Leading term strandings for webs

Bella Mohren; Blaise Marsho; Caitlin Sales; Heather M. Russell; Jacob Martin; Jade Mawn; Junyang Chen; Lance Wong; Madelyn Burns; Michael Bo

REVIEW 2 major objections 5 minor 20 references

Leading term strandings for webs

T0 review · 2 major / 5 minor · reviewed 2026-08-28 · deepseek-v4-flash

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Pith's one-line read Every open strand in a leading-term stranding of an sl_n web runs clockwise, and this constraint builds full web bases from row-strict tableaux.

desk verdict A genuinely new non-recursive web basis construction with a clean leading-term criterion, but two proof gaps—Lemma 49(3) and the edge-flip correspondence in Section 5—need fixing before the paper is fully convincing. read the letter →

arxiv 2608.27425 v1 pith:MG2E2QR2 submitted 2026-08-27 math.CO math.QAmath.RT

Michael Bo , Madelyn Burns , Junyang Chen , Blaise Marsho , Jacob Martin , Jade Mawn , Bella Mohren , Heather M. Russell

show 2 more authors

Caitlin Sales Lance Wong

This is my paper · ORCID

classification math.COmath.QAmath.RT MSC 05E1017B37

keywords webbasesstrandingssl_nwebsleadingtermrow-stricttableauxquantumgroupinvariantsedge-Kempeequivalenceintegraldepth

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Webs are plane graphs whose assigned vectors live in invariant spaces of quantum group representations; a stranding is a system of colored oriented curves on a web that records one monomial of that vector. This paper establishes that the strandings recording the leading term cannot contain any counterclockwise open strand, so their boundary data is always a row-strict tableau. From a row-strict tableau, the paper builds a colored noncrossing matching and then an $\mathfrak{sl}_n$ web whose stranding is provably its leading term, and the webs obtained form a web basis without any recursive construction. For $\mathfrak{sl}_3$ webs without flat vertices, integral face depth alone determines the leading-term stranding, and for all $\mathfrak{sl}_3$ webs any valid stranding can be transformed into any other by reversing strands.

What carries the argument

The load-bearing object is the stranding: a collection of colored oriented curves on a web, each curve contributing one monomial to the web vector, with local validity encoded by alternating-sign binary labels on edges. Two mechanisms carry the proofs. The first is the reversal argument: reversing an open $(i,j)$-strand preserves validity (Lemma 14), and because every stranding contributes with nonzero coefficient, any counterclockwise open strand would produce a strictly smaller monomial, forcing the leading-term stranding to be clockwise. The second is depth: each face has an integral depth from the web's dual graph and a strand depth counting enclosing strands, and Lemma 19 shows strand depth never exceeds integral depth; for the constructed webs these depths coincide, which lets the proof track the $(1,n)$-strands as nested clockwise arcs and drive the infinite-descent contradiction in Theorem 56. The construction from tableau to web passes through a colored noncrossing matching, whose arcs become the simple strands of the web.

What would settle it

Enumerate all valid strandings of the small $\mathfrak{sl}_4$ webs in Example 58 and compare their monomials lexicographically; finding a monomial smaller than that of the constructed stranding $S_T$ would refute Theorem 56, as would exhibiting any web whose leading-term stranding has a counterclockwise open strand.

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Extended reading notes

Core claim

The central claim is that leading-term strandings are rare and readable enough to be used as a tool for constructing and certifying web bases. Lemma 30 and Theorem 33 prove that, in any leading-term stranding, every open strand is clockwise; reversing a counterclockwise strand would make the monomial lexicographically smaller while remaining a valid stranding, and the nonzero-coefficient theorem rules out cancellation. Corollary 35 turns this into a triangularity criterion: a set of webs whose leading-term strandings realize every row-strict tableau as boundary data is a web basis. Theorem 56 shows that Construction 42 realizes every row-strict tableau, producing a web whose constructed stranding is leading and whose tableau is the original $T$; hence the resulting webs form a web basis, a non-recursive counterpart of the previously known recursive $\mathfrak{sl}_n$ web bases. For $\mathfrak{sl}_3$, Theorem 62 identifies the leading-term stranding of a web with no flat vertices as the unique stranding in which strand depths equal integral face depths, and Corollary 69 proves strand-reversal connects any two valid strandings.

Load-bearing premise

The whole argument leans on the framework result that every valid stranding contributes its own monomial with nonzero coefficient to the web vector; if that result were wrong, none of the leading-term comparisons by strand reversal would force a contradiction.

Editorial extensions

If this is right

A valid stranding with any counterclockwise open strand is automatically not a leading-term stranding, so the clockwise condition is a fast necessary test.
The tableau construction gives a non-recursive way to produce a full web basis for every boundary weight vector of $\mathfrak{sl}_n$; at $n=3$ it recovers the nonelliptic web basis.
For $\mathfrak{sl}_3$ webs without flat vertices, the leading term can be read from face depths alone, so leading terms are computable from the dual graph without enumerating strandings.
For every $\mathfrak{sl}_3$ web, a leading-term stranding is reachable from any starting stranding by strand reversals, since the reversal graph is connected; the paper leaves open an algorithmic way to find that sequence.
The construction's flexibility in choosing matchings means the same tableau can produce distinct webs with the same leading term but different full web vectors, as Example 58 illustrates.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Remark 34 leaves open whether 'all clockwise open strands, including induced $(i,j)$-strands' is sufficient for being a leading-term stranding; if it is, leading-term strandings could be recognized by a purely local scan, making web-basis membership a graph-certification problem.
The extra isotopy freedom in choosing the colored noncrossing matching may be enough to satisfy rotation-invariance for the resulting bases; one could test whether some choice of matchings yields bases invariant under boundary rotation for all $n$.
The edge-Kempe proof of transitivity for $\mathfrak{sl}_3$ suggests a route for $n\ge 4$: if valid strandings of $\mathfrak{sl}_n$ webs correspond to edge-colorings of an auxiliary planar graph, the same double-cover argument might settle whether strand reversal is transitive for all $n$.
For webs with flat vertices, the depth-selection theorem fails exactly where three adjacent faces share one depth; a small local modification that breaks flatness could turn the depth rule into a heuristic, and strand reversals could then fix the residue, yielding a testable algorithm for $\mathfrak{sl}_3$ leading terms.

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Formalized claims in Lean

Claim #1: The central claim is that leading-term strandings are rare and readable enough to be used as a tool for constructing and certifying web bases. Lemma 30 and Theorem 33 prove that, in any leading-term stranding, every open strand is clockwise; reversing a counterclockwise strand would make the monomial lexicographically smaller while remaining a valid stranding, and the nonzero-coefficient theorem r

/-- @claim 1 The central claim is that leading-term strandings are rare and readable enough to be used as a tool for constructing and certifying web bases. Lemma 30 and Theorem 33 prove that, in any leading-term stranding, every open strand is clockwise; reversing a counterclockwise strand would make the monomial lexicographically smaller while remaining a valid stranding, and the nonzero-coefficient theorem r -/
def central_claim : Prop :=

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

This paper gives a non-recursive construction of sl_n web bases from row-strict tableaux via colored noncrossing matchings, and proves a clockwise criterion for leading term strandings. That is new and, as far as I can tell, the main machinery is sound. The clockwise lemma (Lemma 30) is clean and likely reusable. The construction in Section 4 is explicit, and Example 58 shows it produces webs beyond Fontaine's recursive basis. If the main theorem holds, this is a useful tool for the web-basis subfield.

The soft spots are in the details, not the overall plan. Lemma 49(3) — every (1,n)-strand of S_T is open — is load-bearing for Theorem 56 and Corollary 57, but the proof is a sketch. After choosing an innermost closed strand, the paper asserts "By Lemma 48, there are no edges of G_T inside it." Lemma 48 does not by itself say that; it only gives the local depth profile at endpoints of an interior same-depth edge. You can likely fill the gap using innermostness plus the fact that an open (1,n)-strand cannot cross the closed strand, but that reasoning is not in the paper. A referee should insist on a complete proof here. This is a conditional, not a rejection: the claim looks true and the intended argument is plausible.

The second soft spot is Corollary 69, the transitivity of strand reversals for sl_3 webs. The proof leans on edge flips to reduce to weight-1 edges, but the bijection between strandings under edge flips is asserted, not shown. That is a real gap, though the result is secondary. Also worth noting: the paper depends on the strandings framework of [13], an unreviewed preprint, for the nonzero-coefficient theorem. That is a burden to keep in mind, not a flaw in this paper.

The math is otherwise careful, the examples are helpful, and the citation pattern looks appropriate. The paper deserves a serious referee. My recommendation: send it out, with an explicit request to verify Lemma 49(3) and to fill the edge-flip/stranding correspondence in Section 5.

Referee Report

2 major / 5 minor

Summary. The paper studies leading term strandings for sl_n webs: curve systems on webs that record the leading monomial of the web's invariant vector with respect to a lexicographic order. The main results are: (1) every open strand of a leading term stranding is clockwise, which implies that the tableau recording the stranding's boundary data is row-strict and yields a sufficient condition for a set of webs to be a web basis; (2) for each row-strict tableau, an explicit non-recursive construction of a web with a prescribed leading term stranding, giving a web basis indexed by row-strict tableaux; (3) for simple sl_3 webs (no flat vertices), the integral depth of faces determines a unique leading term stranding; and (4) for any sl_3 web, any two valid strandings are related by a sequence of strand reversals, obtained via an edge-Kempe equivalence argument. The paper is clearly written and the constructions are explicit and combinatorial.

Significance. If the results are correct, the paper provides a substantial advance: it gives a non-recursive construction of web bases for general sl_n, generalizing the sl_3 constructions of Tymoczko and Russell, and it gives a simple structural criterion for detecting web bases. The clockwise-strand theorem (Lemma 30) and the depth-based stranding for simple sl_3 webs are elegant and likely to be useful. The proofs are mostly self-contained and do not involve parameter fitting or circular reductions; the main external input is the strandings framework of Russell-Tymoczko [13]. The central web-basis theorem (Theorem 56 and Corollary 57) is concrete and checkable, and the sl_3 results extend the class of webs for which leading terms are explicitly known. The main weaknesses are a compressed proof of a load-bearing lemma and an under-specified reduction in Section 5, both of which appear fixable.

major comments (2)

[§4.2, Lemma 49(3)] The proof of Lemma 49(3) is incomplete at a load-bearing point. The sentence "By Lemma 48, there are no edges of G_T inside it" does not follow from Lemma 48 as stated: Lemma 48 only constrains the local depth profile at the endpoints of a single interior edge and does not rule out edges or vertices in the interior region enclosed by an innermost closed (1,n)-strand. The subsequent dichotomy (the strand contains a trunk edge, or every vertex is a crossing) also omits the possibility that the closed strand encloses a small tree of same-depth edges. This matters because Corollary 50 and the monotone head-descent in Theorem 56 both require every (1,n)-strand of S_T to be open. Please supply a complete proof; for example, one can first derive from Lemma 48 that every interior vertex of G_T has its three incident face depths equal to {d-1,d,d+1}, and then show that a closed (1,n)-strand would enclose a region with a face of maximal depth whose neighbours have no larger depth, contradicting that three-consecutive-values property. Alternatively, give a direct planar argument.
[§5, paragraph after Lemma 60] The reduction "By the edge flip relation, we may assume that all sl_3 web edges have weight 1" is used to prove Theorem 62 and Corollary 69 for arbitrary sl_3 webs, but the paper does not state how strandings and strand reversals transfer between G and the edge-flipped web G^{φ(E)}. Lemma 60 only asserts equality of web vectors; it does not by itself provide a bijection between Str(G) and Str(G^{φ(E)}). Without such a bijection, the conclusions of Theorem 62 and Corollary 69 are established only for webs in which every edge has weight 1. Please add an explicit statement of the stranding bijection (for instance, complementing the binary label on every flipped edge) and verify that it intertwines strand reversal, or restrict the statements of the theorems.

minor comments (5)

[§4.2, Lemma 48] The phrase "separating faces of depth d" is ambiguous: it should say explicitly whether the two faces on either side of e have depths d-1 and d+1, or d and d+1, or something else. This ambiguity is especially relevant to the repair of Lemma 49(3).
[§5.2, Proposition 68] In the gluing construction of H', the bipartition of the reflected copy should be explicitly chosen to be swapped, so that each pair of glued boundary vertices lies in opposite bipartition classes; the current proof asserts this without saying how the bipartition is chosen.
[§5.2, Corollary 69] The translation between strandings of a weight-1 sl_3 web and proper edge-3-colorings should be made explicit, stating which binary label corresponds to which color and how each (i,j)-strand is exactly an edge-Kempe chain for the corresponding pair of colors. The parenthetical "red and blue together" is confusing as written.
[§1, Introduction] The introduction claims that for n=3 the construction returns exactly Kuperberg's nonelliptic web basis and that the webs produced by Fontaine's algorithm form a proper subset of the constructed webs. These claims are not proven in the body; Example 58 illustrates only one case. Please either add a proof or soften the statements.
[Abstract and §1] The phrase "non-recursive construction of Fontaine's sl_n web bases" may be misleading, since the constructed webs generally differ from Fontaine's (as Example 58 shows). Consider rephrasing to "a non-recursive construction of web bases of the Fontaine type" or explicitly clarifying the relationship.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: web-basis construction is derived from an external stranding framework and explicit lexicographic arguments, not from fitted inputs or self-referential definitions.

full rationale

The paper's main derivation chain—Construction 36/42 producing G_T and S_T, the structural redrawing in Lemma 49/Corollary 50, and Theorem 56 proving S_T is a leading term stranding by a lexicographic descent—does not reduce any conclusion to its own inputs. No parameter is fitted to data, and no quantity called a prediction is imported from a fit; the web basis is obtained by triangularity from the independent dimension theorem |Inv_n(k)|=|RST_n(k)| (Theorem 22, citing Howe). The leading-term claim is proved by comparing S_T with an arbitrary leading term stranding S_star and deriving an infinite descent of disagreements, rather than by defining the leading term to be the stranding. The only notable self-adjacent dependency is the strandings formalism of [13] (Russell is a co-author): Theorems 10, 25, 27, and Lemma 60 are used as black boxes. These are parameter-free statements about general webs and strandings, not statements about the particular web basis being constructed, so they function as independent support for the present argument rather than as a circular reduction. The skeptic's concern about Lemma 49(3)—that its proof is compressed and the 'no edges inside' claim is asserted rather than fully derived—is a proof-completeness risk, not circularity: a failure of that lemma would be an incorrect derivation step, not a conclusion equivalent to its input by construction. Accordingly, the circularity score is 1, reflecting the shared-author preprint dependency while finding no circular step.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No free parameters or fitted quantities appear; this is a purely combinatorial proof article. The colored noncrossing matching is a new construction, but it is a defined mathematical object, not a postulated physical entity. All substantive assumptions are explicitly cited theorems or stated modeling choices.

assumptions (7)

standard math Theorem 10 of [13]: valid strandings are in bijection with valid binary labelings.
Used throughout Sections 2 and 4 to pass between strandings and binary labels; not proved in this paper.
standard math Theorem 51 of [13]: every stranding contributes its monomial with nonzero coefficient to the web vector.
Used in Lemma 30 and Theorem 56 to ensure lexicographic comparisons between strand monomials are meaningful; load-bearing for the leading-term results.
standard math Howe's dimension formula: dim Inv_n(k) = |RST_n(k)| (Theorem 22).
Used in Corollary 35 to turn a triangularity check into a basis criterion.
standard math Belcastro-Haas theorem: any two proper edge-3-colorings of a planar bipartite cubic graph are edge-Kempe equivalent (Theorem 67).
Used in Proposition 68 and Corollary 69 to connect strand reversals to edge-Kempe switches.
standard math Edge flip relation f^up(G) = f^up(G^{phi(E)}) (Lemma 60 from [13]).
Used in Section 5 to reduce sl_3 webs to weight-1 webs; the induced effect on strandings is not fully spelled out.
domain assumption The paper works with untagged sl_n webs only.
The authors state that tagged webs complicate the combinatorial calculations; all constructions and theorems are for the untagged class.
domain assumption In Section 5 all sl_3 edges are assumed weight 1 after edge flips.
Needed for the source/sink property of interior vertices and for the edge-3-coloring argument in Corollary 69.

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Pith. "Pith review of Leading term strandings for webs." pith.science (2026). https://pith.science/paper/MG2E2QR2

@misc{pith2026260827425,
  author       = {Pith},
  title        = {Pith review of: Leading term strandings for webs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MG2E2QR2}},
  note         = {Machine review of arXiv:2608.27425}
}

abstract

A web is a plane graph encoding an invariant vector in a tensor product of fundamental representations of a quantum group. A stranding of an $\mathfrak{sl}_n$ web is a system of colored oriented curves recording one monomial of the vector it encodes. This article focuses on identifying and constructing leading term strandings, those recording the leading term of a web's vector with respect to a lexicographic order on monomials. We show that every open strand of a leading term stranding is clockwise, which constrains the boundary data of such strandings enough to yield a sufficient criterion for a set of webs to form a web basis. From a row-strict tableau, we construct a web with a prescribed leading term, and the resulting webs form a web basis, giving a non-recursive construction of Fontaine's $\mathfrak{sl}_n$ web bases. For $\mathfrak{sl}_3$ webs with no flat vertices, we identify a leading term stranding using the depths of the faces of the web. Finally, we show a leading term stranding for any $\mathfrak{sl}_3$ web can be reached from an arbitrary stranding via a sequence of operations called strand reversals.

Figures

Figures reproduced from arXiv: 2608.27425 by the authors.

**Figure 1.** An sl4 web. 2.2. Curve systems, strandings, and binary labelings. We are interested in networks of oriented curves that run along the edges of the half-plane graphs described above. Definition 5 (Arcs, loops, and curve systems). Let G be a half-plane graph. • An arc on G is an oriented, simple path in G with both endpoints on the boundary axis. An arc is clockwise (resp. counterclockwise) if its terminal vertex is t… view at source ↗

**Figure 2.** A stranding S together with its binary labeling bS (left) and the (2, 4) strands for S (right). Strandings of a fixed web can be modified locally by reversing a single strand, giving a reconfiguration process on Str(G). Definition 13 (Strand reversal). Let γ be an (i, j)-strand in a stranding S of G ∈ Fn( ⃗k), with 1 ≤ i < j ≤ n. Reversing γ produces the stranding S ′ whose binary labeling bS′ agrees with bS on eve… view at source ↗

**Figure 3.** Reversing the highlighted counterclockwise open strand of stranding S (left) to obtain S ′ (right). 2.3. Depth. In this subsection we introduce two notions of depth for half-plane graphs and study how they interact. The first, integral depth, is a purely combinatorial measure of distance from the boundary. The second, depth with respect to a curve system, measures how many oriented curves enclose a given face. One o… view at source ↗

Figures from the paper (11 more)

**Figure 4.** Figure 4: Integral depth (left) versus strand depth (right) in a web. Vk := ^k q C n q . For a binary vector ⃗b = b1 · · · bn ∈ {0, 1} n with | ⃗b| = k, we define x⃗b := xj1 ∧q · · · ∧q xjk , where {j1 < · · · < jk} = {i : bi = 1}. For example, under this convention, x1010 = x1 …

**Figure 5.** Figure 5: Constructing MT using the algorithm from Lemma 37. Proof. Let T ∈ RST n( ⃗k), and say that steps (1) and (2) of Construction 36 generate t arcs with 2t endpoints. On a horizontal line below the boundary axis, place 2t equally-spaced points. Reading left to right, assig…

**Figure 6.** Figure 6: rq r1 hp h1 [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]

**Figure 7.** Figure 7: Resolving crossings in MT to create web GT stranded by ST with relative strand depths d − 1, d, d + 1. (2) Boundary vertices. Replace a neighborhood of each boundary vertex with more than one incident edge by p+q −1 consecutive vertical edges (a trunk) together with ed…

**Figure 8.** Figure 8: The trunk and branches of GT stranded by ST replacing the neighborhood of a boundary vertex of MT as shown in [PITH_FULL_IMAGE:figures/full_fig_p015_8.png]

**Figure 9.** Figure 9: Converting arc interiors to stranded web edge interiors, with relative strand depths d − 1 and d. Example 43. In [PITH_FULL_IMAGE:figures/full_fig_p016_9.png]

**Figure 10.** Figure 10: A colored noncrossing matching and its corresponding stranded web. Lemma 44. Let T ∈ RST n( ⃗k). Then GT is an sln web, and ST ∈ Str(GT ). Proof. By construction, on each edge of GT the curves of ST alternate direction when ordered by color, so by Lemma 7, every edge …

**Figure 11.** Figure 11: A web drawn in the form of Corollary 50. Remark 52. Note that L(1,n) (ST ) is exactly the band diagram for GT defined in [15, Section 3.1] [PITH_FULL_IMAGE:figures/full_fig_p018_11.png]

**Figure 12.** Figure 12: Write MT and M′ T for the two matchings and GT , G′ T for the webs obtained from them by Construction 42. The recursive construction of [6] returns G′ T . v1 v2 v3 v4 v5 v6 2 3 1 M′ T v1 v2 v3 v4 v5 v6 2 3 1 MT v1 v2 v3 v4 v5 v6 1 2 3 1 3 2 2 1 3 1 1 2 u G′ T v1 v2 v3…

**Figure 13.** Figure 13: Unique edge stranding choices that align integral and strand depth in an sl3 web along with corresponding binary labels. Depths are labeled in brown; Blue and red strands carry colors 1 and 2, respectively. Proof. We first prove (i). Once we show Sdep is valid, it fol…

**Figure 14.** Figure 14: Left: a simple sl3 web (top) and an sl3 web with a flat vertex (bottom). Right: the leading term stranding Sdep from Theorem 62 applied to the simple web (top), and the failure of the algorithm at the flat vertex (bottom). 5.2. Strand reversal and edge-Kempe equivalen…

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Christian Gaetz, Oliver Pechenik, Stephan Pfannerer, Jessica Striker, and Joshua P. Swanson. Rotation-invariant web bases from hourglass plabic graphs.Invent. Math., 243(2):703–804, 2026

work page 2026