Augmentations and ruling polynomials for Legendrian graphs

TL;DR

This paper establishes an equivalence between augmentation numbers and ruling polynomials as invariants of Legendrian graphs, connecting algebraic and combinatorial approaches in Legendrian topology.

Contribution

It demonstrates the equivalence of augmentation numbers and ruling polynomials for Legendrian graphs, bridging algebraic and combinatorial invariants.

Findings

Augmentation number equals ruling polynomial for Legendrian graphs.

Provides a combinatorial interpretation of augmentation invariants.

Connects differential graded algebra invariants with front projection decompositions.

Abstract

In this article, associated to a (bordered) Legendrian graph, we study and show the equivalence between two Legendrian isotopy invariants: augmentation number via point-counting over a finite field, for the augmentation variety of the associated Chekanov-Eliashberg differential graded algebra, and ruling polynomial via combinatorics of the decompositions of the associated front projection.