Computable Sheaf Invariants for Legendrian Rainbow Closures

TL;DR

This paper introduces a new computable Legendrian isotopy invariant for rainbow closures of positive braids, using sheaf theory to encode geometric and algebraic information about Legendrian links.

Contribution

It provides an explicit sheaf-theoretic description of Legendrian invariants for rainbow closures, including parametrization and morphism maps, advancing the computational tools in Legendrian knot theory.

Findings

Parametrization of objects by braid variety points

Linear algebraic characterization of morphisms

Combinatorial rules for morphism compositions

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.