Algebraic Weaves and Braid Varieties

TL;DR

This paper explores the geometric and algebraic structures of braid varieties, introducing new stratifications, symplectic structures, and diagrammatic calculus to deepen understanding of their properties and relations to Legendrian links.

Contribution

It develops a diagrammatic calculus for braid varieties, studies their stratifications, and relates these to symplectic structures and Lagrangian fillings, offering new geometric insights.

Findings

Maximal charts are exponential Darboux charts.

Stratifications relate to Lagrangian fillings.

Holomorphic symplectic structures are established on quotients.

Abstract

In this manuscript we study braid varieties, a class of affine algebraic varieties associated to positive braids. Several geometric constructions are presented, including certain torus actions on braid varieties and holomorphic symplectic structures on their respective quotients. We also develop a diagrammatic calculus for correspondences between braid varieties and use these correspondences to obtain interesting stratifications of braid varieties and their quotients. It is shown that the maximal charts of these stratifications are exponential Darboux charts for the holomorphic symplectic structures, and we relate these strata to exact Lagrangian fillings of Legendrian links.