Computable Sheaf Invariants for Legendrian Rainbow Closures

TL;DR

The paper addresses the computable description of the microlocal-sheaf category attached to Legendrian rainbow closures of positive braids by encoding objects as braid-variety data and morphisms via explicit delta maps. It introduces braid-variety parametrizations $X(\beta,\mathbb{K})$ and path- and braid-matrix formalisms to obtain concrete descriptions of objects and morphisms, proving vanishing of higher Ext groups and giving explicit formulas for $\operatorname{Ext}^{0}$ and $\operatorname{Ext}^{1}$. The composition rules are formulated using Hadamard products and left/right braided compositions, providing a fully computable, braid-aware calculus for morphisms. The framework connects Legendrian microlocal theory with braid varieties and cluster-like structures, enabling applications to exact Lagrangian fillings and related invariants. Overall, the work delivers a detailed, computable invariant for a broad class of Legendrian links via microlocal sheaf theory, with potential implications for Lagrangian-fillings, cluster algebras, and symplectic topology.

Abstract

For any Legendrian link in $\displaystyle \mathbb{R}^{3}$ given by the rainbow closure of a positive braid word, we develop an explicit and computable description of a Legendrian isotopy invariant associated with it, namely the cohomological category of compactly supported, microlocal rank-one sheaves with singular support on the Legendrian link. In particular, we parametrize the objects of the category by points of a braid variety, and for any pair of objects, we provide a linear map that algebraically characterizes their possible non-trivial graded morphism spaces. In addition, we provide combinatorial rules governing the compositions of graded morphisms in the category under consideration. Finally, we present several applications of our results, highlighting the structural features captured by the categorical invariant of interest.

Figures (21)

• Figure 1: Front projection $\Pi_{x,z}(\Lambda(\beta))$ of the Legendrian link $\Lambda(\beta)$.
• Figure 2: Braid diagram on three strands of the positive braid word $\sigma_{1}\sigma_{2}\in\mathrm{Br}^{+}_{3}$.
• Figure 3: Front projection $\Pi_{x,z}(\Lambda(e_{n}))\subset \mathbb{R}^{2}$ of the Legendrian unlink $\Lambda(e_{n})\subset(\mathbb{R}^{3}, \xi_{\mathrm{std}})$ on $n$ strands.
• Figure 8: The front projection $\Pi_{x,z}(\Lambda(e_n)) \subset \mathbb{R}^2$ of the Legendrian link $\Lambda(e_n) \subset (\mathbb{R}^3, \xi_{\mathrm{std}})$, along with the finite open cover $\mathcal{U}_{\Lambda(e_n)} = \{ U_0, U_{\mathrm{B}}, U_{\mathrm{T}} \}$ of $\mathbb{R}^2$. The regions $U_0$, $U_{\mathrm{B}}$, and $U_{\mathrm{T}}$ are depicted in purple, pink, and green, respectively.
• Figure 9: An object $\mathscr{F}$ of the category $H^{\bullet}(\mathcal{S}h_{1}(\Lambda(e_{n}), \mathbb{K})_{0})$ on the region $U_{\mathrm{T}}$.

Theorems & Definitions (69)

• Definition 1.1: Cohomological Category for Rainbow Closures
• Definition 1.2: A-type Braid Variety
• Theorem 1.3: Algebraic Characterization of the Objects, STZ1CGGS1
• Theorem 1.4: Hereditary-Type Property, CNS1
• Definition 1.5: $\delta$--map
• Theorem 1: Algebraic Description of the Lower-Degree Morphism Spaces
• Definition 1.6: Braided Compositions
• Theorem 2: Algebraic Description of the Graded Composition
• Definition 2.7
• Remark 2.8