The Exchange Graphs

The Exchange Graphs of Weakly Separated Collections

Meena Jagadeesan Department of Mathematics, Harvard University, Cambridge, MA 02138 mjagadeesan@college.harvard.edu
Abstract.

Weakly separated collections arise in the cluster algebra derived from the Plücker coordinates on the nonnegative Grassmannian. Oh, Postnikov, and Speyer studied weakly separated collections over a general Grassmann necklace and proved the connectivity of every exchange graph. Oh and Speyer later introduced a generalization of exchange graphs that we call -constant graphs. They characterized these graphs in the smallest two cases. We prove an isomorphism between exchange graphs and a certain class of -constant graphs. We use this to extend Oh and Speyer’s characterization of these graphs to the smallest four cases, and we present a conjecture on a bound on the maximal order of these graphs. In addition, we fully characterize certain classes of these graphs in the special cases of cycles and trees.

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1. Introduction

The notion of weak separation was introduced by Leclerc and Zelevinsky [1] in 1998 in the context of determining combinatorial criterion for the quasicommutativity of two quantum flag minors. The definition of weak separation requires the following definition of a cyclic ordering:

Definition 1.1.

For an integer , the integers are said to be cyclically ordered if there exists such that

Leclerc and Zelevinsky defined weak separation as follows:

Definition 1.2.

Let and be nonnegative integers such that . Two -element subsets are called weakly separated if there do not exist , cyclically ordered, with and , where denotes set-theoretic difference.

Remark 1.3.

This is equivalent to the condition that and are separated by some chord in the circle, i.e. that are cyclically ordered.

Leclerc and Zelevinsky [1] used the notion of weak separation to define certain collections of -elements subsets of for a fixed and . Let denote the collection of all -element subsets of . For nonnegative integers , a subset of the collection is called weakly separated if every two -elements subsets in are weakly separated. The subset is called a maximal weakly separated collection if is not contained in any larger weakly separated collection of .

Leclerc and Zelevinsky conjectured that any two maximal weakly separated collections of have the same cardinality and are linked by a sequence of cardinality-preserving operations called mutations. A mutation is defined as follows:

Definition 1.4.

Consider a set and let be cyclically ordered elements of Suppose that a maximal weakly separated collection contains and . Then the collection is said to be linked to by a mutation.

In 2005, Scott [6] proved that any weakly separated collection satisfies . This motivated him to make the following refined conjecture regarding the cardinality of maximal weakly separated collections: If is a maximal weakly separated collection, then the following is true:

(1)

In 2011, Oh, Postnikov, and Speyer [9] proved a generalization of by linking weak separation to certain graphs called plabic graphs through applying a previous work of Postnikov [5].

In 2006, Postnikov [5] studied total positivity on the Grassmannian. He defined the nonnegative Grassmannian as the part of the real Grassmannian in which all Plücker coordinates are nonnegative. He described a stratification of into positroid strata and combinatorially constructed their parametrization using plabic graphs. Namely, he proved that positroid cells can be parameterized through a certain type of planar bicolored graphs (plabic graphs). These type of graphs play an important role in the study of various mathematical objects, for example, appearing also in Kodama and Williams’s [3] study on KP-solitons.

Postnikov, Oh, and Speyer [9] used plabic graphs to prove Leclerc and Zelevinsky’s conjecture that any two maximal weakly separated collections in are linked by a sequence of mutations, from which followed. The main ingredient of their proof was their bijection between weakly separated collections and reduced plabic graphs. In fact, using this bijection, they were able to prove a generalization of based on extending the notion of a weakly separated collection to a general positroid. Their results had interesting consequences for cluster algebras. In 2003, Scott [7] had proved that the coordinate ring of , in its Plücker embedding, is a cluster algebra where the Plücker coordinates are the cluster variables. Postnikov, Oh, and Speyer [9] proved that the aforementioned clusters are in bijection with the maximal weakly collections of .

Our global aim is to continue the study of the combinatorial properties of the cluster algebra structure on . We specifically study weakly separated collections over a general positroid. In Section 2, we review important existing definitions and known results. In Section 3, we introduce some of our new definitions, state our main results, and outline the rest the paper.

2. Existing Definitions and Known Results

We review the relevant existing definitions, notation, and known results involving weakly separated collections. In Section 2.1, we review the definitions from [9] related to maximal weakly separated collections over a general positroid. In Section 2.2, we review the technology of plabic graphs from [9] and [5]. In Section 2.3, we review exchange graphs from [9] and the subgraphs from [8] that we call -constant graphs.

2.1. Weakly Separated Collections over a General Positroid

We recall the definitions and results from [9] relating to Grassmann necklaces, positroids, decorated permutations, and maximal weakly separated collections.

We define a connected Grassmann necklace.

Definition 2.1.

A connected Grassmann necklace is a sequence of -element subsets of such that, for , the set contains and (where we take the indices modulo ).

Remark 2.2.

All Grassmann necklaces that we will work with will be connected, though we will omit the word connected. For the rest of the paper, we assume that and .

We use the following notion of a linear order on .

Definition 2.3.

Consider positive integers . For , we say that if is a subsequence of For , we say that if and only if or .

Let and be -element subsets of such that and where and . Then we define the partial order

We now define the positroid associated to each Grassmann necklace.

Definition 2.4.

Given a Grassmann necklace , we define the positroid to be

This allows us to extend the notion of a maximal weakly separated collection to a general Grassmann necklace.

Definition 2.5.

For a Grassmann necklace , a weakly separated collection is said be over if is a subset of . The collection is said to be maximal over if is not contained in any larger weakly separated collection over .

Oh, Postnikov, and Speyer [9] proved a generalization of (Scott’s conjecture). Namely, they expressed the cardinality of every maximal weakly separated collection over a Grassmann necklace as a function of the decorated permutation associated to . We review the definitions relating to decorated permutations in the case of connected Grassmann necklaces.

Definition 2.6.

A connected decorated permutation is a permutation over such that there do not exist two circular intervals and such that and .

There is a simple relation between connected Grassmann necklaces and connected decorated permutations.

Proposition 2.7 (Oh, Postnikov, Speyer).

Connected Grassmann necklaces are in bijection with connected decorated permutations over .

Proof.

To go from a Grassmann necklace to a decorated permutation , we set

To go from a decorated permutation to a Grassmann necklace , we set

Oh, Postnikov, and Speyer [9] defined the function (where is the set of permutations of ) as follows:

Definition 2.8.

For , the set forms an alignment in if are cyclically ordered (and all distinct). Let be the number of alignments in .

Oh, Postnikov, and Speyer [9] used the notion of a decorated permutation and the function to formulate their result involving cardinality. We recall the special case that corresponds to connected Grassmann necklaces:

Theorem 2.9 (Oh, Postnikov, Speyer).

Let be a maximal weakly separated collection over a connected Grassmann necklace . Suppose that has associated decorated permutation . Then the following is true:

2.2. Plabic Graphs

We review the technology of plabic graphs discussed in [5] and [9].

We define a plabic graph.

Definition 2.10.

A plabic graph (planar bicolored graph) is a planar undirected graph drawn inside a disk with vertices colored in black or white colors. The vertices on the boundary of the disk are called the boundary vertices. Suppose that there are boundary vertices. Then the boundary vertices are labeled in clockwise order by .

We define the strands in a plabic graph.

Definition 2.11.

A strand in a plabic graph is a directed path that satisfies the rules of the road: At every black vertex, the strand turns right, and at every white vertex, the strand turns left.

We define the criteria for a plabic graph to be reduced.

Definition 2.12.

A plabic graph is called reduced if the following holds:

  • A strand cannot form a closed loop in the interior of .

  • Any strand that passes through itself must be a simple loop that starts and ends at some boundary vertex.

  • For any two strands that have two vertices and in common, one strand must be directed from to , and the other strand must be directed from to .

Let be a reduced plabic graph. Then any strand in connects two boundary vertices. We associate a strand permutation with defined so that if the strand that starts at a boundary vertex ends at a boundary vertex . We label the strand that ends at boundary vertex by .

There are three types of moves on a plabic graph:

(M1)Pick a square with vertices alternating in colors as in Figure 1. Then we can switch the colors of these vertices.

Figure 1. (M1) Square move

(M2)Two adjoint vertices of the same color can be contracted into one vertex as in Figure 2.

Figure 2. (M2)

(M3)We can insert or remove a vertex inside an edge as in Figure 3.

Figure 3. (M3)

Postnikov [5] proved the following relation between strand permutations and the moves (M1), (M2), and (M3).

Theorem 2.13 (Postnikov).

Let and be two reduced plabic graphs with the same number of boundary vertices. Then if and only if can be obtained from by a sequence of moves (M1), (M2), and (M3).

We now describe the relation between reduced plabic graphs and weakly separated collections. We label the faces of a reduced plabic graph as follows: Let be a reduced plabic graph. For every , place inside every face that appears to the left of the strand labeled . Then the label of is the set of all integers placed inside . Let be the collection of sets of labels that occur on each face of the graph . Postnikov [5] showed that all the sets in have the same number of elements. We denote this number by . Oh, Postnikov, and Speyer [9] proved the following:

Theorem 2.14 (Oh, Postnikov, Speyer).

For a decorated permutation corresponding to a Grassmann necklace , a collection is a maximal weakly separated collection over if and only if it has the form for a reduced plabic graph with strand permutation .

We also consider a dual of plabic graphs called plabic tilings described in detail in [8]. The faces and vertices are flipped. In plabic tilings, the faces are colored black and white, and the vertices contain the sets of integer labels.

2.3. Exchange Graphs and -Constant Graphs

We review the definitions and known results regarding exchange graphs and -constant graphs.

For a given Grassmann necklace , the exchange graph is defined by the maximal weakly separated collections over under the mutation operation.

Definition 2.15.

Let be a Grassmann necklace and be the set of maximal weakly separated collections over . The exchange graph is defined as follows:

  1. The vertices of are .

  2. The vertices and are connected by an edge if and only if can be mutated into in one mutation.

In 2011, Oh, Postnikov, and Speyer [9] proved the following result:

Theorem 2.16 (Oh, Postnikov, Speyer).

Any exchange graph is connected.

Given an exchange graph and a weakly separated collection over , we call the -constant graph the subgraph defined by maximal weakly separated collections over that contain with edges defined by mutations of subsets not in .

Definition 2.17.

Let be a Grassmann necklace and be the set of maximal weakly separated collections over that contain . We call the -constant graph the vertex-induced subgraph of generated by . The co-dimension of is defined to be for

Remark 2.18.

Notice that -constant graphs are a generalization of exchange graphs. In fact, every exchange graph is isomorphic to a -constant graph: for any Grassmann necklace , we see that is isomorphic to

In 2014, Oh and Speyer [8] proved the following results:

Theorem 2.19 (Oh, Speyer).

Any -constant graph is connected.

Theorem 2.20 (Oh, Speyer).

The only -constant graph with co-dimension is a path with vertex. The only -constant graphs with co-dimension are a path with vertex and a path with vertices.

3. Main Results

In Section 3.1, we present some of our new notions that are critical to understanding our main results. In Section 3.2, we present our main results along with an outline for the rest of the paper.

3.1. Some New Definitions

In Section 3.1.1, we present some basic definitions. In Section 3.1.2, we define special classes of exchange graphs and -constant graphs. In Section 3.1.3, we define equivalence classes of decorated permutations.

3.1.1. Basic Definitions

We define the following definitions and notation involving adjacency, interior size, and mutations.

We define two -elements subsets to be quasi-adjacent as follows:

Definition 3.1.

We call sets quasi-adjacent if .

Remark 3.2.

Suppose that and are contained in a maximal weakly separated collection . This definition is equivalent to the condition that and are on the same face in the plabic tiling of .

We now define a stronger notion of adjacency that is dependent on the choice of maximal weakly separated collection:

Definition 3.3.

Given a maximal weakly separated collection , we call two -element subsets adjacent if and only if in the plabic tiling of , there exists an edge between and that border a black face and a white face.

Remark 3.4.

Notice that all adjacent subsets are also quasi-adjacent.

We define the following notion with interior size which is closely related to cardinality:

Definition 3.5.

We define the of a maximal weakly separated collection over a Grassmann Necklace to be

Remark 3.6.

Notice that the interior size of is the number of sets in the interior of the plabic tiling of .

In fact, interior size is a property of the Grassmann necklace (and thus the exchange graph ). Let be the decorated permutation associated to . By Theorem 2.9, we know that the interior size of any maximal weakly separated collection in is . Thus, we let the interior size of both the Grassmann necklace and the exchange graph be this value. We denote this value by or .

We use the following notation and definitions to discuss mutations:

Definition 3.7.

Given a maximal weakly separated collection over a Grassmann Necklace , we say that a -element subset is in if in the plabic tiling of , the subset is surrounded by exactly 2 black faces and 2 white faces.

This means that we can mutate into a maximal weakly separated collection over that contains . Suppose that . We then say that can be mutated in into in . If is not surrounded by 2 black faces and 2 white faces, then we say that is in .

3.1.2. Special Classes of Exchange Graphs and -Constant Graphs

We define the applicable, mutation-friendly, and very-mutation-friendly conditions.

Roughly speaking, the applicable condition for a -constant graph requires that the sets in the weakly separated collection are connected to the boundary of the plabic tiling of for any maximal weakly separated collection .

Definition 3.8.

Consider a -constant graph . We say that is if for each set , there exists an integer and a weakly separated collection over satisfying the following properties:

  1. is quasi-adjacent to for ,

  2. and .

Figure 4.

We use the following maximal weakly separated collection and Grassmann necklace as an example throughout the paper:

Example 3.9.

Consider the Grassmann necklace

Consider the maximal weakly separated collection that contains and the following subsets: , , , , , , , , , , , . Figure 4 shows the plabic tiling of .

Example 3.10.

Consider defined as in Example 3.9. Then we define the following weakly separated collections over :

The -constant graph is not applicable and the -constant graph is applicable.

Roughly speaking, the mutation-friendly condition for a Grassmann necklace requires that every subset in that is present in at least one maximal weakly separated collection in is mutatable in some maximal weakly separated collection in .

Definition 3.11.

A Grassmann necklace and its associated decorated permutation are if the intersection of all of the maximal weakly separated collections in is . We call an exchange graph mutation-friendly if and only if the Grassmann necklace is mutation-friendly.

Remark 3.12.

It follows from the definition that for a mutation-friendly exchange graph , we know that

We extend a similar notion to -constant graphs:

Definition 3.13.

A -constant graph is said to be if the intersection of all of the maximal weakly separated collections in is .

We define a stronger notion of the mutation-friendly condition in the case of exchange graphs. This condition, the very-mutation-friendly condition, requires that a certain sequence of -constant graphs of the exchange graph satisfy both the applicable and mutation-friendly conditions.

Definition 3.14.

A mutation-friendly Grassmann necklace with interior size is
if there exists a set of weakly separated collections satisfying the following properties:

  1. is a weakly separated collection over containing for ,

  2. for ,

  3. for ,

  4. and is applicable and mutation-friendly for

We call very-mutation-friendly if and only if is very-mutation-friendly.

3.1.3. Equivalence Classes

In order to simplify our discussion of Grassmann necklaces, we construct an equivalence class of decorated permutations that yield the same exchange graph. We show that exchange graphs are invariant under the following four operations on associated decorated permutations.

We first consider the inverse operation.

Proposition 3.15.

Let be the Grassmann necklace with decorated permutation and be the Grassmann necklace with decorated permutation Then the following is true:

Proof.

We must prove that there is a bijective mapping from the maximal weakly separated collections over to the maximal weakly separated collections over that preserves the mutation operation. Let take to where is obtained by flipping the colors of the vertices in the plabic graph of . ∎

Figure 5.
Figure 6.
Example 3.16.

Let be the permutation . Then . Let be the maximal weakly separated collection corresponding to the plabic graph in Figure 5 and be the maximal weakly separated collection corresponding to the plabic graph in Figure 6. We see that takes to as expected.

We now consider label-reflection operations.

Definition 3.17.

For a positive integer , we define the label-reflection operation as follows. For an integer and a permutation of , we let be the permutation on defined so that for (where all numbers are considered modulo ).

Proposition 3.18.

Given integers , let be the Grassmann necklace with decorated permutation and be the Grassmann necklace with decorated permutation Then the following is true:

The proof is analogous to the proof of Proposition 3.15, except that is defined by reflecting the plabic graph of about the vertex labeled .

Figure 7.
Example 3.19.

Let be the permutation , and let . Then . Let be the maximal weakly separated collection corresponding to the plabic graph in Figure 5 and be the maximal weakly separated collection corresponding to the plabic graph in Figure 7. We see that takes to as expected.

We now consider between-label-reflection operations.

Definition 3.20.

For a positive integer , we define the between-label-reflection operation as follows. For an even integer and a permutation of , we let be the permutation on defined so that for (where all numbers are considered modulo ).

Proposition 3.21.

Given integers such that is even, let be the Grassmann necklace with decorated permutation and be the Grassmann necklace with decorated permutation Then the following is true:

The proof is analogous to the proof of Proposition 3.15, except that is defined by reflecting the plabic graph of about the perpendicular bisector of the edge between the vertex labeled begins and the vertex labeled .

Figure 8.
Example 3.22.

Let be the permutation , and let . Then . Let be the maximal weakly separated collection corresponding to the plabic graph in Figure 5 and be the maximal weakly separated collection corresponding to the plabic graph in Figure 8. We see that takes to as expected.

We now consider rotation operations.

Definition 3.23.

For a positive integer , we define the rotation operation as follows. For an even integer and a permutation of , we let be the permutation on defined so that for (where all numbers are considered modulo ).

Proposition 3.24.

Given integers , let be the Grassmann necklace with decorated permutation and be the Grassmann necklace with decorated permutation Then the following is true:

The proof is analogous to the proof of Proposition 3.15, except that is defined by rotating the vertex labels on the plabic graph of so that the vertex is now labeled .

Figure 9.
Example 3.25.

Let be the permutation , and let . Then . Let be the maximal weakly separated collection corresponding to the plabic graph in Figure 5 and be the maximal weakly separated collection corresponding to the plabic graph in Figure 9. We see that takes to as expected.

This motivates the following definition of an equivalence class of decorated permutations:

Definition 3.26.

For a given decorated permutation , we define the to contain all permutations that can be obtained from one another by a sequence of inverse operations, label-reflection operations, between-label reflection operations, and rotation operations. We also let denote the class of corresponding Grassmann necklaces.

We prove that any two decorated permutations in the same equivalence class yield the same exchange graph.

Lemma 3.27.

Let be an equivalence class, and consider . Let be the Grassmann necklace with decorated permutation and be the Grassmann necklace with decorated permutation Then the following is true:

Proof.

This follows from Proposition 3.15, Proposition 3.18, Proposition 3.21, and Proposition 3.24. ∎

3.2. Statement of Main Results

We continue the study of -constant graphs and exchange graphs for maximal weakly separated collections over a general positroid with connected Grassmann necklaces.

In Section 4, we prove our main result: an isomorphism between exchange graphs and applicable -constant graphs.

Theorem 3.28.

For any , the set of possible applicable -constant graphs of co-dimension is isomorphic to the set of the possible exchange graphs with interior size . An applicable -constant graph of co-dimension that is mutation-friendly is isomorphic to a mutation-friendly exchange graph with interior size .

A consequence of this result is that all properties of the exchange graphs apply to applicable -constant graphs and vice versa. For , we show that we can eliminate the applicable condition:

Corollary 3.29.

For any , the set of possible constant graphs of co-dimension is isomorphic to the set of the possible exchange graphs with interior size . A -constant graph of co-dimension that is mutation-friendly is isomorphic to a mutation-friendly exchange graph with interior size .

In Section 5, we characterize all exchange graphs with interior size . We also generalize Theorem 2.20 (Oh and Speyer’s characterization result for -constant graphs) by characterizing all -constant graphs of co-dimension . These results involve the information in Table 1 and Table 2. These tables require the graphs which are defined in Table 4 and the direct product of graphs which is defined as follows:

Definition 3.30.

Given graphs and , we define the direct product as follows:

  1. The vertices of are

  2. There is an edge between and if and only if either

    • and is adjacent to in ,

    • or is adjacent to in and

Interior Size Exchange Graph Orders Exchange Graphs
0 1 A
1 1, 2 A, B
2 1, 2, 3, 4, 5 A, B, C, D, B B
3 1, 2, 3, 4, 5, 6, 7, 8, 10, 14 A, B, C, D, E, F, G, H, I,
B C, B B B, B D
4 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F, G, H, I, J, K, L, M, N,
10, 11, 12, 13, 14, 15, 16, 17 O, P, Q, R, S, T, U, V, W, X, Y, Z1, Z2,
19, 20, 25, 26, 28, 34, 42 Z3, Z4, Z5, Z6, B C, B B B, B D,
B E, B G, B F, B H, B I,
B B B B, B B C, B B D
C C, D D
Table 1.

We have the following two results:

Theorem 3.31.

The information is Table 1 is true.

Theorem 3.32.

The information in Table 2 is true.

Co-dimension -Constant Graph Orders -Constant Graphs
0 1 A
1 1, 2 A, B
2 1, 2, 3, 4, 5 A, B, C, D, B B
3 1, 2, 3, 4, 5, 6, 7, 8, 10, 14 A, B, C, D, E, F, G, H, I,
B C, B B B, B D
Table 2.

In Section 6, we present the following conjecture on a bound for the maximal order of an exchange graph with a given interior size:

Conjecture 3.33.

For any , the maximum possible order of an exchange graph with interior size is the Catalan number (using the convention that , , and ).

Remark 3.34.

Theorem 3.31 proves Conjecture 3.33 for .

In Section 7, we consider very-mutation-friendly exchange graphs in the special cases of cycles and trees. We fully characterize the very-mutation-friendly exchange graphs that are trees. We require the following equivalence classes:

Definition 3.35.

We define to be the equivalence class with the permutation which is defined as follows:

  • For , we have .

  • For , we let be a permutation on defined as follows:

Theorem 3.36.

Any very-mutation-friendly exchange graph that is a tree must be a path. For an integer , a very-mutation-friendly exchange graph with interior size is a path if and only if it is a part of the equivalence class (path with vertices).

We also fully characterize the equivalence classes of very-mutation-friendly exchange graph with interior size that are single cycles.

Theorem 3.37.

If a very-mutation-friendly exchange graph is a single cycle, then it must have , , , or vertices. If is prime and very-mutation friendly, then must be part of the equivalence class with (cycle with vertex), with (cycle with vertices), or with (cycle with vertices). If is very-mutation-friendly and not prime, then must be a part of the equivalence class with (cycle with vertices).

4. Proof of Theorem 3.28 and Corollary 3.29

Theorem 3.28 is an isomorphism between applicable -constant graphs and exchange graphs. Corollary 3.29 is a special case of Theorem 3.28 in the case of where we show that the applicable condition is no longer necessary.

One direction of the proof of Theorem 3.28 is easy. It is clear that an exchange graph with interior size is isomorphic to the applicable -constant graph which has co-dimension . This proves that the set of possible applicable -constant graphs of co-dimension contains an isomorphic copy of the set of possible exchange graphs with interior size . In this section, we prove the other direction of Theorem 3.28: namely, that every applicable -constant graph is isomorphic to an exchange graph with appropriate interior size. We also prove Corollary 3.29.

In Section 4.1, we introduce with some modifications the approaches of Oh, Postnikov, and Speyer [9, Section 9] as well as Danilov, Karzanov, and Koshevoy [2] involving domains inside special weakly separated collections. In Section 4.2, we construct the machinery of interior-reduced plabic graphs and plabic tilings. In Section 4.3, we construct a decomposition set of a Grassmann necklace. In Section 4.4, we construct adjacency graphs and clusters. In Section 4.5, we prove a special case of Theorem 3.28. In Section 4.6, we use this to prove Theorem 3.28 in the general case. In Section 4.7, we show that all -constant graphs with co-dimension 4 are applicable, which when applied to Theorem 3.28, proves Corollary 3.29.

4.1. Domains inside and outside of cyclic patterns

We introduce with some modifications the approaches in [9] and [2] regarding domains inside non self-intersecting closed curves. Fix unit vectors in the upper half-plane, so that go in this order clockwise around the origin and are -independent. Define:

A subset is identified with the point in .

Now, suppose we have a weakly separated collection of subsets of equal size such that (where indices are considered modulo ). Then we call a sequence. Note that might have repeated subsets. We can associate with a clockwise-oriented piecewise linear closed curve obtained by concatenating the line-segments connecting consecutive points and for .

We consider sequences with the following additional constraint on :

Definition 4.1.

We call a closed curve prong-closed if is the concatenation of curves , , and that satisfy the following conditions:

  1. The curve is a clockwise-oriented non self-intersecting closed curve.

  2. The intersection is a single point .

  3. satisfies one of the following:

    • or the following conditions hold:

      1. is a non self-intersecting open curve,

      2. has an endpoint at ,

      3. has an endpoint at a point in the interior of ,

      4. and .

We also impose the following constraints on the sequence . Sequences that satsify certain constraints involving weak separation are called generalized cyclic patterns [2]. We consider the following variant of a generalized cyclic pattern:

Definition 4.2.

A quasi-generalized cyclic pattern is a sequence of subsets of where such that

  1. is weakly separated,

  2. the sets in all have the same size,

  3. and

Figure 10.
Figure 11.

We show two examples of quasi-generalized cyclic patterns such that is prong-closed.

Example 4.3.

Suppose that

Then we have that , shown in Figure 10, is prong-closed.

Example 4.4.

Then we have that , shown in Figure 11, is prong-closed.

For an quasi-generalized cyclic pattern such that the curve is prong-closed, let

Remark 4.5.

Our definition of differs slightly from the definitions in [2]. The definition in [2] includes in , while we do not include .

We describe how to decompose into two pure domains: and . Let be the curve obtained by concatenating and . Then, by definition, we know that subdivides into two closed regions and such that

Then we let

and

4.2. Interior-Reduced Plabic Graphs

We construct the interior-reduced plabic tiling machinery that is helpful in proving Theorem 3.28.

Consider a plabic tiling of a maximal weakly separated collection . Take a sequence of subsets . We call the boundary curve. Consider the sub-plabic tiling consisting of the sets and faces in the plabic tiling of on and within . Suppose that

  1. The boundary curve is prong-closed,

  2. and there do not exist with such that there is a face containing both and in the aformentioned sub-plabic tiling.

Then we call the sub-plabic tiling together with the boundary curve an interior-reduced plabic tiling.

Suppose that there is an interior-reduced plabic tiling that consists of sets of the weakly separated multi-collection (that may have repeated subsets as permitted by ). We omit the phrase “multi” for the remainder of the paper. Then, we denote the interior-reduced plabic tiling of as . Let the sequence of subsets on the boundary be . We let be with the set labels removed.

We now define the interior-reduced plabic graph to be the dual plabic graph (without strand labels) of with the following alteration: Suppose that has repeated elements so that is nonempty. For every two quasi-adjacent sets and connected by , consider the faces in that contain both and . Consider the dual vertices of these faces in the dual plabic graph. If there is only one such vertex, then we split this vertex into two vertices by the move (M2). Now, we let the two vertices be and . We make and into boundary vertices as follows: For , consider the strand that goes from to . We break this strand into two strands so that the first strand ends at and the second strand begins at .

In the case that , we omit the subscripts.

Figure 12.
Figure 13.

We consider the following examples:

Example 4.6.

Consider as described in Example 3.9. Consider the weakly separated collection that contains , , , , , , , , , , , , , and . Notice that is an interior-reduced plabic tiling. Figure 12 shows , , and .

Example 4.7.

Consider as described in Example 3.9. Consider the weakly separated (multi-)collection that contains , , , , , , , , , , , , , , and . The difference between this and the in Example 4.6 in the additional copy of . Notice that is an interior-reduced plabic tiling. Figure 12 shows , , and .

We prove the following property of interior-reduced plabic graphs:

Proposition 4.8.

Every interior-reduced plabic graph is a reduced plabic graph without strand labels.

Proof.

We consider the interior-reduced plabic graph . First, notice that the condition (2) in the definition of an interior-reduced plabic tiling guarantees that the each boundary vertex of will only have one notch coming out of it on the outside. This means that is a plabic graph. Suppose that we label the strands of the in clockwise order. It suffices to prove that these strands satisfy the properties of a reduced plabic graph. We know that this is true, because the strands of are contained in the strands of the reduced plabic graph of (up to breaking and relabeling). ∎

Notice that the sequence of sets on the boundary curve of an interior-reduced plabic tiling form a modified Grassmann-like necklace which is defined as follows:

Definition 4.9.

Let be a quasi-generalized cyclic pattern. We call a modified Grassmann-like necklace if the following properties are satisfied:

  1. The curve is prong-closed.

  2. There is no such that (where indices are modulo ) and the integers are cyclically ordered.

Remark 4.10.

It follows from the definition that

We associate modified Grassmann-like necklaces with weakly separated multi-collections.

Definition 4.11.

Let be a modified Grassmann-like necklace contained in a maximal weakly separated collection . Then, we define the -enclosed collection of to be

where repeated subsets in continue to be repeated subsets in .

Remark 4.12.

Notice that is always an interior-reduced plabic tiling. We thus know that for a fixed maximal weakly separated collection , the -enclosed collections are in bijection with the interior-reduced plabic tilings/graphs of .

We show two examples of -enclosed collections.

Example 4.13.

Let be as defined in Example 4.3. Then is a modified-like Grassmann necklace. Consider and as defined in Example 4.6. Then .

Example 4.14.

Let be as defined in Example 4.4. Then is a modified-like Grassmann necklace. Consider and as defined in Example 4.7. Then .

Now, consider a modified Grassmann-like necklace . Suppose that is contained in a maximal weakly separated collection . We map to a Grassmann necklace in such a way that is independent of choice of . We call the