On The Augmentation Categories of Positive Braid Closures

TL;DR

This paper demonstrates that 0-resolution of crossings in positive braid Legendrian closures induces a cohomologically faithful $A_infty$ functor, and computes the bilinearized Legendrian contact cohomology for these knots.

Contribution

It introduces a new functorial relationship between crossing resolutions and augmentation categories in Legendrian knot theory.

Findings

0-resolution induces a cohomologically faithful $A_infty$ functor

Computed bilinearized Legendrian contact cohomology for these knots

Established a link between crossing resolutions and contact homology invariants

Abstract

In this paper we show that 0-resolution of a crossing in the Legendrian closure of a positive braid induces a cohomologically faithful $A_\infty$ functor on augmentation categories. In particular, we compute the bilinearized Legendrian contact cohomology of these knots for augmentations induced by 0-resolution.