Hourglass Plabic Graphs

• Hourglass plabic graphs are planar bicolored networks that extend Postnikov’s framework by incorporating symmetric reflection properties and edge multiplicities.
• They model both the symmetric locus of Gr₍≥0₎(n,2n) and U_q(sl_r)-web invariants through distinct combinatorial constructions including trip permutations and move calculus.
• Their study unifies diverse areas such as cluster algebras, tableau correspondences, and six-vertex models, offering rotation-invariant bases and explicit combinatorial criteria.

Searching arXiv for the cited papers and closely related work on hourglass plabic graphs. Hourglass plabic graphs are a family of planar bicolored networks that extend Postnikov’s plabic-graph technology in two distinct directions recorded in the literature. In one direction, they are symmetric plabic graphs embedded in a disk and invariant under reflection across a diameter, up to swapping black and white vertex colors; these graphs encode the symmetric locus of $\Gr_{\ge 0}(n,2n)$ and, with suitable conventions, the totally nonnegative Lagrangian Grassmannian (Karpman et al., 2015). In the other direction, they are bipartite plabic graphs with edge multiplicities drawn as “hourglasses,” designed to model $U_q(\mathfrak{sl}_r)$-web invariants, rotation-invariant web bases, and related combinatorics of tableaux, six-vertex configurations, and cluster structures (Gaetz et al., 2023, Gaetz et al., 2024). The two usages share the plabic-graph framework, trip combinatorics, and move calculus, but they serve different ambient problems.

1. Terminology and basic definitions

A plabic graph is a planar bicolored graph embedded in a disk with boundary vertices labeled clockwise. In Postnikov’s setting, all boundary vertices have degree one, edges incident to boundary vertices are called legs, and leaves adjacent to boundary vertices are lollipops. Reduced plabic graphs are those satisfying the standard trip conditions: no cycle-trips, no interior leaves, no trip traverses an edge twice unless it starts at a lollipop, and no pair of trips share two edges in the same order (Karpman et al., 2015).

The expression “hourglass plabic graph” is used in two main senses.

Usage	Defining feature	Ambient setting
Symmetric hourglass graph	Reflection across a diameter with color reversal	$\Gr_{\ge 0}(n,2n)$, $LG(n,2n)$
Multiplicity hourglass graph	Edge multiplicities drawn as parallel twisted strands	$U_q(\mathfrak{sl}_r)$-webs

In the symmetric usage, one works with $2n$ boundary vertices and a distinguished diameter $d$. Writing $i' = 2n+1-i$, a symmetric plabic graph is one whose underlying uncolored network is invariant under reflection across $d$, has no vertices on $d$, and whose reflected interior vertices have opposite colors (Karpman et al., 2015).

In the multiplicity usage, an hourglass graph consists of an embedded planar graph $U_q(\mathfrak{sl}_r)$0 together with a positive integer multiplicity $U_q(\mathfrak{sl}_r)$1 on each edge. An edge with $U_q(\mathfrak{sl}_r)$2 is drawn as $U_q(\mathfrak{sl}_r)$3 parallel twisted strands; when $U_q(\mathfrak{sl}_r)$4 it is a simple edge. An $U_q(\mathfrak{sl}_r)$5-hourglass plabic graph is a bipartite hourglass graph in a disk with all internal vertices of degree $U_q(\mathfrak{sl}_r)$6 counted with multiplicity and all boundary vertices of simple degree one (Gaetz et al., 2024). In the $U_q(\mathfrak{sl}_r)$7 formulation, each internal vertex has degree $U_q(\mathfrak{sl}_r)$8 counted with multiplicity and each edge has multiplicity in $U_q(\mathfrak{sl}_r)$9, so hourglass edges encode $\Gr_{\ge 0}(n,2n)$0 while simple edges encode $\Gr_{\ge 0}(n,2n)$1 or $\Gr_{\ge 0}(n,2n)$2 depending on boundary color (Enugandla et al., 9 Dec 2025).

A further broader usage appears in the study of triangular-grid billiards and plabic graphs of essential dimension $\Gr_{\ge 0}(n,2n)$3: the paper does not use the term “hourglass plabic graphs,” but notes that connected reduced plabic graphs of essential dimension $\Gr_{\ge 0}(n,2n)$4 align with hourglass-type graphs dual to alternating triangulations by the two orientations of triangles in the triangular grid (Defant et al., 2022).

2. Symmetric hourglass plabic graphs and the Grassmannian

For symmetric plabic graphs, the central combinatorial involution is

$\Gr_{\ge 0}(n,2n)$5

If $\Gr_{\ge 0}(n,2n)$6 is an almost perfect matching with boundary $\Gr_{\ge 0}(n,2n)$7, reflection gives a matching $\Gr_{\ge 0}(n,2n)$8 with boundary $\Gr_{\ge 0}(n,2n)$9. Under symmetric edge weights, mirrored edges receive equal weights, and the boundary measurements satisfy

$LG(n,2n)$0

This forces $LG(n,2n)$1, so symmetric plabic graphs of this type map naturally to $LG(n,2n)$2 (Karpman et al., 2015).

The boundary measurement map is defined on a reduced plabic graph $LG(n,2n)$3 with edge weights $LG(n,2n)$4 by

$LG(n,2n)$5

where the sum runs over almost perfect matchings $LG(n,2n)$6 with boundary $LG(n,2n)$7. It is surjective onto the corresponding totally nonnegative positroid cell, but not injective because of gauge transformations at interior vertices (Karpman et al., 2015).

The symmetric theory is organized by a characterization theorem. For a positroid cell $LG(n,2n)$8 with positroid $LG(n,2n)$9, bounded affine permutation $U_q(\mathfrak{sl}_r)$0, Grassmann necklace $U_q(\mathfrak{sl}_r)$1, and dual Grassmann necklace $U_q(\mathfrak{sl}_r)$2, the following are equivalent: $U_q(\mathfrak{sl}_r)$3 is representable by a symmetric plabic graph; $U_q(\mathfrak{sl}_r)$4 is closed under $U_q(\mathfrak{sl}_r)$5; the bounded affine permutation satisfies

$U_q(\mathfrak{sl}_r)$6

and the Grassmann and dual Grassmann necklaces satisfy

$U_q(\mathfrak{sl}_r)$7

These conditions give necessary and sufficient combinatorial criteria for symmetric realizability (Karpman et al., 2015).

Construction proceeds by symmetric bridge addition. Bridges appear either centrally at $U_q(\mathfrak{sl}_r)$8 or in mirrored pairs at $U_q(\mathfrak{sl}_r)$9 and $2n$0, with equal weights on the paired bridges. The resulting symmetric bridge graph has as many independent positive parameters as the number of steps in the symmetric construction, matching the dimension of the symmetric part of the positroid cell. Set-theoretically, that symmetric part is obtained by imposing the linear constraints $2n$1 (Karpman et al., 2015).

This symmetric locus has a geometric interpretation: with suitable conventions, the symmetric part of $2n$2 coincides with the totally nonnegative part of $2n$3, the moduli of maximal isotropic subspaces for a symplectic form. The note emphasizes that the linear equalities $2n$4 give the set-theoretic equations of the symmetric locus, while the detailed isotropic or Lagrangian Plücker relations are deferred to later work (Karpman et al., 2015).

3. Hourglass multiplicities, trips, and web invariants

In the representation-theoretic literature, hourglass plabic graphs refine ordinary plabic graphs by allowing multiplicity on edges and replacing single zig-zag strands by several trip systems. For $2n$5, the $2n$6-th trip permutation $2n$7 is defined by launching from a boundary vertex and following edges while taking the $2n$8-th leftmost turn at white vertices and the $2n$9-th rightmost turn at black vertices. The inverse relation

$d$0

is part of the basic formalism (Gaetz et al., 2024).

In the $d$1 framework, the trip data $d$2 is supplemented by a separation labeling $d$3 and a boundary word

$d$4

where barred letters are complements in $d$5. For fully reduced graphs, this boundary word is a balanced lattice word, and the separation labeling is a distinguished proper labeling among all proper labelings $d$6 assigning $d$7-subsets of $d$8 to edges so that labels around each internal vertex union to $d$9 (Enugandla et al., 9 Dec 2025).

The web evaluation is a weighted sum over proper edge labelings. In the $i' = 2n+1-i$0 clasped-web formulation,

$i' = 2n+1-i$1

More generally, an $i' = 2n+1-i$2-hourglass plabic graph with boundary vertices $i' = 2n+1-i$3 corresponds to a $i' = 2n+1-i$4-web $i' = 2n+1-i$5 whose evaluation lies in

$i' = 2n+1-i$6

and in Plücker degree two the relevant invariant space is

$i' = 2n+1-i$7

identified with the span of products $i' = 2n+1-i$8 with $i' = 2n+1-i$9 in the coordinate ring of $d$0 (Gaetz et al., 2024).

The multiplicity formalism is not merely decorative. Hourglass edges are intrinsic to $d$1-webs and, in higher-rank degree-two theory, to the uniform treatment of $d$2-web bases. This is why the later papers present hourglass plabic graphs as a new avatar of webs rather than as a minor variant of Postnikov’s networks (Gaetz et al., 2023, Gaetz et al., 2024).

4. Fully reducedness, local moves, and six-vertex models

The theory depends on a strengthened reducedness condition. In the degree-two arbitrary-r formulation, an hourglass plabic graph is fully reduced if it has no isolated components, no trip segment with a self-intersection, no pair of $d$3 segments with an oriented double crossing, and no pair of $d$4 and $d$5 segments with an oriented double crossing. Oriented double crossings between equal or adjacent $d$6 are “bad double crossings” (Gaetz et al., 2024). In the $d$7 formulation, a leafless hourglass plabic graph is fully reduced if no move-equivalent graph contains a forbidden $d$8-cycle containing an hourglass edge; for $d$9, this is equivalent to a monotonicity condition on $d$0 and its interactions with $d$1 (Gaetz et al., 2023).

The local move calculus comprises contraction moves and square moves throughout the theory. In the $d$2 setting there is also a benzene move, a local flip on a hexagonal face. Contraction and square moves preserve the associated invariant $d$3, whereas benzene moves do not (Enugandla et al., 9 Dec 2025). This distinction is important: benzene moves are indispensable in the combinatorics of move-equivalence and reducedness, but they are not invariant-preserving in the same sense as square and contraction moves.

Square faces have a particularly rigid structure. For an $d$4-hourglass plabic graph, a square face $d$5 is fully reduced if and only if

$d$6

If $d$7, iterated square skein relations simplify the corresponding web invariant; and the square move preserves the tuple $d$8, which is the combinatorial datum indexing the basis (Gaetz et al., 2024).

A second reformulation uses a symmetrized six-vertex model. In the $d$9 theory, contracted hourglass plabic graphs of oscillating type are in bijection with symmetrized six-vertex configurations, and fully reducedness corresponds to well-orientedness. The six-vertex move calculus matches the plabic one: square moves correspond to ASM moves, benzene moves correspond to Yang–Baxter moves, and the trip-monotonicity criterion is restated as conditions on trip$U_q(\mathfrak{sl}_r)$00 strands, including the absence of self-intersections and double crossings (Gaetz et al., 2023, Enugandla et al., 9 Dec 2025).

This move theory is also where several common misconceptions are corrected. Hourglass plabic graphs are not defined solely by the presence of multi-edges; the essential structure is the interaction between multiplicities, trip systems, and admissible local moves. Likewise, “fully reduced” is stronger than ordinary reducedness of the underlying plabic graph, although fully reduced graphs do have reduced underlayers in Postnikov’s sense (Gaetz et al., 2024).

5. Rotation-invariant bases, tableaux, and clasping

The principal representation-theoretic achievement of the subject is the construction of rotation-invariant web bases. For $U_q(\mathfrak{sl}_r)$01, the set

$U_q(\mathfrak{sl}_r)$02

is a basis of the relevant invariant space and is rotation-invariant (Gaetz et al., 2023). In Plücker degree two for arbitrary $U_q(\mathfrak{sl}_r)$03, the collection $U_q(\mathfrak{sl}_r)$04 of tensor invariants of fully reduced $U_q(\mathfrak{sl}_r)$05-hourglass plabic graphs of Plücker degree two is a rotation-invariant web basis for $U_q(\mathfrak{sl}_r)$06 (Gaetz et al., 2024).

These bases are controlled by tableaux. In the degree-two theory, the Fraser map

$U_q(\mathfrak{sl}_r)$07

gives a bijection, up to square moves, between two-column rectangular standard Young tableaux and move-equivalence classes of fully reduced $U_q(\mathfrak{sl}_r)$08-hourglass plabic graphs of standard type and Plücker degree two, with compatibility

$U_q(\mathfrak{sl}_r)$09

Promotion and evacuation of tableaux intertwine rotation and reflection of hourglass plabic graphs, and Fraser’s degree-two basis agrees with the hourglass basis (Gaetz et al., 2024).

The earlier $U_q(\mathfrak{sl}_r)$10 paper develops this correspondence through rectangular fluctuating tableaux, growth rules, and a bijection between move-equivalence classes of contracted fully reduced hourglass plabic graphs and tableaux. The slogan is that trip data on graphs matches promotion data on tableaux, so rotation on graphs corresponds to promotion and reflection corresponds to evacuation (Gaetz et al., 2023).

A further development is the clasped theory. For $U_q(\mathfrak{sl}_r)$11, start with a fully reduced web basis $U_q(\mathfrak{sl}_r)$12 for $U_q(\mathfrak{sl}_r)$13, then choose a clasp sequence $U_q(\mathfrak{sl}_r)$14 that groups consecutive boundary factors into irreducible summands $U_q(\mathfrak{sl}_r)$15. The induced map

$U_q(\mathfrak{sl}_r)$16

has the property that, for basis webs $U_q(\mathfrak{sl}_r)$17, the following are equivalent: $U_q(\mathfrak{sl}_r)$18; $U_q(\mathfrak{sl}_r)$19 is non-convex; and $U_q(\mathfrak{sl}_r)$20 has no trips that start and end in the same clasp. Moreover, the nonzero images $U_q(\mathfrak{sl}_r)$21 form a basis of $U_q(\mathfrak{sl}_r)$22 (Enugandla et al., 9 Dec 2025).

For sorted clasp sequences there are two further equivalent descriptions: no clasp contains a bad local boundary configuration, and the lattice word $U_q(\mathfrak{sl}_r)$23 has no $U_q(\mathfrak{sl}_r)$24-descents. The counting of surviving basis vectors is then identified with Littlewood–Richardson tableaux through a bijection

$U_q(\mathfrak{sl}_r)$25

giving a direct combinatorial dimension formula (Enugandla et al., 9 Dec 2025).

6. Applications, related structures, and limitations

Hourglass plabic graphs connect several lines of research. In the symmetric setting, they parametrize the symmetric part of $U_q(\mathfrak{sl}_r)$26, and with appropriate conventions this coincides with the totally nonnegative Lagrangian Grassmannian. The practical criteria for symmetric realizability can be stated in any of the equivalent languages of positroids, bounded affine permutations, or Grassmann necklaces (Karpman et al., 2015).

In the web-theoretic setting, the framework unifies all known rotation-invariant $U_q(\mathfrak{sl}_r)$27-web bases in degree two. For $U_q(\mathfrak{sl}_r)$28, it recovers Kuperberg’s $U_q(\mathfrak{sl}_r)$29 spider bases and the $U_q(\mathfrak{sl}_r)$30 basis of earlier work; in the degree-two arbitrary-r setting it provides a single combinatorial model simultaneously generalizing the Tamari lattice, the alternating sign matrix lattice, and the lattice of plane partitions (Gaetz et al., 2024). In the $U_q(\mathfrak{sl}_r)$31 theory, the symmetrized six-vertex correspondence likewise organizes ASM and plane-partition combinatorics inside the move-equivalence theory of hourglass graphs (Gaetz et al., 2023).

Cluster-algebra applications appear in $U_q(\mathfrak{sl}_r)$32. The cluster algebra $U_q(\mathfrak{sl}_r)$33 is not finite type; its cluster variables include $U_q(\mathfrak{sl}_r)$34 quadratic variables and $U_q(\mathfrak{sl}_r)$35 cubic variables. Using the hourglass basis and a growth algorithm from $U_q(\mathfrak{sl}_r)$36 tableaux, web diagrams and dual webs were computed for representative quadratic and cubic cluster variables, with all others obtained by dihedral symmetry (Zhang et al., 24 Jul 2025). The same paper combines hourglass web construction on the $U_q(\mathfrak{sl}_r)$37 side with Lam’s compatibility method and the immanant map on the dual-web side, producing explicit web representatives for cluster variables in $U_q(\mathfrak{sl}_r)$38 (Zhang et al., 24 Jul 2025).

A different but related geometric appearance occurs in triangular-grid billiards. There, a polygon in the triangular grid determines a billiards permutation $U_q(\mathfrak{sl}_r)$39, and connected reduced plabic graphs of essential dimension $U_q(\mathfrak{sl}_r)$40 arise by dualizing the triangular cells inside the polygon. For such a graph with $U_q(\mathfrak{sl}_r)$41 cycles in its trip permutation, $U_q(\mathfrak{sl}_r)$42 internal vertices, and $U_q(\mathfrak{sl}_r)$43 marked boundary points, one has

$U_q(\mathfrak{sl}_r)$44

The vertex bound is tight, with equality precisely for graphs dual to trees of unit hexagons (Defant et al., 2022).

Several limitations are explicit in the literature. The detailed isotropic or Lagrangian Plücker relations for the symmetric locus are deferred beyond the set-theoretic equations $U_q(\mathfrak{sl}_r)$45 (Karpman et al., 2015). The arbitrary-r web-basis theorem currently covers Plücker degree two rather than all degrees (Gaetz et al., 2024). The clasped-basis theorem is proved for $U_q(\mathfrak{sl}_r)$46, and extension to $U_q(\mathfrak{sl}_r)$47 is stated to require new diagrammatics and likely new combinatorial structures generalizing hourglass edges and six-vertex models (Enugandla et al., 9 Dec 2025). These limitations mark the boundary between a mature combinatorial formalism and a still-expanding representation-theoretic program.