+t))$
		$\displaystyle=\left((-1)^{(-1)^{s+1}s+s+2}s,\,(-1)^{(-1)^{s+1}s}\,(2(-1)^{s+1}% s+(-1)^{s}(2s+t))\right)=(s,t).$

		$\displaystyle\left(1-\frac{1-x_{2i+s+(-s)}}{1-x_{2i+s+(-s)}x_{2i+s+(-s)+(2s+t)% }}\right)\cdot\left(\frac{1-x_{2i+s+t-(-s)}}{1-x_{2i+s+t-(-s)}x_{2i+s+t-(-s)-(% 2s+t)}}\right)^{-1}$
		$\displaystyle=\left(1-\frac{1-x_{2i}}{1-x_{2i}x_{2i+2s+t}}\right)\left(\frac{1% -x_{2i+2s+t}x_{2i}}{1-x_{2i+2s+t}}\right)=x_{2i}.$

	$\displaystyle b_{2i}$	$\displaystyle=\frac{1-a_{2i-1}}{1-a_{2i-1}a_{2i-2}}=\left(1-\frac{1-x_{2i-4}}{% x_{2i-1}}\right)\cdot\left(1-\frac{(1-x_{2i-4})(1-x_{2i+1})}{x_{2i-1}x_{2i-2}}% \right)^{-1}$
		$\displaystyle=\frac{x_{2i-2}(x_{2i-1}+x_{2i-4}+1)}{x_{2i-1}x_{2i-2}-(1-x_{2i-4% })(1-x_{2i+1})}.$

	$\displaystyle\tilde{x}_{2i}\tilde{x}_{2i+3}$	$\displaystyle=\chi(l_{1,-1}\cap l_{-1,2},l_{1,-2}\cap l_{-1,2},l_{0,3}\cap l_{% -1,2},l_{0,2}\cap l_{-1,2})$
		$\displaystyle=\chi(\hat{P}_{i-1,0},\hat{P}_{i,1},\hat{P}_{i+1,1},\hat{P}_{i+2,% 0})=-y_{i,0},$

$\displaystyle y_{i+1,-1}$	$\displaystyle=-\frac{\tilde{x}_{2i+4}(1-\tilde{x}_{2i+3}\tilde{x}_{2i+6})(1-% \tilde{x}_{2i+1})}{(1-\tilde{x}_{2i+1}\tilde{x}_{2i+4})(1-\tilde{x}_{2i+3})}% \cdot\frac{\tilde{x}_{2i+3}(1-\tilde{x}_{2i+1}\tilde{x}_{2i+4})(1-\tilde{x}_{2% i+6})}{(1-\tilde{x}_{2i+3}\tilde{x}_{2i+6})(1-\tilde{x}_{2i+4})}$	(33)
	$\displaystyle=-\frac{\tilde{x}_{2i+3}\tilde{x}_{2i+4}(1-\tilde{x}_{2i+1})(1-% \tilde{x}_{2i+6})}{(1-\tilde{x}_{2i+3})(1-\tilde{x}_{2i+4})};$
$\displaystyle y_{i+1,1}$	$\displaystyle=-\frac{\tilde{x}_{2i}(1-\tilde{x}_{2i-2}\tilde{x}_{2i+1})(1-% \tilde{x}_{2i+3})}{(1-\tilde{x}_{2i}\tilde{x}_{2i+3})(1-\tilde{x}_{2i+1})}% \cdot\frac{\tilde{x}_{2i+7}(1-\tilde{x}_{2i+6}\tilde{x}_{2i+9})(1-\tilde{x}_{2% i+4})}{(1-\tilde{x}_{2i+4}\tilde{x}_{2i+7})(1-\tilde{x}_{2i+6})}$
	$\displaystyle=-\frac{\tilde{x}_{2i}\tilde{x}_{2i+7}(1+y_{i-1,0})(1+y_{i+3,0})(% 1-\tilde{x}_{2i+3})(1-\tilde{x}_{2i+4})}{(1+y_{i,0})(1+y_{i+2,0})(1-\tilde{x}_% {2i+1})(1-\tilde{x}_{2i+6})}.$

	$\displaystyle y_{i+1,-1}\,y_{i+1,1}$	$\displaystyle=\frac{\tilde{x}_{2i}\tilde{x}_{2i+3}\tilde{x}_{2i+4}\tilde{x}_{2% i+7}(1+y_{i-1,0})(1+y_{i+3,0})}{(1+y_{i,0})(1+y_{i+2,0})}$
		$\displaystyle=\frac{y_{i,0}y_{i+2,0}(1+y_{i-1,0})(1+y_{i+3,0})}{(1+y_{i,0})(1+% y_{i+2,0})}=\frac{(1+y_{i-1,0})(1+y_{i+3,0})}{(1+y_{i,0}^{-1})(1+y_{i+2,0}^{-1% })}.$

	$\displaystyle\mathcal{F}_{3}^{\prime}$	$\displaystyle=\mathcal{F}_{3}\cdot\prod_{i=0}^{n-1}\frac{x_{2i-4}+x_{2i-1}-1}{% x_{2i+2}+x_{2i+5}-1}\cdot\prod_{i=0}^{n-1}\frac{x_{2i+2}x_{2i+3}-(1-x_{2i+5})(% 1-x_{2i})}{x_{2i-2}x_{2i-1}-(1-x_{2i+1})(1-x_{2i-4})}$
		$\displaystyle=\mathcal{F}_{3}\cdot\frac{\prod_{i={-3}}^{n-4}(x_{2i+2}+x_{2i+5}% -1)}{\prod_{i=0}^{n-1}(x_{2i+2}+x_{2i+5}-1)}\cdot\frac{\prod_{i=2}^{n+1}(x_{2i% -2}x_{2i-1}-(1-x_{2i+1})(1-x_{2i-4}))}{\prod_{i=0}^{n-1}(x_{2i-2}x_{2i-1}-(1-x% _{2i+1})(1-x_{2i-4}))}=\mathcal{F}_{3}.$

	$\displaystyle x_{2i}$	$\displaystyle=\frac{(s_{1,-2}-s_{1,-1})(s_{1,0}-s_{1,2})}{(s_{1,-2}-s_{1,0})(s% _{1,-1}-s_{1,2})}=-\frac{s_{1,-2}-1}{1-s_{1,2}}>0;$		(35)
	$\displaystyle x_{2i+1}$	$\displaystyle=\frac{(s_{-1,2}-s_{-1,1})(s_{-1,0}-s_{-1,-2})}{(s_{-1,2}-s_{-1,0% })(s_{-1,1}-s_{-1,-2})}=\frac{-s_{-1,-2}(s_{-1,2}-1)}{s_{-1,2}(1-s_{-1,-2})}<0.$		(35)

	$\displaystyle O_{1}(P)$	$\displaystyle=\sum_{i=0}^{n-1}(-x_{2i+1}+x_{2i-1}x_{2i}x_{2i+1});$	$\displaystyle O_{n}(P)$	$\displaystyle=\prod_{i=0}^{n-1}x_{2i+1};$
	$\displaystyle E_{1}(P)$	$\displaystyle=\sum_{i=0}^{n-1}(-x_{2i}+x_{2i-2}x_{2i-1}x_{2i});$	$\displaystyle E_{n}(P)$	$\displaystyle=\prod_{i=1}^{n}x_{2i}.$

$\displaystyle P_{i}=(1-s_{1})P^{\prime}_{i-k}+s_{1}P^{\prime}_{i-k-1};$

$\displaystyle P_{i}=(1-t_{1})P^{\prime}_{i}+t_{1}P^{\prime}_{i-1};$

$\displaystyle P_{i+1}=(1-s_{2})P^{\prime}_{i-k+1}+s_{2}P^{\prime}_{i-k};$

$\displaystyle P_{i+1}=(1-t_{2})P^{\prime}_{i+1}+t_{3}P^{\prime}_{i};$

$\displaystyle P_{i+2}=(1-s_{3})P^{\prime}_{i-k+2}+s_{3}P^{\prime}_{i-k+1};$

$\displaystyle P_{i+k}=(1-s_{4})P^{\prime}_{i}+s_{4}P^{\prime}_{i-1};$

$\displaystyle P_{i+k+1}=(1-s_{5})P^{\prime}_{i+1}+s_{5}P^{\prime}_{i},$