Plabic Tangles: Diagrams and Promotion Maps

• Plabic tangles are planar bicolored graphs with inner disks (blobs) that extend reduced plabic graphs by enabling the insertion of subgraphs and inducing rational maps between Grassmannians.
• The framework employs m-vector-relation configurations (m-VRCs) to reconstruct internal data from boundary information using path formulas, ensuring m-generic solvability.
• Promotion maps and quasi-cluster homomorphisms derived from plabic tangles connect amplituhedron geometry with total positivity in Grassmannian settings.

Searching arXiv for the primary paper on plabic tangles and closely related foundational work. Searching arXiv for foundational plabic-graph and hourglass-plabic literature relevant to the diagrammatic and cluster-algebraic context. Plabic tangles are planar bicolored graph-theoretic objects introduced to formalize “a plabic graph with holes into which other plabic graphs may be inserted,” and to use such objects to define rational maps between products of Grassmannians called promotions (Even-Zohar et al., 4 Aug 2025). The framework was motivated by the graphical form of the BCFW recurrence for tilings of the amplituhedron, where one starts with a fixed “core” plabic graph and inserts smaller plabic graphs into selected faces, thereby inducing a map

$C(\widehat{\Gr}_{4,N_L})\otimes C(\widehat{\Gr}_{4,N_R})\to C(\widehat{\Gr}_{4,n}).$

Its central new ingredient is the theory of $m$-vector-relation configurations ($m$-VRCs), which makes it possible to reconstruct internal vectors from generic boundary data and then read off induced data on the inner disks of the tangle (Even-Zohar et al., 4 Aug 2025).

1. Definition and diagrammatic setting

A plabic tangle $(G,D)$ consists of a plabic graph $G=((B,W),E)$ drawn inside an outer disk with boundary vertices $bd=\{1,\dots,n\}$, together with $\ell$ inner disks inside faces of $G$, called blobs (Even-Zohar et al., 4 Aug 2025). Each inner disk $D^{(i)}$ has boundary vertices $D^{(i)}$, and each vertex $m$0 is connected by a segment to a unique black vertex $m$1, such that the resulting picture is planar. Each disk, inner and outer, has a marked interval $m$2, and the boundary vertices of that disk are labeled clockwise in increasing order starting just after the $m$3 (Even-Zohar et al., 4 Aug 2025).

The plabic graph $m$4 is called the core, and the inner disks are called blobs (Even-Zohar et al., 4 Aug 2025). The paper explicitly presents this as a device modeled on Jones’ planar tangles, with composition by insertion of one tangle into a blob of another (Even-Zohar et al., 4 Aug 2025). In that sense, plabic tangles extend the older use of reduced plabic graphs as planar bicolored boundary diagrams for positroid cells of the positive Grassmannian (Paulos et al., 2014).

This definition sits naturally in the broader plabic-graph literature. A plabic graph is a planar bicolored graph embedded in a disk, and reduced plabic graphs encode positroid cells, decorated permutations, and cluster seeds through their dual quivers (Oh et al., 2011). The plabic-tangle framework keeps that planar-bicolored core but adds inner disks and a mechanism for extracting maps between Grassmannians rather than only coordinates on a single positroid cell (Even-Zohar et al., 4 Aug 2025).

2. $m$5-vector-relation configurations and solvability

The mechanism underlying promotion is the notion of an $m$6-vector-relation configuration. For a bipartite plabic graph $m$7 with black boundary vertices and $m$8, an $m$9-VRC is an assignment of a vector $m$0 to each black vertex $m$1, and a nonzero scalar $m$2 to each edge $m$3, such that the boundary vectors $m$4 span $m$5, and for each white vertex $m$6,

$m$7

The boundary matrix is

$m$8

and a VRC is non-degenerate if every internal black vector is nonzero (Even-Zohar et al., 4 Aug 2025).

The theory includes two symmetries. The group $m$9 acts by left multiplication on all vectors, and a gauge group acts at internal vertices. Gauge transformation at an internal vertex $(G,D)$0 by $(G,D)$1 changes adjacent edge weights $(G,D)$2, and if $(G,D)$3 is black also changes $(G,D)$4 (Even-Zohar et al., 4 Aug 2025). The corresponding moduli set is denoted $(G,D)$5, and for $(G,D)$6,

$(G,D)$7

If $(G,D)$8 has an acyclic reverse perfect orientation $(G,D)$9 with source set $G=((B,W),E)$0, then every black-vertex vector can be reconstructed from the source vectors by the path formula

$G=((B,W),E)$1

Here the numerator is over edges of $G=((B,W),E)$2 oriented black-to-white and the denominator over edges oriented white-to-black (Even-Zohar et al., 4 Aug 2025). This formula is the basic reason internal data can be recovered from boundary data.

A plabic graph $G=((B,W),E)$3 is $G=((B,W),E)$4-generically solvable if for generic boundary data $G=((B,W),E)$5 or $G=((B,W),E)$6, there exists a unique $G=((B,W),E)$7-VRC with boundary $G=((B,W),E)$8, up to gauge (Even-Zohar et al., 4 Aug 2025). For a tangle $G=((B,W),E)$9, solvability means that its core is solvable and every blob has at least $bd=\{1,\dots,n\}$0 boundary vertices (Even-Zohar et al., 4 Aug 2025). The paper proves that for reduced $bd=\{1,\dots,n\}$1, there is at most one $bd=\{1,\dots,n\}$2 for which $bd=\{1,\dots,n\}$3 can be solvable, namely

$bd=\{1,\dots,n\}$4

3. Promotion maps and the quasi-cluster conjecture

For a solvable plabic tangle $bd=\{1,\dots,n\}$5, promotion is first defined on configuration spaces: $bd=\{1,\dots,n\}$6 Starting from generic outer boundary data $bd=\{1,\dots,n\}$7, one takes the unique VRC $bd=\{1,\dots,n\}$8 with boundary $bd=\{1,\dots,n\}$9, assigns the line $\ell$0 to each blob vertex $\ell$1, and then reads blob vertices clockwise from the marked interval $\ell$2 to obtain an element of $\ell$3 for each blob (Even-Zohar et al., 4 Aug 2025).

To pass from configuration spaces to Grassmannians, one uses a pinning. A pinning is a collection of rational functions

$\ell$4

such that for generic $\ell$5, the unique VRC with boundary $\ell$6 has a representative

$\ell$7

Given a pinned plabic tangle, one obtains geometric promotion

$\ell$8

and algebraic promotion

$\ell$9

defined by replacing blob columns with the corresponding vectors $G$0 (Even-Zohar et al., 4 Aug 2025).

The central conjecture of the paper states that for dominant solvable plabic tangles there exists a brushing $G$1 and signs such that geometric promotion sends totally positive points to totally positive points, and algebraic promotion is a quasi-cluster homomorphism after freezing some variables on the target side (Even-Zohar et al., 4 Aug 2025). In the paper’s terminology, an algebra homomorphism $G$2 is a quasi-cluster homomorphism if, for suitable seeds $G$3, every mutable variable maps proportionally to a mutable variable,

$G$4

where proportionality means equality up to a Laurent monomial in frozen variables, and the exchange ratios satisfy

$G$5

(Even-Zohar et al., 4 Aug 2025).

The paper proves this conjecture for several infinite families: star promotion, spurion promotion, chain-tree promotion, and forest promotion (Even-Zohar et al., 4 Aug 2025). This places plabic tangles squarely inside the cluster-algebraic geometry of Grassmannians, while still allowing the possibility of algebraic, non-rational maps in higher intersection-number cases (Even-Zohar et al., 4 Aug 2025).

4. Amplituhedron geometry, intersection number, and amplitrees

A major structural result of the theory is that $G$6-VRCs count fibers of the amplituhedron map on positroid varieties (Even-Zohar et al., 4 Aug 2025). For $G$7, the amplituhedron map is

$G$8

and the paper also uses the twistor embedding

$G$9

writing

$D^{(i)}$0

For a positroid variety $D^{(i)}$1, the restricted map is

$D^{(i)}$2

If $D^{(i)}$3, the $D^{(i)}$4-intersection number $D^{(i)}$5 is defined to be the generic degree of $D^{(i)}$6; otherwise it is $D^{(i)}$7 (Even-Zohar et al., 4 Aug 2025). The paper proves that if $D^{(i)}$8, then for generic $D^{(i)}$9,

$D^{(i)}$0

Equivalently, the number of $D^{(i)}$1-VRCs with generic boundary equals the generic number of points in the fiber of the amplituhedron map (Even-Zohar et al., 4 Aug 2025). In particular,

$D^{(i)}$2

The paper gives a complete classification for plabic trees. A bipartite plabic tree $D^{(i)}$3 of type $D^{(i)}$4 is $D^{(i)}$5-balanced if for every edge $D^{(i)}$6, when

$D^{(i)}$7

each component $D^{(i)}$8 satisfies

$D^{(i)}$9

Then

$m$00

and if $m$01 is not $m$02-balanced, then

$m$03

The $m$04-balanced trees are called $m$05-amplitrees (Even-Zohar et al., 4 Aug 2025).

For amplitrees, the paper constructs explicit VRCs using the Grassmann-Cayley algebra. If $m$06 is a vertex in a rooted tree, then

$m$07

for a boundary vertex $m$08,

$m$09

for a white vertex with children $m$10, and

$m$11

for a black vertex with children $m$12 (Even-Zohar et al., 4 Aug 2025). This supplies an explicit recursive realization of VRC vectors in the degree-one case.

5. Dominance, brushings, operads, and examples

A solvable tangle is dominant if each blob can attain a Zariski-dense subset of generic configurations: for each blob $m$13, the image of

$m$14

is dense in $m$15 (Even-Zohar et al., 4 Aug 2025). The paper identifies the correct combinatorial structure controlling this property: a solvable plabic tangle is dominant if and only if it admits a brushing (Even-Zohar et al., 4 Aug 2025).

A brushed plabic tangle consists of a tangle $m$16 together with, for each blob $m$17, a reverse acyclic perfect orientation $m$18, a collection of oriented vertex-disjoint paths $m$19 from outer boundary vertices $m$20 to the black vertices $m$21 adjacent to blob vertices $m$22, and signs $m$23 (Even-Zohar et al., 4 Aug 2025). Given any pinning, the brushing normalizes it by dividing by the path weight

$m$24

and setting

$m$25

A lemma shows that this normalized value is independent of the initial pinning (Even-Zohar et al., 4 Aug 2025).

Plabic tangles also form an operad. If $m$26 denotes plabic tangles with outer boundary size $m$27 and blob sizes $m$28, there is a unit $m$29, given by the trivial tangle of $m$30 parallel segments, and composition

$m$31

by inserting one tangle into the $m$32-th blob of another (Even-Zohar et al., 4 Aug 2025). The paper proves that dominant solvable plabic tangles form a suboperad $m$33, and brushed dominant solvable plabic tangles form a suboperad $m$34 (Even-Zohar et al., 4 Aug 2025).

Several classes of promotion are worked out explicitly. Star promotion gives a unary map $m$35 (Even-Zohar et al., 4 Aug 2025). Spurion promotion, chain-tree promotion, and forest promotion supply further infinite families of quasi-cluster homomorphisms (Even-Zohar et al., 4 Aug 2025). The BCFW promotion is the motivating binary example and recovers the quasi-cluster map arising in the amplituhedron recursion (Even-Zohar et al., 4 Aug 2025).

The paper also treats the $m$36-mass box, where the intersection number is $m$37. In that case promotion is no longer single-valued and rational, but has two branches

$m$38

with

$m$39

and the paper proves that if $m$40 is a cluster variable for $m$41, then $m$42 and $m$43 are positive on $m$44 (Even-Zohar et al., 4 Aug 2025). This is used to point toward positivity phenomena beyond ordinary cluster-algebra positivity (Even-Zohar et al., 4 Aug 2025).

6. Antecedents, neighboring frameworks, and scope

The explicit notion of plabic tangles appears only in the 2025 paper “Plabic Tangles and Cluster Promotion Maps” (Even-Zohar et al., 4 Aug 2025). Earlier work developed the surrounding structures but did not use the term. In particular, plabic graphs themselves were already identified with positroid cells of the positive Grassmannian, decorated permutations, local square moves, and cluster seeds through dual quivers (Paulos et al., 2014). Maximal weakly separated collections in a positroid were shown to be in bijection with reduced plabic graphs, establishing a precise planar-combinatorial foundation for their boundary and face data (Oh et al., 2011).

Several later frameworks are especially close in spirit to plabic tangles. Hourglass plabic graphs add edge multiplicities and trip strands, giving a planar, boundary-attached, strand-carrying diagrammatics for $m$45-webs in Plücker degree two (Gaetz et al., 2024). “Flip cycles in plabic graphs” studies the loops of local rewrites in the plabic flip graph and proves that the fundamental group is generated by cycles of sizes $m$46, $m$47, and $m$48, supplying a coherence theory for local plabic moves (Balitskiy et al., 2019). “3D plabic graphs” generalize ordinary planar plabic graphs to braid-sensitive objects in $m$49, with relative cycles, surfaces, quivers, and cluster seeds; this is one of the closest preexisting analogues of a genuinely tangle-like plabic theory, although the paper does not use that name (Galashin et al., 2022). “Plabic links, quivers, and skein relations” attaches links $m$50 to plabic graphs and shows that local plabic modifications can realize the HOMFLY skein relation, thereby producing a strong link-theoretic local calculus from plabic data (Galashin et al., 2022).

These neighboring theories clarify the scope of plabic tangles. The 2025 framework is not a theory of tangles in the knot-theoretic sense, but a theory of plabic graphs with inner disks, VRCs, promotion maps, and operadic insertion (Even-Zohar et al., 4 Aug 2025). Its strongest direct claim is that dominant solvable plabic tangles should yield quasi-cluster homomorphisms and preserve total positivity, with several infinite families already proved and the $m$51-mass box suggesting a further positive algebraic structure beyond cluster algebras (Even-Zohar et al., 4 Aug 2025).