2.1. Topological setting

Let $\mathfrak {S}$ be an oriented punctured surface of finite topological type, namely $\mathfrak {S}$ is diffeomorphic to the space obtained by removing a finite subset P, called the set of punctures, from a closed oriented surface $\overline {\mathfrak {S}}$ . In particular, note that $\mathfrak {S}$ has empty boundary, $\partial \mathfrak {S} = \varnothing $ . We require that there is at least one puncture and that the Euler characteristic $\chi (\mathfrak {S})$ of the punctured surface $\mathfrak {S}$ is strictly less than zero, $\chi (\mathfrak {S}) < 0$ . These topological conditions guarantee the existence of an ideal triangulation $\lambda $ of the punctured surface $\mathfrak {S}$ , namely a triangulation $\overline {\lambda }$ of the closed surface $\overline {\mathfrak {S}}$ whose vertex set is equal to the set of punctures P. See Figure 1 for some examples of ideal triangulations.

Figure 1 Ideal triangulations.

To simplify the exposition, we always assume that $\lambda $ does not contain any self-folded triangles, meaning that each triangle $\mathfrak {T}$ of $\lambda $ has three distinct edges. Such a $\lambda $ always exists. Our results should generalize, essentially without change, to allow for self-folded triangles.

2.2. Webs

We will often be working with embedded curves, where there are no self-intersections.

Figure 2 Web.

For an example, in Figure 2 we show a web on the once punctured torus, which has four components consisting of two trivalent graphs and two curves.

Figure 3 Global parallel-move.

Intuitively, we think of parallel equivalent as meaning homotopic on the surface.

2.3. Faces

For an example, the web shown in Figure 2 above has one disk-face, one bigon-face, two square-faces and two hexagon-faces; these faces are shaded in the figure. Notice that one of the hexagon-faces consists of five edges of the web, one edge being counted twice.

By orientation considerations, faces must have an even number of sides.

Bigon- and square-faces always consist of exactly two and four edges, respectively, of W. See Figure 4.

Figure 4 Prohibited square-face.

In figures, we often omit the web orientations, as in Figure 5.

Figure 5 Local webs.

2.4. Nonelliptic webs

If W is nonelliptic and if $W^\prime $ is parallel equivalent to W, then $W^\prime $ is nonelliptic. Denote the set of nonelliptic webs by $\mathscr {W}_{\mathfrak {S}}$ , and the set of parallel equivalence classes of nonelliptic webs by $[\mathscr {W}_{\mathfrak {S}}]$ . The empty web $W = \varnothing $ represents a class with one element in $[\mathscr {W}_{\mathfrak {S}}]$ .

3.1. Ideal polygons

For a nonnegative integer $k \geqslant 0$ , an ideal k-polygon $\mathfrak {D}_k$ is the surface $\mathfrak {D}_0 - P$ obtained by removing k punctures $P \subseteq \partial \mathfrak {D}_0$ from the boundary of the closed disk $\mathfrak {D}_0$ .

Observe that, when $k> 0$ , the boundary $\partial \mathfrak {D}_k$ of the ideal polygon consists of k ideal arcs.

3.2. Local webs

Recall the notion of a curve (Definition 2.1).

For some examples of local webs, see Figure 5. There, $k=4$ .

3.3. External faces

Definition 3.2. A face D of a local web W in an ideal polygon $\mathfrak {D}_k$ ( $k \geqslant 0$ ) is a contractible component of the complement $W^c \subseteq \mathfrak {D}_k$ of W that is puncture-free, meaning that D does not limit to any punctures $p \in P$ . A n-face $D_n$ is a face with n sides. Here, a maximal segment $\alpha \subseteq (\partial \mathfrak {D}_k) \cap D_n$ of the boundary $\partial \mathfrak {D}_k$ contained in the face $D_n$ is counted as a side, called a boundary side. An external face $D^{\mathrm {ext}}$ (resp. internal face $D^{\mathrm {int}}$ ) of the local web W is a face having at least one (resp. no) boundary side.

In contrast to internal faces, external faces can have an odd number of sides. An alternative name for an external two-face $D^{\mathrm {ext}}_2$ , three-face $D^{\mathrm {ext}}_3$ , four-face $D^{\mathrm {ext}}_4$ with one boundary side and five-face $D^{\mathrm {ext}}_5$ with one boundary side is a cap-, fork-, H- and half-hexagon-face, respectively; see Figure 6. Also, as for global webs (see Definition 2.4), an alternative name for an internal zero-face $D^{\mathrm {int}}_0$ , two-face $D^{\mathrm {int}}_2$ , four-face $D^{\mathrm {int}}_4$ and six-face $D^{\mathrm {int}}_2$ is a disk-, bigon-, square- and hexagon-face.

Figure 6 Cap-, fork-, H-, external four-, and half-hexagon-face.

For example, the connected local web in Figure 5a has one fork-face, two H-faces, one half-hexagon-face, one external six-face, one bigon-face, one square-face and one internal eight-face. Also, the disconnected local web in Figure 5b has one cap-face, one fork-face, one H-face, one half-hexagon-face, one external six-face, two disk-faces, one bigon-face and one square-face.

3.4. Combinatorial identity

Proposition 3.3. (compare [Reference KuperbergKup96, §6.1])

Let W be a connected local web in the closed disk $\mathfrak {D}_0$ with nonempty boundary $\partial W \neq \varnothing $ . Then,

Proof. Since W is connected, its complement $W^c \subseteq \mathfrak {D}_0$ contains at most one annulus, which faces the boundary $\partial \mathfrak {D}_0$ . Such an annulus does not exist, since $\partial W \neq \varnothing $ . Thus, every component D of $W^c$ is contractible, and of course puncture-free, so D is a face.

It follows that the closed disk $\mathfrak {D}_0$ can be tiled by the dual graph of W. More precisely, the vertices of the dual graph are the faces of W, and the complement of the dual graph consists of triangles. In Figure 7, we demonstrate this tiling procedure for the local web W that we saw in Figure 5a above (after forgetting the punctures).

Figure 7 Tiling the closed disk with the dual graph of a local web.

This triangular tiling gives rise to a flat Riemannian metric with conical singularities and piecewise-geodesic boundary on the closed disk $\mathfrak {D}_0$ , by requiring that each triangle is Euclidean equilateral. Apply the Gauss–Bonnet theorem to this singular flat surface.

3.5. Nonellipticity

Definition 3.4. As for global webs, a local web W in an ideal polygon $\mathfrak {D}_k$ is nonelliptic if W has no disk-, bigon-, or square-faces. Otherwise, W is called elliptic; see Figure 8.

Figure 8 Nonelliptic local webs in the closed disk.

Lemma 3.6 plus a small argument allows us to relax the hypotheses of Lemma 3.5 as follows.

3.6. Essential and rungless local webs

Note that essential local webs cannot have any cap- or fork-faces but can have H-faces. Later, we will need the operation of adding or removing an H-face, depicted in Figure 11.

Figure 11 Adding or removing an H-face.

Figure 12 More essential webs.

3.7. Ladder-webs in ideal biangles

Another name for an ideal two-polygon $\mathfrak {D}_2$ is an ideal biangle or just biangle, denoted by $\mathfrak {B}$ . The boundary $\partial \mathfrak {B}$ consists of two ideal arcs $E^\prime $ and $E^{\prime \prime }$ , called the boundary edges of the biangle. We want to characterize essential local webs W in the biangle $\mathfrak {B}$ ; compare (1) in Remark 3.10.

In the current section, components $\gamma _i$ of a multicurve $\Gamma $ will always be embedded, but different components might intersect. This will not be the case later on in §8.

A pair of arcs $\gamma _1$ and $\gamma _2$ each intersecting both boundary edges in $\mathfrak {B}$ are oppositely oriented if $\gamma _1$ and $\gamma _2$ go into (resp. out of) and out of (resp. into) $E^\prime $ , respectively. Similarly, the arcs $\gamma _1$ and $\gamma _2$ are same-oriented if $\gamma _1$ and $\gamma _2$ both go into (resp. out of) $E^\prime $ , respectively.

Definition 3.12. A symmetric strand-set pair $S = (S^\prime , S^{\prime \prime })$ for the biangle $\mathfrak {B}$ is a pair of finite collections $S = (S^\prime , S^{\prime \prime }) = ( \{s^\prime \}, \{s^{\prime \prime }\})$ of disjoint oriented strands located on the boundary $\partial \mathfrak {B} = E^\prime \cup E^{\prime \prime }$ such that the strands $s^\prime $ (resp. $s^{\prime \prime }$ ) lie on the boundary edge $E^\prime $ (resp. $E^{\prime \prime }$ ) and such that the number of in-strands (resp. out-strands) on $E^\prime $ is equal to the number of out-strands (resp. in-strands) on $E^{\prime \prime }$ ; see the leftmost picture in Figure 13.

Figure 13 Construction of a ladder-web.

Given a symmetric strand-set pair $S = (S^\prime , S^{\prime \prime })$ , in the following definition we associate to S a multicurve in the biangle $\mathfrak {B}$ , denoted $\left< W(S) \right>$ .

Observe, in the local picture $\left< W(S) \right>$ , that $\gamma _1$ and $\gamma _2$ intersect if and only if (1) they are oppositely oriented, and (2) they intersect exactly once. We denote by $\mathscr {P}(S) \subseteq \mathfrak {B}$ the set of intersection points p of pairs of oppositely oriented arcs in the local picture $\left< W(S) \right>$ .

Finally, we say how to associate a local web $W(S)$ in $\mathfrak {B}$ to a symmetric pair $S=(S^\prime , S^{\prime \prime })$ .

Definition 3.14. The ladder-web $W(S)$ in the biangle $\mathfrak {B}$ obtained from a symmetric strand-set pair $S= (S^\prime , S^{\prime \prime })$ is the unique (up to ambient isotopy of $\mathfrak {B}$ ) local web obtained by resolving each intersection point $p \in \mathscr {P}(S)$ into two vertices connected by a horizontal edge, relative to the biangle, called a rung; see Figures 13 and 14.

Figure 14 Replacing a local crossing with an H (also called a rung).

The following statement is implicit in [Reference KuperbergKup96, Lemma 6.7] and also appears in [Reference Frohman and SikoraFS22, §8].

Proof. For the first statement, the nonellipticity of $W(S)$ follows because two oppositely oriented curves in the local picture $\left< W(S) \right>$ do not cross more than once (if there were a square-face, a pair of curves would cross twice), and the tautness of $W(S)$ is immediate.

Conversely, let W be an essential local web in $\mathfrak {B}$ . The collection of ends of W located on the boundary edges $E^\prime \cup E^{\prime \prime }$ determines a strand-set pair $S= (S^\prime , S^{\prime \prime })$ . We show that S is symmetric and $W = W(S)$ . In particular, S is uniquely determined.

If W has a trivalent vertex, let $\overline {W}$ denote the induced local web in the closed disk $\mathfrak {D}_0$ underlying $\mathfrak {B}$ , obtained by filling in the two punctures of $\mathfrak {B}$ . Applying Proposition 3.7 to $\overline {W}$ guarantees that $\overline {W}$ (possibly minus some arc components) has at least three fork- and/or H-faces. At most two of these faces can straddle the two punctures of $\mathfrak {B}$ , so we gather W has one fork- or H-face $D^{\mathrm {ext}}$ lying on $E^\prime $ or $E^{\prime \prime }$ . Since W is taut, $D^{\mathrm {ext}}$ is an H-face.

We can then remove this H-face from $\mathfrak {B}$ (recall Figure 11), obtaining a local web $W_1$ that is essential and has strictly fewer trivalent vertices than W. Repeating this process, we obtain a sequence of essential local webs $W=W_0, W_1, \dots , W_n$ such that $W_n$ has no trivalent vertices and is obtained from W by removing finitely many H-faces. By nonellipticity, $W_n$ consists of a collection of arcs $\gamma _i^{(n)}$ (as opposed to loops), and since $W_n$ is taut, each arc $\gamma _i^{(n)}$ connects to both boundary edges $E^\prime $ and $E^{\prime \prime }$ of the biangle  $\mathfrak {B}$ .

Replacing the removed H-faces with local crossings (Figure 14), we obtain a multicurve $\Gamma $ in $\mathfrak {B}$ consisting of arcs $\gamma _i^{(0)}$ , each intersecting both edges $E^\prime $ and $E^{\prime \prime }$ , such that only oppositely oriented arcs $\gamma _i^{(0)}$ intersect; see Figure 15. In particular, the pair $(S^\prime , S^{\prime \prime })$ is symmetric.

Figure 15 Essential local web $W_{\mathfrak {B}}$ in the biangle, and its corresponding local picture $ \langle W_{\mathfrak {B}}\rangle $ .

We claim $\Gamma $ is the local picture $\left< W(S) \right>$ . Since only oppositely oriented arcs intersect, $\Gamma $ is order preserving (Definition 3.13). It remains to show $\Gamma $ is minimally intersecting, namely that no arcs intersect more than once. Suppose they did. Then, because only oppositely oriented arcs intersect, there would be an embedded bigon B in the complement $\Gamma ^c \subseteq \mathfrak {B}$ ; see the right side of Figure 16. Such an embedded bigon B corresponds in the local web W to a square-face, violating the nonellipticity of W. We gather $\Gamma = \left< W(S) \right>$ , as claimed.

Figure 16 Prohibited ladder-webs and local pictures.

By definition of the multicurve $\Gamma $ and the local web $W(S)$ , it follows that $W = W(S)$ .

For technical reasons, in §8 we will need the following concept.

3.8. Honeycomb-webs in ideal triangles

Another name for an ideal three-polygon $\mathfrak {D}_3$ is an ideal triangle $\mathfrak {T}$ . We want to characterize rungless essential local webs W in triangles $\mathfrak {T}$ .

For example, in Figure 17 we show the five-out-honeycomb-web $H_5^{\mathrm {out}}$ .

Figure 17 Honeycomb-web.

The following statement is implicit in [Reference KuperbergKup96, Lemma 6.8] and also appears in [Reference Frohman and SikoraFS22, §9].

Proposition 3.18. A honeycomb-web $H_n$ in the triangle $\mathfrak {T}$ is rungless and essential. Conversely, given a connected rungless essential local web W in $\mathfrak {T}$ having at least one trivalent vertex, there exists a unique honeycomb-web $H_n = H_n^{\mathrm {out}}$ or $=H_n^{\mathrm {in}}$ such that $W = H_n$ . Consequently, a (possibly disconnected) rungless essential local web W in $\mathfrak {T}$ consists of a unique (possibly empty) honeycomb $H_n$ together with a collection of disjoint oriented arcs located on the corners of $\mathfrak {T}$ ; see the left-hand side of Figure 18.

Figure 18 Rungless essential local web $W_{\mathfrak {T}}$ in the triangle and its corresponding local picture $\left < W_{\mathfrak {T}} \right>$ in the holed triangle.

Proof. The first statement is immediate.

Step 1. Let W be as in the second statement. Just like the proof of Proposition 3.15, applying Proposition 3.7 to the induced web $\overline {W}$ in the underlying closed disk $\mathfrak {D}_0$ guarantees that $\overline {W}$ has at least three fork- and/or H-faces, at most three of which can straddle the three punctures of $\mathfrak {T}$ . Since W is taut and rungless, W has no fork- or H-faces. Thus, $\overline {W}$ has exactly three fork- and/or H-faces, each of which straddles a puncture. Since these three faces are the only ones with a positive contribution in the formula of Proposition 3.3, they must be fork-faces. Moreover, since the total contribution of these three fork-faces is $2 \pi $ , every other face has exactly zero contribution. We gather that each interior face of W is a hexagon-face and each external face of W is a half-hexagon-face.

Step 2. To prove that W is a honeycomb-web $H_n$ , we argue by induction on n, showing that the triangle $\mathfrak {T}$ can be tiled by W face-by-face, starting from a corner of $\mathfrak {T}$ .

(2.a) Assume inductively that some number of half-hexagon-faces have been laid down as part of the bottom layer of faces sitting on the bottom edge E, illustrated in Figure 19.

Figure 19 Laying down a honeycomb: 1 of 2.

The strand labeled s either: (1) ends on the right edge $E^\prime $ of the triangle $\mathfrak {T}$ , thereby creating a fork straddling the rightmost puncture and completing the bottom layer of faces; (2) ends at a vertex disjoint from the vertices previously laid, hence the strand s is part of the boundary of the next half-hexagon-face; (3) ends at one of the vertices previously laid.

If (1), we continue to the next step of the induction, which deals with laying down the middle layers. If (2), we repeat the current step. Lastly, we argue (3) cannot occur. Indeed, suppose it did. The strand s is part of the boundary of the next half-hexagon-face $D_5^{\mathrm {ext}}$ . But, as can be seen from the figure, the external face $D_5^{\mathrm {ext}}$ has $\geqslant $ six sides, which is a contradiction.

(2.b) Assume inductively that the bottom layer and some number of middle layers have been laid down and moreover that some number of faces have been laid down as part of the current layer, illustrated in Figure 20. Consider the next face D shown in the figure.

Figure 20 Laying down a honeycomb: 2 of 2.

The face D is either external or internal. If it is internal, then D is a hexagon-face. In this case, the strands s and $s^\prime $ end at the fifth and sixth vertices of the hexagon-face, and we repeat the current step. Otherwise, D is external, so it is a half-hexagon-face, $D = D_{5}^{\mathrm {ext}}$ . However, we see from the figure that in this case $D_5^{\mathrm {ext}}$ has $\geqslant $ six sides, which is a contradiction.

To finish the induction, we repeat this step until the strand $s^\prime $ does not exist, in which case the strand s is part of a nonexternal side of a half-hexagon-face lying on the boundary edge $E^\prime $ .

Step 3. The last statement of the proposition follows since each honeycomb-web $H_n$ attaches to all three boundary edges of the triangle $\mathfrak {T}$ .

Later, in order to assign coordinates to webs, we will need to consider rungless essential local webs $W_{\mathfrak {T}}$ in a triangle $\mathfrak {T}$ up to a certain equivalence relation. Say that a local parallel-move applied to $W_{\mathfrak {T}}$ is a move swapping two arcs on the same corner of $\mathfrak {T}$ ; see Figure 21.

Figure 21 Local parallel-move.

For technical reasons, in §8 we will need the following concept.

4.1. Generic isotopies

Whenever there is an ideal triangulation $\lambda $ present, we always assume that ‘web’ means ‘generic web’. However, we distinguish between isotopies and generic isotopies.

4.2. Minimal position

Recall the notion of two parallel equivalent webs; see Definition 2.3.

Definition 4.2. Given a web W on the surface $\mathfrak {S}$ and given an edge E of the ideal triangulation $\lambda $ , the local geometric intersection number of the web W with the edge E is

$$ \begin{align*} I(W, E) = \min_{W^\prime}(\iota(W^\prime, E)) \quad \in \mathbb{Z}_{\geqslant 0} \quad\quad \left( W^\prime \text{ is parallel equivalent to } W \right), \end{align*} $$

where $\iota (W^\prime , E)$ is the number of intersection points of $W^\prime $ with E.

The web W is in minimal position with respect to the ideal triangulation $\lambda $ if

$$ \begin{align*} \iota(W, E) = I(W, E) \quad \in \mathbb{Z}_{\geqslant 0} \quad\quad \left( \text{for all edges } E \text{ of } \lambda \right). \end{align*} $$

(If this is the case, W minimizes the intersection number $\iota (W, \lambda )$ with the ideal triangulation $\lambda $ .)

Let $W^\prime $ be a web, let $\mathfrak {T}$ be a triangle in the ideal triangulation $\lambda $ , and let $W^\prime _{\mathfrak {T}} = W^\prime \cap \mathfrak {T}$ be the restriction of $W^\prime $ to $\mathfrak {T}$ . Suppose that the local web $W^\prime _{\mathfrak {T}}$ is not taut; see Definition 3.8. Then there is an edge E of $\lambda $ and a compact arc $\alpha $ ending on E such that $\iota (W^\prime , E)> \iota (W^\prime , \alpha )$ ; see Figure 22. We can then isotope the part of $W^\prime $ that is inside the bigon B, which is bounded by $\alpha $ and the segment $\overline {E}$ of E delimited by $\partial \alpha $ , into the adjacent triangle, resulting in a new web W. This is called a tightening-move. Similarly, if the restriction $W^\prime _{\mathfrak {T}}$ has an H-face, then we may apply an H-move to push the H into the adjacent triangle; see again Figure 22.

Figure 22 Tightening- and H-moves.

Note that tightening- and H-moves can be achieved with an isotopy of the web but not a generic isotopy. Also, by definition, in order to apply an H-move, we assume that the shaded region shown at the bottom of Figure 22 is empty, namely it does not intersect the web.

We borrow the following result from [Reference Frohman and SikoraFS22, §6] (and give essentially the same proof in the arXiv version [Reference Douglas and SunDS20a] of this article).

4.3. Split ideal triangulations

A split ideal triangulation $\widehat {\lambda }$ with respect to the ideal triangulation $\lambda $ is a collection of bi-infinite arcs obtained by doubling every edge E of $\lambda $ . In other words, we fatten each edge E into a biangle $\mathfrak {B}$ ; see Figure 23.

Figure 23 Split ideal triangulation.

The notions of generic web and generic isotopy for webs with respect to the split ideal triangulation $\widehat {\lambda }$ are the same as those for webs with respect to the ideal triangulation $\lambda $ . We always assume that webs are generic with respect to $\widehat {\lambda }$ .

To avoid cumbersome notation, we identify the triangles $\mathfrak {T}$ of the ideal triangulation $\lambda $ to the triangles $\mathfrak {T}$ of the split ideal triangulation $\widehat {\lambda }$ .

4.4. Good position

Definition 4.5. For a fixed split ideal triangulation $\widehat {\lambda }$ , a web W on $\mathfrak {S}$ is in good position with respect to $\widehat {\lambda }$ if the restriction $W_{\mathfrak {B}} = W \cap \mathfrak {B}$ (resp. $W_{\mathfrak {T}} = W \cap {\mathfrak {T}}$ ) of W to each biangle $\mathfrak {B}$ (resp. triangle $\mathfrak {T}$ ) of $\widehat {\lambda }$ is an essential (resp. rungless essential) local web; see Figure 24.

Figure 24 (Part of) a web in good position.

Note that for a web W in good position, each restriction $W_{\mathfrak {B}}$ to a biangle $\mathfrak {B}$ of $\widehat {\lambda }$ is a ladder-web; see Definition 3.14, Proposition 3.15 and Figures 13 and 15. Also, each restriction $W_{\mathfrak {T}}$ to a triangle $\mathfrak {T}$ of $\widehat {\lambda }$ is a (possibly empty) honeycomb-web $H_n$ together with a collection of disjoint oriented corner arcs; see Definition 3.17, Proposition 3.18 and Figures 17 and 18.

If W is a web in good position, then a modified H-move carries an H-face in a biangle $\mathfrak {B}$ to an H-face in an adjacent biangle $\mathfrak {B}^\prime $ , thereby replacing W with a new web $W^\prime $ ; see Figure 25. If, in addition, W is nonelliptic, then $W^\prime $ is also in good position. The nonelliptic condition for W is required to ensure that the new local web restriction $W^\prime _{\mathfrak {B}^\prime }$ is nonelliptic.

Figure 25 Modified H-move.

Once more, the following result is implicit in [Reference KuperbergKup96, Lemma 6.5 and the proof of Theorem 6.2, pp. 139-140] (in the setting of an ideal k-polygon $\mathfrak {D}_k$ ) and also appears in [Reference Frohman and SikoraFS22, §10].

Proof. We will keep track of isotopies by moving the split triangulation $\widehat {\lambda }$ instead of webs.

By Proposition 4.3, we can replace $W^\prime $ with a nonelliptic web W that is isotopic to $W^\prime $ and that is in minimal position with respect to the ideal triangulation $\lambda $ . We proceed to construct the split ideal triangulation $\widehat {\lambda }$ .

Let us begin by splitting each edge E of $\lambda $ into two edges $E^\prime $ and $E^{\prime \prime }$ that are very close to E. These split edges form a preliminary split ideal triangulation $\widehat {\lambda }$ , whose triangles (resp. biangles) are denoted by $\widehat {\mathfrak {T}}$ (resp. $\mathfrak {B}_E$ ); see the left-hand side of Figure 26.

Figure 26 Enlarging a biangle.

By definition of minimal position, the restriction $W_{\mathfrak {T}}$ of W to a triangle $\mathfrak {T}$ of the ideal triangulation $\lambda $ is taut. Since, in addition, W is nonelliptic, we have that $W_{\mathfrak {T}}$ is essential. If the preliminary split ideal triangulation $\widehat {\lambda }$ is sufficiently close to $\lambda $ , then the restriction $W_{\widehat {\mathfrak {T}}}$ of W to the triangle $\widehat {\mathfrak {T}} \subseteq \mathfrak {T}$ associated to $\mathfrak {T}$ is also an essential local web. If all of the local webs $W_{\widehat {\mathfrak {T}}}$ are rungless, then W is in good position with respect to $\widehat {\lambda }$ .

Otherwise, assume $W_{\widehat {\mathfrak {T}}}$ has an H-face on an edge of $\widehat {\lambda }$ , say the edge $E^\prime $ . Then by isotopy we can enlarge the biangle $\mathfrak {B}_E$ until it just envelops this H-face. In other words, we can isotope the edge $E^\prime $ so that it cuts out this H-face from the triangle $\widehat {\mathfrak {T}}$ ; see Figure 26. The result of this step is a new split ideal triangulation $\widehat {\lambda }$ , retaining the property that the local web restrictions $W_{\widehat {\mathfrak {T}}}$ are essential. Repeating this process until all of the local webs $W_{\widehat {\mathfrak {T}}}$ are rungless, we obtain the desired split ideal triangulation $\widehat {\lambda }$ . Notice it might be the case that there is more than one biangle into which an H-face can be moved; see again Figure 26.

For the second statement of the proposition, note that if a nonelliptic web W is in good position with respect to $\widehat {\lambda }$ , then W is minimal with respect to the ideal triangulation $\lambda $ (which, for the sake of argument, we can take to be contained in $\widehat {\lambda }$ , that is $\lambda \subseteq \widehat {\lambda }$ ). (Indeed, this follows by the proof of the first part of Proposition 4.3, provided in [Reference Frohman and SikoraFS22, Reference Douglas and SunDS20a], and uses the fact that adding a ladder web $W_{\mathfrak {B}}$ to a rungless essential web $W_{\mathfrak {T}}$ preserves the tautness property.) Similarly, $W^\prime $ is in minimal position. Thus, applying the second part of Proposition 4.3, we gather that W can be taken to $W^\prime $ by a finite sequence of H-moves, global parallel-moves and generic isotopies. The result follows by the definition of good position and modified H-moves.

5.1. Dotted ideal triangulations

Consider a surface $\widehat {\mathfrak {S}} = \mathfrak {S}$ or $= \mathfrak {T}$ equipped with an ideal triangulation $\lambda $ , where, in this subsection, $\lambda = \mathfrak {T}$ when $\widehat {\mathfrak {S}} = \mathfrak {T}$ . The associated dotted ideal triangulation is the pair consisting of $\lambda $ together with $N^\prime =N$ or $=7$ distinct dots attached to the one- and two-cells of $\lambda $ , where there are two edge-dots attached to each one-cell and there is one triangle-dot attached to each two-cell; see Figure 27. Given a triangle $\mathfrak {T}$ of $\lambda $ and an edge E of $\mathfrak {T}$ , it makes sense to talk about the left-edge-dot and right-edge-dot as viewed from $\mathfrak {T}$ ; see Figure 27b. Choosing an ordering for the $N^\prime $ dots lying on the dotted ideal triangulation $\lambda $ defines a one-to-one correspondence between functions $\{ \text {dots} \} \to \mathbb {Z}$ and elements of $\mathbb {Z}^{N^\prime }$ . We always assume that such an ordering has been chosen.

Figure 27 Dotted ideal triangulations.

5.2. Local coordinate functions

Consider a dotted ideal triangle $\mathfrak {T}$ ; see Figure 27b. Recall (Definition 3.19) that $\mathscr {W}_{\mathfrak {T}}$ denotes the collection of rungless essential local webs $W_{\mathfrak {T}}$ in $\mathfrak {T}$ and that $[\mathscr {W}_{\mathfrak {T}}]$ denotes the set of corner-ambiguity classes $[ W_{\mathfrak {T}} ]$ of local webs $W_{\mathfrak {T}}$ in $\mathscr {W}_{\mathfrak {T}}$ .

Definition 5.1. An integer local coordinate function or just local coordinate function,

$$ \begin{align*} \Phi_{\mathfrak{T}} : \mathscr{W}_{\mathfrak{T}} \longrightarrow \mathbb{Z}^7 \end{align*} $$

is a function assigning to each local web $W_{\mathfrak {T}}$ in $\mathscr {W}_{\mathfrak {T}}$ one integer coordinate per dot lying on the dotted triangle $\mathfrak {T}$ , satisfying the following properties:

1. 1. if a local web $W_{\mathfrak {T}}$ in $\mathscr {W}_{\mathfrak {T}}$ can be written $W_{\mathfrak {T}} = W^\prime _{\mathfrak {T}} \sqcup W^{\prime \prime }_{\mathfrak {T}}$ as the disjoint union of two local webs, each in $\mathscr {W}_{\mathfrak {T}}$ , then

   $$ \begin{align*} \Phi_{\mathfrak{T}}(W_{\mathfrak{T}}) = \Phi_{\mathfrak{T}}(W^\prime_{\mathfrak{T}}) + \Phi_{\mathfrak{T}}(W^{\prime\prime}_{\mathfrak{T}}) \quad \in \mathbb{Z}^7; \end{align*} $$

2. 2. for an edge E of $\mathfrak {T}$ , the ordered pair of coordinates $(a^L_E, a^R_E)$ of the function $\Phi _{\mathfrak {T}}$ assigned to the left- and right-edge-dots lying on E, respectively, depends only on the pair $(n^{\text {in}}_E, n^{\text {out}}_E)$ of numbers of in- and out-strands of the local web $ W_{\mathfrak {T}}$ on the edge E; moreover, different pairs $(n_E^{\mathrm {in}}, n_E^{\mathrm {out}})$ yield different pairs of coordinates $(a^L_E, a^R_E)$ ;

3. 3. there are two symmetries; the first is that $\Phi _{\mathfrak {T}}$ respects the rotational symmetry of the triangle (see Remark 5.2 below for a more precise statement), and the second is that if the numbers $n^{\text {in}}_E$ and $n^{\text {out}}_E$ of in- and out-strands on an edge E are exchanged, then the coordinates $a^L_E$ and $a^R_E$ are exchanged as well;

4. 4. observe, by property (1), the function $\Phi _{\mathfrak {T}}(W_{\mathfrak {T}}) = \Phi _{\mathfrak {T}}(W^\prime _{\mathfrak {T}})$ agrees on local webs $W_{\mathfrak {T}}$ and $W^\prime _{\mathfrak {T}}$ in $\mathscr {W}_{\mathfrak {T}}$ representing the same corner-ambiguity class $[W_{\mathfrak {T}}] = [W^\prime _{\mathfrak {T}}]$ in $[ \mathscr {W}_{\mathfrak {T}} ]$ (because $W_{\mathfrak {T}}$ and $W^\prime _{\mathfrak {T}}$ differ only by permutations of oriented corner arcs), thus inducing

   $$ \begin{align*} \Phi_{\mathfrak{T}} : [\mathscr{W}_{\mathfrak{T}}] \longrightarrow \mathbb{Z}^7, \end{align*} $$
   also called $\Phi _{\mathfrak {T}}$ ; we require that this induced function $\Phi _{\mathfrak {T}}$ is an injection.

The coordinates assigned by $\Phi _{\mathfrak {T}}$ to edge-dots (resp. triangle-dots) are called edge-coordinates (resp. triangle-coordinates).

Figure 28 Property (1): $a_i = a_i^\prime + a_i^{\prime \prime }$ .

Figure 29 Properties (2) and (3).

We illustrate properties (1), (2), (3) in Figures 28 and 29.

5.3. Local coordinates from Fock–Goncharov theory

We define an explicit Fock–Goncharov local coordinate function $\Phi _{\mathfrak {T}}^{\mathrm {FG}}: \mathscr {W}_{\mathfrak {T}} \to \mathbb {Z}_{\geqslant 0}^7$ valued in nonnegative integers.

By property (1) in Definition 5.1, it suffices to define $\Phi _{\mathfrak {T}}^{\mathrm {FG}}$ on connected local webs in $\mathscr {W}_{\mathfrak {T}}$ . By Proposition 3.18, these come in one of exactly eight types $H_n^{\text {out}}$ , $H_n^{\text {in}}$ , $R_1$ , $L_1$ , $R_2$ , $L_2$ , $R_3$ , $L_3$ illustrated in Figure 30. In the figure, note that in the two top left triangles we have, for visibility, drawn the local pictures $\left < H_n^{\text {out}} \right>$ and $\left < H_n^{\text {in}} \right>$ as a shorthand for the actual n-out-honeycomb-web $H_n^{\text {out}}$ and n-in-honeycomb-web $H_n^{\text {in}}$ , respectively; see Definition 3.20. It is immediate that $\Phi _{\mathfrak {T}}^{\mathrm {FG}}$ satisfies property (3) and the first part of (2). The second part of (2) follows by the invertibility of the matrix $\left ( \begin {smallmatrix} 2 & 1 \\ 1 & 2 \end {smallmatrix} \right )$ . We will check property (4) in §6.

Figure 30 Fock–Goncharov local coordinate function $\Phi _{\mathfrak {T}}^{\mathrm {FG}}$ .

5.4. Global coordinates from local coordinate functions

Assume that, for an abstract dotted triangle $\mathfrak {T}$ , we have chosen an arbitrary local coordinate function $\Phi _{\mathfrak {T}}: \mathscr {W}_{\mathfrak {T}} \to \mathbb {Z}^7$ . We show that this induces a global coordinate function $\Phi _{\lambda } : [\mathscr {W}_{\mathfrak {S}}] \to \mathbb {Z}^N$ that is well adapted to the choice of $\Phi _{\mathfrak {T}}$ . The argument uses only properties (1), the first part of (2) and (3) of $\Phi _{\mathfrak {T}}$ .

As a guiding example of the construction to come, reference Figure 33, which uses the Fock–Goncharov local coordinate function $\Phi _{\mathfrak {T}}^{\mathrm {FG}}$ . This is an example on the once punctured torus $\mathfrak {S}$ . Note that the web W in the example has one hexagon-face. All of the other components of $W^c$ are not contractible. So W is nonelliptic.

Step 1. Consider the split ideal triangulation $\widehat {\lambda }$ (§4.3). We put dots on each triangle $\mathfrak {T}$ of $\widehat {\lambda }$ . The chosen local coordinate function $\Phi _{\mathfrak {T}}$ can be associated to each of these dotted triangles $\mathfrak {T}$ ; see the left-hand side of Figure 31.

Figure 31 Local coordinates $\Phi _{\mathfrak {T}}$ attached to the triangles $\mathfrak {T}$ of $\widehat {\lambda }$ (left), and the corresponding global coordinates $\Phi _{\lambda }$ attached to $\lambda $ (right).

Step 2. Fix a nonelliptic web W on $\mathfrak {S}$ that is in good position (Definition 4.5) with respect to the split ideal triangulation $\widehat {\lambda }$ . We assign to W one integer coordinate per dot lying on the dotted ideal triangulation $\lambda $ , namely an element $\Phi _{\lambda }(W)$ in $\mathbb {Z}^N$ .

By good position, the local web restriction $W_{\mathfrak {T}} = W \cap \mathfrak {T}$ is in $\mathscr {W}_{\mathfrak {T}}$ for each triangle $\mathfrak {T}$ of $\widehat {\lambda }$ . So, we may evaluate the local coordinate function $\Phi _{\mathfrak {T}}$ on $W_{\mathfrak {T}}$ , obtaining coordinates for each of the seven dots lying on the dotted triangle $\mathfrak {T}$ of $\widehat {\lambda }$ . For instance, in this way we assign coordinates to all of the dots shown on the left-hand side of Figure 31 above. We claim that these coordinates glue together along each biangle $\mathfrak {B}$ of $\widehat {\lambda }$ in such a way that we obtain one coordinate per dot lying on the dotted ideal triangulation $\lambda $ ; see Figure 31.

Indeed, suppose $\mathfrak {B}$ is a biangle between two triangles $\mathfrak {T}^\prime $ and $\mathfrak {T}^{\prime \prime }$ of $\widehat {\lambda }$ . Let $E^\prime $ and $E^{\prime \prime }$ be the corresponding boundary edges of $\mathfrak {B}$ , and let $a^{ L}_{E^\prime }$ and $a^{ R}_{E^\prime }$ (resp. $a^{ L}_{E^{\prime \prime }}$ and $a^{ R}_{E^{\prime \prime }}$ ) be the coordinates assigned by $\Phi _{\mathfrak {T}^\prime }$ (resp. $\Phi _{\mathfrak {T}^{\prime \prime }}$ ) to the left- and right-edge-dots, respectively, lying on $E^\prime $ (resp. $E^{\prime \prime }$ ) as viewed from $\mathfrak {T}^\prime $ (resp. $\mathfrak {T}^{\prime \prime }$ ). Also, denote by $n^{\text {in}}_{E^\prime }$ and $n^{\text {out}}_{E^\prime }$ (resp. $n^{\text {in}}_{E^{\prime \prime }}$ and $n^{\text {out}}_{E^{\prime \prime }}$ ) the numbers of in- and out-strands of the local web restriction $W_{\mathfrak {T}^\prime }$ (resp. $W_{\mathfrak {T}^{\prime \prime }}$ ) lying on the edge $E^\prime $ (resp. $E^{\prime \prime }$ ); see Figure 32.

Figure 32 Local coordinates attached to a biangle: $a^L_{E^\prime } = a^R_{E^{\prime \prime }}$ and $a^R_{E^\prime } = a^L_{E^{\prime \prime }}$ .

Figure 33 Tropical Fock–Goncharov $\mathcal {A}$ -coordinates for a nonelliptic web.

Since, by good position, the restriction $W_{\mathfrak {B}} = W \cap \mathfrak {B}$ is a ladder-web, we have $n^{\text {out}}_{E^\prime } = n^{\text {in}}_{E^{\prime \prime }}$ and $ n^{\text {in}}_{E^{\prime }} = n^{\text {out}}_{E^{\prime \prime }}$ . It follows immediately from properties (3) and the first part of (2) that the coordinates across from each other agree $a^L_{E^\prime } = a^R_{E^{\prime \prime }}$ and $a^R_{E^\prime } = a^L_{E^{\prime \prime }}$ . So, we may glue together the two pairs of coordinates into two coordinates lying on the edge E of $\lambda $ , as desired.

Step 3. For a general nonelliptic web $W^\prime $ on $\mathfrak {S}$ , by the first part of Proposition 4.7 there exists a nonelliptic web W that is parallel equivalent to $W^\prime $ and that is in good position with respect to the split ideal triangulation $\widehat {\lambda }$ . Define $\Phi _{\lambda }(W^\prime ) = \Phi _{\lambda }(W)$ in $\mathbb {Z}^N$ .

To show $\Phi _{\lambda }(W^\prime )$ is well defined, suppose $W_2$ is another web as W. By the second part of Proposition 4.7, the nonelliptic webs W and $W_2$ are related by a sequence of modified H-moves and global parallel-moves. The effect of either of these moves on a web in good position is to swap, possibly many, parallel oppositely oriented corner arcs in the triangles $\mathfrak {T}$ of $\widehat {\lambda }$ ; recall Figures 25 and 3 above, respectively. By property (1) of $\Phi _{\mathfrak {T}}$ , we have $\Phi _{\lambda }(W) = \Phi _{\lambda }(W_2)$ .

From this point on, our approach diverges from that in [Reference Frohman and SikoraFS22]. In particular, our coordinates are different from theirs.

Definition 5.4. The Fock–Goncharov global coordinate function

$$ \begin{align*} \Phi_{\lambda}^{\mathrm{FG}} : [\mathscr{W}_{\mathfrak{S}}] \longrightarrow {\mathbb{Z}}_{\geqslant 0}^N \end{align*} $$

is the well defined global coordinate function on $[\mathscr {W}_{\mathfrak {S}}]$ , valued in nonnegative integers, induced by the Fock–Goncharov local coordinate function $\Phi _{\mathfrak {T}}^{\mathrm {FG}}$ . In §7-8, we prove:

6.1. Integer cones

We define notions of independence for cones.

Note:

• • strongly independent over $\Omega \Longrightarrow $ weakly independent over $\Omega $ ;

• • strongly independent over $\mathbb {Z}_{\geqslant 0}$ $\Longleftrightarrow $ weakly independent over $\mathbb {Z}$ $\Longleftrightarrow $ linearly independent over $\mathbb {Q}$ (the usual definition from linear algebra).

The following technical fact is immediate from the definitions.

6.2. Local Knutson–Tao cone

Let $\mathfrak {T}$ be a dotted ideal triangle (§5.1); recall Figure 27b above. In this section, we are going to order the dots on $\mathfrak {T}$ so that if the dots are labeled as in the left-hand side of Figure 34, then a point $c \in \mathbb {Z}^7$ will be written

(*) $$ \begin{align} c = \left(a_{11}, a_{12}, a_{21}, a_{22}, a_{31}, a_{32}, a\right) \quad \in \mathbb{Z}^7. \end{align} $$

Figure 34 Rhombus numbers.

Let $\mathbb {Z}/3 \subseteq \mathbb {Q}$ denote the set of integer thirds within the rational numbers, namely $\mathbb {Z}/3$ is the image of the map $\mathbb {Z} \to \mathbb {Q}$ sending $n \mapsto n/3$ . Note that $\mathbb {Z} \subseteq \mathbb {Z}/3$ .

To each point $c \in \mathbb {Z}^7$ , as in Equation (*), associate a nine-tuple of rhombus numbers

$$ \begin{align*} r(c) = \left( r_{11}, r_{12}, r_{13}, r_{21}, r_{22}, r_{23}, r_{31}, r_{32}, r_{33}\right) \quad \in \left(\mathbb{Z}/3\right)^9 \end{align*} $$

by the linear equations (see Figure 34 above)

$$ \begin{align*} r_{12} &= \left(a + a_{32} - a_{11} - a_{31}\right)/3, \quad\quad r_{11} = \left(a_{22} + a_{31} - a - 0\right)/3, \end{align*} $$

$$ \begin{align*} r_{13} &= \left(a_{21} + a - a_{12} - a_{22}\right)/3; \qquad\qquad\qquad\qquad\qquad\qquad\quad \ \end{align*} $$

$$ \begin{align*} r_{22} &= \left(a + a_{12} - a_{21} - a_{11}\right)/3, \quad\quad r_{21} = \left(a_{32} + a_{11} - a-0\right)/3, \end{align*} $$

$$ \begin{align*} r_{23} &= \left(a_{31} + a - a_{22} -a_{32}\right)/3; \qquad\qquad\qquad\qquad\qquad\qquad\quad \ \end{align*} $$

$$ \begin{align*} r_{32} &= \left(a+a_{22} -a_{31} -a_{21}\right)/3, \quad\quad r_{31} = \left(a_{12} + a_{21} - a -0\right)/3, \end{align*} $$

$$ \begin{align*} r_{33} &= \left(a_{11}+a-a_{32} -a_{12}\right)/3. \qquad\qquad\qquad\qquad\qquad\qquad\quad \ \end{align*} $$

Definition 6.4. The local Knutson–Tao positive cone, or just local Knutson–Tao cone or local cone, $\mathscr {C}^+_{\mathfrak {T}}$ associated to the dotted ideal triangle $\mathfrak {T}$ is defined by

$$ \begin{align*} \mathscr{C}^+_{\mathfrak{T}} = \left\{ c \in \mathbb{Z}^7; \quad r(c) = \left(r_{11}, r_{12}, r_{13}, r_{21}, r_{22}, r_{23}, r_{31}, r_{32}, r_{33}\right) \quad\in \mathbb{Z}_{\geqslant 0}^9 \subseteq \left(\mathbb{Z}/3\right)^9 \right\}. \end{align*} $$

By linearity, this indeed defines a cone contained in $\mathbb {Z}^7$ . We will prove below in this subsection that $\mathscr {C}^+_{\mathfrak {T}} \subseteq \mathbb {Z}_{\geqslant 0}^7$ is, in fact, a positive cone.

To see that $\mathscr {C}^+_{\mathfrak {T}}$ is nontrivial, one checks that the image $\Phi _{\mathfrak {T}}^{\mathrm {FG}}(\mathscr {W}_{\mathfrak {T}})$ of the Fock-Goncharov local coordinate function $\Phi _{\mathfrak {T}}^{\mathrm {FG}} : \mathscr {W}_{\mathfrak {T}} \to \mathbb {Z}_{\geqslant 0}^7$ (§5.3) lies in the local cone $\Phi _{\mathfrak {T}}^{\mathrm {FG}}(\mathscr {W}_{\mathfrak {T}}) \subseteq \mathscr {C}^+_{\mathfrak {T}}$ . By property (1) in Definition 5.1, it suffices to check this on the connected local webs in $\mathscr {W}_{\mathfrak {T}}$ ; recall Figure 30 above. Specifically, using the convention in Equation (*), we have

$$ \begin{align*} c(R_1) &= \Phi_{\mathfrak{T}}^{\mathrm{FG}}(R_1) = \left(0, 0, 1, 2, 2, 1, 1\right), \quad\quad\quad\quad\quad c(L_1) = \Phi_{\mathfrak{T}}^{\mathrm{FG}}(L_1) = \left(0, 0, 2, 1, 1, 2, 2\right),\\c(R_2) &= \Phi_{\mathfrak{T}}^{\mathrm{FG}}(R_2) = \left(2, 1, 0, 0, 1, 2, 1\right), \quad\quad\quad\quad\quad c(L_2) = \Phi_{\mathfrak{T}}^{\mathrm{FG}}(L_2) =\left(1, 2, 0, 0, 2, 1, 2\right),\\c(R_3) &= \Phi_{\mathfrak{T}}^{\mathrm{FG}}(R_3) =\left(1, 2, 2, 1, 0, 0, 1\right), \quad\quad\quad\quad\quad c(L_3) = \Phi_{\mathfrak{T}}^{\mathrm{FG}}(L_3) =\left(2, 1, 1, 2, 0, 0, 2\right),\\c(H_n^{\mathrm{in}}) &= \Phi_{\mathfrak{T}}^{\mathrm{FG}}(H_n^{\mathrm{in}}) =\left(2n, n, 2n, n, 2n, n, 3n\right), \hphantom{\quad\quad\quad\quad\quad c(L_1) = \Phi_{\mathfrak{T}}^{\mathrm{FG}}(L_1) = \left(0, 0, 2, 1, 1, 2, 2\right),}\\c(H_n^{\mathrm{out}}) &= \Phi_{\mathfrak{T}}^{\mathrm{FG}}(H_n^{\mathrm{out}}) =\left(n, 2n, n, 2n, n, 2n, 3n\right). \hphantom{\quad\quad\quad\quad\quad c(L_1) = \Phi_{\mathfrak{T}}^{\mathrm{FG}}(L_1) = \left(0, 0, 2, 1, 1, 2, 2\right),}\end{align*} $$

The associated nine-tuples of rhombus numbers are

$$ \begin{align*} r(c(R_1)) &= \left(1, 0, 0, 0, 0, 0, 0, 0, 0\right), \quad\quad r(c(L_1)) = \left(0, 1, 1, 0, 0, 0, 0, 0, 0\right),\\r(c(R_2)) &= \left(0, 0, 0, 1, 0, 0, 0, 0, 0\right), \quad\quad r(c(L_2)) = \left(0, 0, 0, 0, 1, 1, 0, 0, 0\right),\\r(c(R_3)) &= \left(0, 0, 0, 0, 0, 0, 1, 0, 0\right), \quad\quad r(c(L_3)) = \left(0, 0, 0, 0, 0, 0, 0, 1, 1\right), \\r(c(H_n^{\mathrm{in}})) &= \left(0, 0, n, 0, 0, n, 0, 0, n\right), \hphantom{\quad\quad r(c(L_1)) = \left(0, 1, 1, 0, 0, 0, 0, 0, 0\right),}\\r(c(H_n^{\mathrm{out}})) &= \left(0, n, 0, 0, n, 0, 0, n, 0\right). \hphantom{\quad\quad r(c(L_1)) = \left(0, 1, 1, 0, 0, 0, 0, 0, 0\right),} \end{align*} $$

By rank considerations, the eight cone points $c(R_1)$ , $c(L_1)$ , $c(R_2)$ , $c(L_2)$ , $c(R_3)$ , $c(L_3)$ , $c(H_n^{\mathrm {in}})$ , $c(H_n^{\mathrm {out}})$ have a linear dependence relation over $\mathbb {Z}$ . For instance (see Figure 35),

$$ \begin{align*} c(H_n^{\mathrm{out}}) + c(H_n^{\mathrm{in}}) = n \left( c(L_1) + c(L_2) + c(L_3) \right) \quad \in \mathscr{C}^+_{\mathfrak{T}}. \end{align*} $$

Figure 35 Linear dependence relation over $\mathbb {Z}$ .

Nevertheless, we can say the following:

Proposition 6.6. The collection of eight cone points

$$ \begin{align*} c(R_1), c(L_1), c(R_2), c(L_2), c(R_3), c(L_3), c(H_1^{\mathrm{in}}), c(H_1^{\mathrm{out}}) \quad \in \Phi_{\mathfrak{T}}^{\mathrm{FG}}(\mathscr{W}_{\mathfrak{T}}) \subseteq \mathscr{C}^+_{\mathfrak{T}} \end{align*} $$

forms a weak basis of the Knutson–Tao local cone $\mathscr {C}^+_{\mathfrak {T}}$ .

Among these eight cone points, the seven points

$$ \begin{align*} c(R_1), c(L_1), c(R_2), c(L_2), c(R_3), c(L_3), c(H_1^{\mathrm{in}}) \end{align*} $$

are strongly independent over $\mathbb {Z}_{\geqslant 0}$ , and the seven points

$$ \begin{align*} c(R_1), c(L_1), c(R_2), c(L_2), c(R_3), c(L_3), c(H_1^{\mathrm{out}}) \end{align*} $$

are strongly independent over $\mathbb {Z}_{\geqslant 0}$ . Moreover, each cone point c in $\mathscr {C}^+_{\mathfrak {T}}$ can be uniquely expressed in exactly one of the following three forms:

$$ \begin{align*} c &= n_1 c(R_1) + n_2 c(L_1) + \cdots + n_6 c(L_3), \\ c &= n_1 c(R_1) + n_2 c(L_1) + \cdots + n_6 c(L_3) + n c(H_1^{\mathrm{in}}), \\ c &= n_1 c(R_1) + n_2 c(L_1) + \cdots + n_6 c(L_3) + n c(H_1^{\mathrm{out}}), \quad\quad\left(n_i \in \mathbb{Z}_{\geqslant 0}, \quad n \in \mathbb{Z}_{>0}\right). \end{align*} $$

Because the spanning set $c(R_1), c(L_1), c(R_2), c(L_2), c(R_3), c(L_3), c(H_1^{\mathrm {in}}), c(H_1^{\mathrm {out}})$ consists of positive points, we immediately obtain:

Proof. Assume $\Phi _{\mathfrak {T}}^{\mathrm {FG}}(W_{\mathfrak {T}}) = \Phi _{\mathfrak {T}}^{\mathrm {FG}}(W^\prime _{\mathfrak {T}}) \in \mathscr {C}^+_{\mathfrak {T}}$ . This cone point falls into one of the three families in Proposition 6.6. For the sake of argument, suppose

$$ \begin{align*} \Phi_{\mathfrak{T}}^{\mathrm{FG}}(W_{\mathfrak{T}}) = \Phi_{\mathfrak{T}}^{\mathrm{FG}}(W^\prime_{\mathfrak{T}}) = n_1 c(R_1) + n_2 c(L_1) + \cdots + n_6 c(L_3) + n c(H_1^{\mathrm{in}}) \quad\left(n_i \in {\mathbb{Z}}_{\geqslant 0}, \quad n \in {\mathbb{Z}}_{>0}\right). \end{align*} $$

Note that $n c(H_1^{\mathrm {in}}) = c(H_n^{\mathrm {in}})$ in $\mathscr {C}^+_{\mathfrak {T}}$ ; see Figure 30. By the uniqueness property in Proposition 6.6 together with property (1) in Definition 5.1, we gather that $W_{\mathfrak {T}}$ and $W^\prime _{\mathfrak {T}}$ have $1 + \sum _{i=1}^6 n_i$ connected components, one of which is a n-in-honeycomb $H_n^{\mathrm {in}}$ , and $n_1$ (resp. $n_2$ , $n_3$ , $n_4$ , $n_5$ , $n_6$ ) of which are corner arcs $R_1$ (resp. $L_1$ , $R_2$ , $L_2$ , $R_3$ , $L_3$ ). The only ambiguity is how these corner arcs are permuted on their respective corners, that is $[W_{\mathfrak {T}}] = [W^\prime _{\mathfrak {T}}]$ in $[\mathscr {W}_{\mathfrak {T}}]$ .

Proof of Proposition 6.6

Define two subsets $\overline {(\mathscr {C}^+_{\mathfrak {T}})^{\mathrm {in}}}$ and $(\mathscr {C}^+_{\mathfrak {T}})^{\mathrm {out}}$ of $\mathscr {C}^+_{\mathfrak {T}}$ by

(#) $$ \begin{align} \overline{(\mathscr{C}^+_{\mathfrak{T}})^{\mathrm{in}}} &= \mathrm{Span}_{\mathbb{Z}_{\geqslant 0}}\left(c(R_1), c(L_1), c(R_2), c(L_2), c(R_3), c(L_3)\right) + \mathbb{Z}_{\geqslant 0} \cdot c(H_1^{\mathrm{in}}), \end{align} $$

(##) $$ \begin{align} (\mathscr{C}^+_{\mathfrak{T}})^{\mathrm{out}} &= \mathrm{Span}_{\mathbb{Z}_{\geqslant 0}}\left(c(R_1), c(L_1), c(R_2), c(L_2), c(R_3), c(L_3)\right) + {\mathbb{Z}}_{> 0} \cdot c(H_1^{\mathrm{out}}). \end{align} $$

(Here, $A+B = \{ a+b; \quad a \in A \text { and } b \in B \}$ .) Put

$$ \begin{align*} c_1 &= c(R_1), & c_2 &= c(L_1), & c_3 &= c(R_2), & c_4 &= c(L_2), \ \ \end{align*} $$

$$ \begin{align*} c_5 &= c(R_3), & c_6 &= c(L_3), & c_7 &= c(H_1^{\mathrm{in}}), & c_8 &= c(H_1^{\mathrm{out}}). \end{align*} $$

By Lemma 6.3, with $\mathscr {C} = \mathscr {C}^+_{\mathfrak {T}}$ , in order to prove Proposition 6.6 it suffices to establish:

Claim 6.9. There exists

1. 1. a cone $\mathscr {C}^\prime \subseteq \mathbb {Z}^7$ ;

2. 2. a collection of cone points $c^{\prime }_1, \dots , c^{\prime }_8$ in $\mathscr {C}^\prime $ ;

3. 3. a partition $\mathscr {C}^\prime = \overline {(\mathscr {C}^\prime )^{>0}} \sqcup (\mathscr {C}^\prime )^{< 0}$ ;

4. 4. a $\mathbb {Z}_{\geqslant 0}$ -linear bijection $\psi \colon \mathscr {C}^\prime \to \mathscr {C}^+_{\mathfrak {T}}$ ;

5. 5. an extension $\widetilde {\psi }$ of $\psi $ to a $\mathbb {Q}$ -linear isomorphism $\widetilde {\psi } \colon \mathbb {Q}^7 \to \mathbb {Q}^7$ ;

such that

1. 1. we have $\psi (c^{\prime }_i) = c_i$ ;

2. 2. we have $\psi (\overline {(\mathscr {C}^\prime )^{>0}}) = \overline {(\mathscr {C}^+_{\mathfrak {T}})^{\mathrm {in}}}$ and $\psi ((\mathscr {C}^\prime )^{<0}) = (\mathscr {C}^+_{\mathfrak {T}})^{\mathrm {out}}$ ;

3. 3. the eight cone points $c^{\prime }_1, \dots , c^{\prime }_6, c^{\prime }_7, c^{\prime }_8$ form a weak basis of the cone $\mathscr {C}^\prime $ ;

4. 4. the seven cone points $c^{\prime }_1, \dots , c^{\prime }_6, c^{\prime }_7$ are strongly independent over $\mathbb {Z}_{\geqslant 0}$ ;

5. 5. the seven cone points $c^{\prime }_1, \dots , c^{\prime }_6, c^{\prime }_8$ are strongly independent over $\mathbb {Z}_{\geqslant 0}$ .

We prove the claim. Define $\mathscr {C}^\prime \subseteq \mathbb {Z}_{\geqslant 0}^6 \times \mathbb {Z} \subseteq \mathbb {Z}^7$ by

(**) $$ \begin{align} \mathscr{C}^\prime = \left\{ \left(r_{11}, r_{12}, r_{21}, r_{22}, r_{31}, r_{32}, x\right) \in \mathbb{Z}_{\geqslant 0}^6 \times \mathbb{Z}; \quad -x \leqslant \mathrm{min}\left(r_{12}, r_{22}, r_{32}\right) \right\}. \end{align} $$

It follows from the definition that $\mathscr {C}^\prime $ is a cone. Put

$$ \begin{align*} c^{\prime}_1 &= \left(1, 0, 0, 0, 0, 0, 0\right), & c^{\prime}_2 &= \left(0, 1, 0, 0, 0, 0, 0\right), \\ c^{\prime}_3 &= \left(0, 0, 1, 0, 0, 0, 0\right), & c^{\prime}_4 &= \left(0, 0, 0, 1, 0, 0, 0\right), \\ c^{\prime}_5 &= \left(0, 0, 0, 0, 1, 0, 0\right), & c^{\prime}_6 &= \left(0, 0, 0, 0, 0, 1, 0\right), \\ c^{\prime}_7 &= \left(0, 0, 0, 0, 0, 0, 1\right), & c^{\prime}_8 &= \left(0,1,0,1,0,1,-1\right). \end{align*} $$

One checks that $c^{\prime }_1, \dots , c^{\prime }_8$ are in $\mathscr {C}^\prime $ . Define

$$ \begin{align*} \overline{(\mathscr{C}^\prime)^{>0}} = \mathscr{C}^\prime \cap \left( \mathbb{Z}_{\geqslant 0}^6 \times \mathbb{Z}_{\geqslant 0}\right), \quad\quad (\mathscr{C}^\prime)^{<0} = \mathscr{C}^\prime \cap \left( \mathbb{Z}_{\geqslant 0}^6 \times \mathbb{Z}_{<0}\right). \end{align*} $$

Then $\mathscr {C}^\prime = \overline {(\mathscr {C}^\prime )^{>0}} \sqcup (\mathscr {C}^\prime )^{<0}$ is a partition.

First, we show $c^{\prime }_1, \dots , c^{\prime }_6, c^{\prime }_7, c^{\prime }_8$ spans $\mathscr {C}^\prime $ . We see that

(†) $$ \begin{align} \overline{(\mathscr{C}^\prime)^{>0}} = \mathrm{Span}_{\mathbb{Z}_{\geqslant 0}}\left(c^{\prime}_1, \dots, c^{\prime}_6\right) + \mathbb{Z}_{\geqslant 0} \cdot c^{\prime}_7 \quad\left(= \mathbb{Z}_{\geqslant 0}^6 \times \mathbb{Z}_{\geqslant 0} \right). \end{align} $$

If $c' \in (\mathscr {C}^\prime )^{<0}$ , then its last coordinate is $x \leqslant -1$ . Since $-x> 0$ and $-x \leqslant \mathrm {min}(r_{12}, r_{22}, r_{32})$ ,

$$ \begin{align*} c' = \left(r_{11}, -x+r^{\prime}_{12}, r_{21}, -x+r^{\prime}_{22}, r_{31}, -x+r^{\prime}_{32}, -x \cdot -1\right) \end{align*} $$

for some $r_{11}, r^{\prime }_{12}, r_{21}, r^{\prime }_{22}, r_{31}, r^{\prime }_{32} \in \mathbb {Z}_{\geqslant 0}$ and $-x \in \mathbb {Z}_{>0}$ . That is,

$$ \begin{align*} c' = r_{11} c^{\prime}_1 + r^{\prime}_{12} c^{\prime}_2 + r_{21} c^{\prime}_3 + r^{\prime}_{22} c^{\prime}_4 + r_{31} c^{\prime}_5 + r^{\prime}_{32} c^{\prime}_6 + \left(-x\right) c^{\prime}_8 \quad \in \mathrm{Span}_{\mathbb{Z}_{\geqslant 0}}\left(c^{\prime}_1, \dots, c^{\prime}_6\right) + \mathbb{Z}_{>0} \cdot c^{\prime}_8. \end{align*} $$

Thus,

(††) $$ \begin{align} (\mathscr{C}^\prime)^{<0} = \mathrm{Span}_{\mathbb{Z}_{\geqslant 0}}\left(c^{\prime}_1, \dots, c^{\prime}_6\right) + \mathbb{Z}_{>0} \cdot c^{\prime}_8, \end{align} $$

where the $\supseteq $ containment follows since $\mathrm {Span}_{\mathbb {Z}_{\geqslant 0}} (c^{\prime }_1, \dots , c^{\prime }_6 ) + \mathbb {Z}_{>0} \cdot c^{\prime }_8 \subseteq \mathbb {Z}_{\geqslant 0}^6 \times \mathbb {Z}_{<0}$ .

Next, we show $c^{\prime }_1, \dots , c^{\prime }_6, c^{\prime }_7, c^{\prime }_8$ are weakly independent over $\mathbb {Z}_{\geqslant 0}$ . Indeed, if $n_1 c^{\prime }_1 + \cdots + n_7 c^{\prime }_7 + n_8 c^{\prime }_8 = 0$ , then $n_1 = n_3 = n_5 = 0$ and $n_2 + n_8, n_4 + n_8, n_6 + n_8, n_7 - n_8 = 0$ . Since all $n_i \in \mathbb {Z}_{\geqslant 0}$ , it follows that $n_2 = n_4 = n_6 = n_8 = 0$ , and so $n_7 = n_8 = 0$ , as desired.

We gather that $c^{\prime }_1, \dots , c^{\prime }_6, c^{\prime }_7, c^{\prime }_8$ form a weak basis of $\mathscr {C}^\prime $ .

Next, we show $c^{\prime }_1, \dots , c^{\prime }_6, c^{\prime }_7$ are strongly independent over $\mathbb {Z}_{\geqslant 0}$ . This is equivalent to being linearly independent over $\mathbb {Q}$ , which follows from the definitions. Similarly, it follows from the definitions that $c^{\prime }_1, \dots , c^{\prime }_6, c^{\prime }_8$ are strongly independent over $\mathbb {Z}_{\geqslant 0}$ .

We now define a $\mathbb {Z}_{\geqslant 0}$ -linear bijection $\varphi \colon \mathscr {C}^+_{\mathfrak {T}} \to \mathscr {C}^\prime $ . Its inverse will be the desired $\mathbb {Z}_{\geqslant 0}$ -linear bijection $\psi = \varphi ^{-1} \colon \mathscr {C}^\prime \to \mathscr {C}^+_{\mathfrak {T}}$ . Let c be a cone point in $\mathscr {C}^+_{\mathfrak {T}}$ , written as in Equation (*). Put

$$ \begin{align*} x &= \left(a_{11} - a_{12} + a_{21} - a_{22} + a_{31} - a_{32}\right)/3 \end{align*} $$

$$ \begin{align*} &= r_{13} - r_{12} = r_{23} - r_{22} = r_{33} - r_{32}\qquad \end{align*} $$

$$ \begin{align*} &\geqslant -r_{12} \text{ and} -r_{22} \text{ and} -r_{32},\qquad\qquad\quad \end{align*} $$

where the rhombus numbers $r_{ij}$ are in $\mathbb {Z}_{\geqslant 0}$ since $c \in \mathscr {C}^+_{\mathfrak {T}}$ ; see Figure 36 (we think of x as the tropical Fock–Goncharov $\mathcal {X}$ -coordinate for the triangle). Thus,

$$ \begin{align*} x \geqslant \mathrm{max}(-r_{12}, -r_{22}, -r_{32}) = - \mathrm{min}(r_{12}, r_{22}, r_{32}). \end{align*} $$

Therefore, recalling $\mathscr {C}^\prime \subseteq \mathbb {Z}_{\geqslant 0}^6 \times \mathbb {Z}$ (Equation (**)), we may define the function $\varphi \colon \mathscr {C}^+_{\mathfrak {T}} \to \mathscr {C}^\prime $ by

$$ \begin{align*} \varphi(c) = (r_{11}, r_{12}, r_{21}, r_{22}, r_{31}, r_{32}, x). \end{align*} $$

Figure 36 Four ways to view the tropical Fock–Goncharov $\mathcal {X}$ -coordinate.

It follows from the definition that $\varphi \colon \mathscr {C}^+_{\mathfrak {T}} \to \mathscr {C}^\prime $ is $\mathbb {Z}_{\geqslant 0}$ -linear. One checks that $\varphi (c_i) = c^{\prime }_i$ . Since the $c^{\prime }_i$ span $\mathscr {C}^\prime $ , we have $\varphi $ is surjective. In particular, by Equations (#),(##),(†),(††),

$$ \begin{align*} \varphi\left(\overline{(\mathscr{C}^+_{\mathfrak{T}})^{\mathrm{in}}}\right) = \overline{(\mathscr{C}^\prime)^{>0}} \quad \text{and} \quad \varphi\left((\mathscr{C}^+_{\mathfrak{T}})^{\mathrm{out}}\right) = (\mathscr{C}^\prime)^{<0}. \end{align*} $$

The formula for $\varphi $ extends to define a $\mathbb {Q}$ -linear isomorphism $\widetilde {\varphi } \colon \mathbb {Q}^7 \to \mathbb {Q}^7$ , and its inverse is the desired $\mathbb {Q}$ -linear isomorphism $\widetilde {\psi } = (\widetilde {\varphi })^{-1} \colon \mathbb {Q}^7 \to \mathbb {Q}^7$ . Indeed, the bijectivity of $\widetilde {\varphi }$ follows by computing the values on the standard column basis of $\mathbb {Q}^7$ , giving the invertible matrix

$$ \begin{align*} \widetilde{\varphi}(\vec{e}_1, \vec{e}_2, \vec{e}_3, \vec{e}_4, \vec{e}_5, \vec{e}_6, \vec{e}_7) = \frac{1}{3} \left( \begin{smallmatrix} 0&0&0&1&1&0&-1 \\ -1&0&0&0&-1&1&1 \\ 1&0&0&0&0&1&-1 \\ -1&1&-1&0&0&0&1 \\ 0&1&1&0&0&0&-1 \\ 0&0&-1&1&-1&0&1 \\ 1&-1&1&-1&1&-1&0 \\ \end{smallmatrix} \right). \end{align*} $$

So $\widetilde {\psi } = (\widetilde {\varphi })^{-1}$ is defined. Since $\widetilde {\varphi }$ is an injection so is its restriction $\varphi \colon \mathscr {C}^+_{\mathfrak {T}} \to \mathscr {C}^\prime $ . Also, since, as we argued above, $\varphi $ is a surjection, we gather $\varphi $ is a bijection. Thus, $\psi = \varphi ^{-1} : \mathscr {C}^\prime \to \mathscr {C}^+_{\mathfrak {T}}$ is defined. This completes the proof of the claim, thereby establishing the proposition.

6.3. Global Knutson–Tao cone

Given the dotted ideal triangulation $\lambda $ on the surface $\mathfrak {S}$ , an element c of $\mathbb {Z}^N$ corresponds to a function $\{\text {dots on } \lambda \} \to \mathbb {Z}$ ; see §5.1. If $\mathfrak {T}$ is a dotted triangle of $\lambda $ , then an element c of $\mathbb {Z}^N$ induces a function $\{ \text {dots on } \mathfrak {T} \} \to \mathbb {Z}$ , which likewise corresponds to an element $c_{\mathfrak {T}}$ of $\mathbb {Z}^7$ .

Definition 6.10. The global Knutson–Tao positive cone, or just Knutson–Tao cone or global cone, $\mathscr {C}^+_{\lambda } \subseteq \mathbb {Z}_{\geqslant 0}^N$ is defined by

$$ \begin{align*} \mathscr{C}^+_{\lambda} = \left\{ c \in \mathbb{Z}^N; \quad c_{\mathfrak{T}} \text{ is in } \mathscr{C}^+_{\mathfrak{T}} \text{ for all triangles } \mathfrak{T} \text{ of } \lambda \right\}. \end{align*} $$

It follows from Corollary 6.7 that $\mathscr {C}^+_{\lambda } \subseteq \mathbb {Z}_{\geqslant 0}^N$ is indeed a positive cone.

In §5, we defined the global coordinate function $\Phi _{\lambda }^{\mathrm {FG}} : [\mathscr {W}_{\mathfrak {S}}] \to \mathbb {Z}_{\geqslant 0}^N$ ; see Definition 5.4. Since the image $\Phi _{\mathfrak {T}}^{\mathrm {FG}}([\mathscr {W}_{\mathfrak {T}}]) \subseteq \mathscr {C}^+_{\mathfrak {T}}$ (which is, in fact, an equality by Corollary 6.7), it follows by the construction of $\Phi _{\lambda }^{\mathrm {FG}}$ that the image $\Phi _{\lambda }^{\mathrm {FG}}([\mathscr {W}_{\mathfrak {S}}]) \subseteq \mathscr {C}^+_{\lambda }$ ; recall, for instance, Figure 33.

Proposition 6.11. Moreover, we have (see §7-8 for a proof)

$$ \begin{align*} \Phi_{\lambda}^{\mathrm{FG}}([\mathscr{W}_{\mathfrak{S}}]) = \mathscr{C}^+_{\lambda}. \end{align*} $$

7.1. Inverse mapping

Our strategy for proving Propositions 5.5 and 6.11 (equivalently, Theorem 7.1) is to construct an explicit inverse mapping

$$ \begin{align*} \Psi_{\lambda}^{\mathrm{FG}}: \mathscr{C}^+_{\lambda} \longrightarrow [\mathscr{W}_{\mathfrak{S}}], \end{align*} $$

namely a function that is both a left and a right inverse for the function $\Phi _{\lambda }^{\mathrm {FG}}$ . The definition of the mapping $\Psi _{\lambda }^{\mathrm {FG}}$ is relatively straightforward, and it will be automatic that it is an inverse for $\Phi _{\lambda }^{\mathrm {FG}}$ . The more challenging part will be to show that $\Psi _{\lambda }^{\mathrm {FG}}$ is well defined.

7.2. Inverse mapping: ladder gluing construction

Recall that for a triangle $\mathfrak {T}$ we denote by $\mathscr {W}_{\mathfrak {T}}$ the collection of rungless essential local webs $W_{\mathfrak {T}}$ in $\mathfrak {T}$ ; see Definition 3.19. We will once again make use of the split ideal triangulation $\widehat {\lambda }$ ; see §4.3.

For example, see the third row of Figure 37, an example on a once-punctured torus.

Figure 37 Ladder gluing construction (on the once-punctured torus). Shown are two different ways of assigning the local webs, differing by permutations of corner arcs. On the left, the result of the gluing is a nonelliptic web. On the right, the result is an elliptic web, which has to be resolved by removing a square before becoming a nonelliptic web. The two nonelliptic webs obtained in this way are equivalent.

To a compatible collection $\{ W_{\mathfrak {T}} \}_{\mathfrak {T} \in \widehat {\lambda } }$ of local webs, we will associate a global web W on $\mathfrak {S}$ that need not be nonelliptic and that is in good position with respect to $\widehat {\lambda }$ ; recall Definition 4.5. The global web W is well defined up to ambient isotopy of $\mathfrak {S}$ respecting $\widehat {\lambda }$ .

Construction of W. Consider a biangle $\mathfrak {B}$ sitting between two triangles $\mathfrak {T}^\prime $ and $\mathfrak {T}^{\prime \prime }$ . The local webs $W_{\mathfrak {T}^\prime }$ and $W_{\mathfrak {T}^{\prime \prime }}$ determine strand sets $S^\prime $ and $S^{\prime \prime }$ on the boundary edges $E^\prime $ and $E^{\prime \prime }$ , respectively. By the compatibility property, the strand-set pair $S = (S^\prime , S^{\prime \prime })$ is symmetric; see Definition 3.12. Let $W_{\mathfrak {B}} = W_{\mathfrak {B}}(S)$ be the induced ladder-web in $\mathfrak {B}$ ; see Definition 3.14.

Define W to be the global web obtained by gluing together the local webs $\{ W_{\mathfrak {T}} \}_{\mathfrak {T} \in \widehat {\lambda }}$ and $\{ W_{\mathfrak {B}} \}_{\mathfrak {B} \in \widehat {\lambda }}$ in the obvious way; see the fourth row and the left side of the fifth row of Figure 37.

The following statement is immediate.

If the global web W is obtained via the ladder gluing construction, then W could be (1) nonelliptic, for example, see the left side of the fifth row of Figure 37, or (2) elliptic, for example, see the fourth row of Figure 37.

7.3. Inverse mapping: resolving an elliptic web

Recall the notion of a local parallel-move; see Figure 21. Note that if $\{ W_{\mathfrak {T}}^\prime \}_{\mathfrak {T} \in \widehat {\lambda }}$ is a compatible collection of local webs and if $W_{\mathfrak {T}}$ is related to $W^\prime _{\mathfrak {T}}$ by a sequence of local parallel-moves, then $\{ W_{\mathfrak {T}}\}_{\mathfrak {T} \in \widehat {\lambda }}$ is also compatible.

Proof. Suppose that the global web $W^\prime $ obtained by applying the ladder gluing construction to the local webs $\{ W^\prime _{\mathfrak {T}} \}_{\mathfrak {T} \in \widehat {\lambda }}$ is elliptic.

Step 1. We show that the elliptic global web $W^\prime $ has no disk- or bigon-faces. If there were a disk- or bigon-face, then it could not lie completely in a triangle $\mathfrak {T}$ or biangle $\mathfrak {B}$ of $\widehat {\lambda }$ , for this would violate that the local web restriction $W^\prime _{\mathfrak {T}}$ or $W^\prime _{\mathfrak {B}}$ is essential (in particular, nonelliptic) by Lemma 7.5. Consequently, there is a cap- or fork-face lying in some $\mathfrak {T}$ or $\mathfrak {B}$ , contradicting that the local web restriction $W^\prime _{\mathfrak {T}}$ or $W^\prime _{\mathfrak {B}}$ is essential (in particular, taut).

Step 2. We consider the possible positions of square-faces relative to the split ideal triangulation $\widehat {\lambda }$ . We claim that a square-face can only appear as demonstrated at the top of Figure 38, namely having two H-faces in two (possibly identical) biangles $\mathfrak {B}$ and, in between, having opposite sides traveling parallel through the intermediate triangles $\mathfrak {T}$ and biangles $\mathfrak {B}$ . Indeed, otherwise there would be a square-, cap-, or fork-face, similar to Step 1.

Figure 38 Resolving a square-face.

Step 3. We remove a square-face. Since the square-faces are positioned in this way, given a fixed square-face there is a well defined state into which the square-face can be resolved, illustrated in Figure 38. The resulting global web $W_1$ is in good position with respect to $\widehat {\lambda }$ . Also, $W_1$ is less complex than $W^\prime $ , where the complexity of a global web in good position is measured by the total number of vertices lying in the union $\cup _{\mathfrak {B}} \mathfrak {B}$ of all of the biangles $\mathfrak {B}$ . Note that resolving a square-face decreases the complexity by four.

The effect of resolving a square-face is to perform, in each triangle $\mathfrak {T}$ , some number (possibly zero) of local parallel-moves, replacing the original local webs $\{ W^\prime _{\mathfrak {T}} \}_{\mathfrak {T} \in \widehat {\lambda }}$ with new local webs $\{ (W_1)_{\mathfrak {T}} \}_{\mathfrak {T} \in \widehat {\lambda }}$ such that $(W_1)_{\mathfrak {T}}$ is equivalent to $W^\prime _{\mathfrak {T}}$ up to corner-ambiguity; see Figure 38.

Step 4. By a complexity argument, we can repeat the previous step until we obtain a sequence $W^\prime = W_0, W_1, W_2, \dots , W_n = W$ of global webs in good position such that $\{ (W_{i+1})_{\mathfrak {T}} \}_{\mathfrak {T} \in \widehat {\lambda }}$ is related to $\{ (W_{i})_{\mathfrak {T}} \}_{\mathfrak {T} \in \widehat {\lambda }}$ by a sequence of local parallel-moves and such that W has no square-faces. By Lemma 7.5, W is recovered by applying the ladder gluing construction to $\{ W_{\mathfrak {T}} \}_{\mathfrak {T} \in \widehat {\lambda }}$ . By Step 1, W has no disk- or bigon-faces. Thus, W is nonelliptic.

We refer to the algorithm used in the proof of Lemma 7.6 as the square removing algorithm. For example, see the fourth row and the right side of the fifth row of Figure 37.

Note that the algorithm removes the square-faces at random, thus the local webs $\{ W_{\mathfrak {T}} \}_{\mathfrak {T} \in \widehat {\lambda }}$ satisfying the conclusion of Lemma 7.6 are not necessarily unique. For example, see Figure 39.

Figure 39 Elliptic web resulting from the ladder gluing construction (top) and two different applications of the square removing algorithm, yielding different, but parallel equivalent, nonelliptic webs (bottom).

7.4. Inverse mapping: definition

Let c be a point in the global cone $\mathscr {C}^+_{\lambda }$ ; see Definition 6.10. Our goal is to associate to c a parallel equivalence class $\Psi _{\lambda }^{\mathrm {FG}}(c) \in [\mathscr {W}_{\mathfrak {S}}]$ of global nonelliptic webs on $\mathfrak {S}$ . Equivalently, we want to associate to c a nonelliptic web $\widetilde {\Psi }_{\lambda }^{\mathrm {FG}}(c)$ on $\mathfrak {S}$ well defined up to parallel equivalence; see Definition 2.3. Recall that we identify the triangles $\mathfrak {T}$ of the ideal triangulation $\lambda $ with the triangles $\mathfrak {T}$ of the split ideal triangulation $\widehat {\lambda }$ .

Construction of $\widetilde {\Psi }_{\lambda }^{\mathrm {FG}}(c)$ . The global cone point c determines a local cone point $c_{\mathfrak {T}}$ in the local cone $\mathscr {C}^+_{\mathfrak {T}}$ for each triangle $\mathfrak {T}$ of $\lambda $ ; see just before Definition 6.10. By the triangle identifications between $\lambda $ and $\widehat {\lambda }$ , the local cone point $c_{\mathfrak {T}} \in \mathscr {C}^+_{\mathfrak {T}}$ is assigned to each triangle $\mathfrak {T}$ of $\widehat {\lambda }$ ; see the first and second rows of Figure 37.

Note, by construction, corresponding edge-coordinates located across a biangle $\mathfrak {B}$ take the same value. More precisely, if $\mathfrak {B}$ sits between two triangles $\mathfrak {T}^\prime $ and $\mathfrak {T}^{\prime \prime }$ and if the boundary edges of $\mathfrak {B}$ are $E^\prime $ and $E^{\prime \prime }$ , respectively, then the coordinate $a^L_{E^\prime }$ (resp. $a^R_{E^\prime }$ ) lying on the left-edge-dot (resp. right-edge-dot) as viewed from $\mathfrak {T}^\prime $ agrees with the coordinate $a^R_{E^{\prime \prime }}$ (resp. $a^L_{E^{\prime \prime }}$ ) lying on the right-edge-dot (resp. left-edge-dot) as viewed from $\mathfrak {T}^{\prime \prime }$ ; see Figure 37.

By Corollaries 6.7 and 6.8, for each local cone point $c_{\mathfrak {T}} \in \mathscr {C}^+_{\mathfrak {T}}$ assigned to a triangle $\mathfrak {T}$ of $\widehat {\lambda }$ , there exists a unique corner-ambiguity class $[W_{\mathfrak {T}}]$ of local webs $W_{\mathfrak {T}}$ in $\mathscr {W}_{\mathfrak {T}}$ such that $\Phi _{\mathfrak {T}}^{\mathrm {FG}}(W_{\mathfrak {T}}) = c_{\mathfrak {T}}$ for any representative $W_{\mathfrak {T}}$ of $[W_{\mathfrak {T}}]$ .

We now make a choice of such a representative $W_{\mathfrak {T}}$ for each $\mathfrak {T}$ . Two different choices $W_{\mathfrak {T}}$ and $W^\prime _{\mathfrak {T}}$ of local webs representing $[W_{\mathfrak {T}}] = [W^\prime _{\mathfrak {T}}]$ are, by definition, related by local parallel-moves; see the third row of Figure 37.

Since corresponding edge-coordinates across biangles agree, the collection $\{ W_{\mathfrak {T}} \}_{\mathfrak {T} \in \widehat {\lambda }}$ of local webs is compatible (Definition 7.3). This follows by Figure 30. (There is also a general argument, by properties (2) and (3) in Definition 5.1, which uses the fact that if $W_{\mathfrak {T}} \in \mathscr {W}_{\mathfrak {T}}$ , then the opposite web $W^{\mathrm {op}}_{\mathfrak {T}}$ obtained by reversing all of the orientations of $W_{\mathfrak {T}}$ is also in $\mathscr {W}_{\mathfrak {T}}$ ).

By Lemma 7.6, this choice of a compatible collection $\{ W_{\mathfrak {T}} \}_{\mathfrak {T} \in \widehat {\lambda }}$ of local webs can be made (in a nonunique way) such that the global web W on $\mathfrak {S}$ obtained by applying the ladder gluing construction to $\{ W_{\mathfrak {T}} \}_{\mathfrak {T} \in \widehat {\lambda }}$ is nonelliptic. Finally, we define $\widetilde {\Psi }_{\lambda }^{\mathrm {FG}}(c) = W$ . In order for the global web $\widetilde {\Psi }_{\lambda }^{\mathrm {FG}}(c)$ to be well defined up to parallel equivalence, we require:

Definition 7.8. The inverse mapping

$$ \begin{align*} \Psi_{\lambda}^{\mathrm{FG}} : \mathscr{C}^+_{\lambda} \longrightarrow [\mathscr{W}_{\mathfrak{S}}] \end{align*} $$

is defined by sending a cone point c in the global Knutson–Tao cone $\mathscr {C}^+_{\lambda }$ to the parallel equivalence class in $[\mathscr {W}_{\mathfrak {S}}]$ of the global nonelliptic web $\widetilde {\Psi }_{\lambda }^{\mathrm {FG}}(c)$ on $\mathfrak {S}$ .

In summary, we have reduced the proof of Theorem 7.1 to proving the main lemma.