Positroid Links and Braid varieties

TL;DR

This paper explores the relationship between braid varieties and open positroid varieties, demonstrating their equivalence via Legendrian links and connecting them to Richardson varieties and brick manifolds.

Contribution

It introduces four different Legendrian front descriptions of open positroid strata and establishes their Legendrian isotopy, linking braid varieties to Richardson varieties and brick manifolds.

Findings

Legendrian links associated with different braid types are isotopic.

Open positroid strata can be represented as augmentation varieties in four ways.

Brick manifolds serve as projective compactifications of braid varieties.

Abstract

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.