# Decompositions of augmentation varieties via weaves and rulings

## TL;DR

This paper establishes a correspondence between different decompositions of braid and augmentation varieties, linking geometric, combinatorial, and microlocal perspectives, and provides methods to compute cluster variables from Legendrian links.

## Contribution

It proves the equivalence of multiple decompositions of augmentation and braid varieties and connects them through sheaf theory and Morse complexes, offering new computational tools.

## Key findings

- Decompositions of braid and augmentation varieties are shown to agree.
- The decompositions coincide with Deodhar and microlocal sheaf decompositions.
- Cluster variables can be computed from Legendrian links using Morse complex sequences.

## Abstract

The braid variety of a positive braid and the augmentation variety of a Legendrian link both admit decompositions coming from weaves and rulings, respectively. We prove that these decompositions agree under an isomorphism between the braid variety and the augmentation variety. We also prove that both decompositions coincide with a Deodhar decomposition and another decomposition coming from the microlocal theory of sheaves. Our proof relies on a detailed comparison between weaves and Morse complex sequences. Among other things, we show that the cluster variables of the maximal cluster torus of the augmentation variety can be computed from the Legendrian via Morse complex sequences.

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Source: https://tomesphere.com/paper/2508.20226