Positroid Links and Braid varieties

We study braid varieties and their relation to open positroid varieties. We discuss four different types of braids associated to open positroid strata and show that their associated Legendrian links are all Legendrian isotopic. In particular, we prove that each open positroid stratum can be presented as the augmentation variety for four different Legendrian fronts described in terms of either permutations, juggling patterns, cyclic rank matrices or Le diagrams. We also relate braid varieties to open Richardson varieties and brick manifolds, showing that the latter provide projective compactifications of braid varieties, with normal crossing divisors at infinity.

💡 Research Summary

The paper investigates the interplay between positroid strata in the Grassmannian and several constructions coming from braid theory, contact topology, and algebraic geometry. A positroid stratum Π⊂Gr(k,n) can be encoded in four equivalent combinatorial ways: a pair of permutations (u,w) with u≤w in Bruhat order and w a k‑Grassmannian permutation, a k‑bounded affine permutation f, a cyclic rank matrix r, or a Le diagram L. For each encoding the authors define a specific braid:

• Rₙ(u,w) – an n‑strand braid built from the pair (u,w);
• Jₖ(f) – a k‑strand braid associated to the affine permutation f;
• Mₖ(r) – a k‑strand braid obtained from the cyclic rank matrix;
• Dₖ(L) – a k‑strand braid derived from the Le diagram.

Although these braids have different numbers of strands and different crossing patterns, the first main result (Theorem 1.1) shows that they are equivalent up to positive Markov stabilizations, additions of trivial unlinked components, and a new “Markov‑type destabilization” move specially designed to compare braids of different strand counts. This establishes a precise combinatorial bridge among the four presentations of the same positroid type.

Next, each braid is closed and lifted to a Legendrian link in the standard contact ℝ³. The four Legendrian links Λ(u,w), Λ(f), Λ(r), Λ(L) are shown (Theorem 1.2) to be Legendrian isotopic, again up to disjoint maximal‑tb unknots. Consequently, the Legendrian contact differential graded algebras (DG‑algebras) attached to these links are stable‑tame isomorphic, and their degree‑zero homology spectra coincide with the coordinate ring of the corresponding positroid stratum (up to torus factors). In other words, every open positroid variety can be realized as an augmentation variety of any of the four Legendrian fronts. This provides a contact‑geometric reconstruction of positroid strata.

The paper then turns to braid varieties X(β;w), introduced in earlier work as affine varieties defined by matrix equations attached to a braid word β and a target permutation w. Theorem 1.3 proves that a positroid variety Π_{u,w} can be expressed as a braid variety using either the n‑strand braid Rₙ(u,w) or the k‑strand braid Jₖ(f). The two descriptions differ only by a torus factor (ℂ*)^{n‑k‑φ}, where φ counts the fixed points of the affine permutation f. This links open positroid varieties to open Richardson varieties and shows that the braid‑variety perspective captures the same geometry.

A further major contribution is the identification of brick manifolds (as defined by Pech–Rietsch–Williams) as smooth projective compactifications of braid varieties. For a positive braid word β, the authors consider its Demazure product δ(β) and the associated brick manifold brick(β). Theorem 1.4 establishes an explicit algebraic isomorphism between the affine braid variety X(β;δ(β)) and the open dense stratum brick∘(β) of the brick manifold, and proves that the complement is a normal‑crossing divisor whose components correspond to all possible deletions of a letter from β while preserving the Demazure product. Hence brick(β) furnishes a smooth projective SNC compactification of X(β;δ(β)). This result clarifies the relationship between braid varieties and the combinatorics of subword complexes, and allows the transfer of known properties (e.g., shellability, homology calculations) from subword complexes to braid varieties.

Finally, the authors compute the torus‑equivariant homology of braid varieties attached to positroids, discuss a “curious Lefschetz” property for open Richardson varieties, and outline conjectural connections to cluster algebra structures on these varieties.

Overall, the paper unifies four different combinatorial models of a positroid stratum, shows that they give rise to the same Legendrian link and the same augmentation variety, relates these objects to braid varieties and open Richardson varieties, and provides smooth projective compactifications via brick manifolds. The work opens new avenues for studying positroid varieties through contact topology, braid group actions, and algebraic geometry, and suggests further exploration of cluster structures and homological invariants in this unified framework.