# A Chekanov-Eliashberg algebra for Legendrian graphs

## TL;DR

This paper introduces a new combinatorial differential graded algebra invariant for Legendrian graphs and tangles in contact Euclidean space, extending Legendrian contact homology to include vertices.

## Contribution

It defines a novel algebraic invariant for Legendrian graphs that incorporates vertices, with a generalized notion of equivalence and a van Kampen type theorem.

## Key findings

- The invariant distinguishes Legendrian graphs with vertices.
- The construction recovers known Legendrian link algebras via vertex operations.
- A van Kampen type theorem for the algebra is established.

## Abstract

We define a differential graded algebra for Legendrian graphs and tangles in the standard contact Euclidean three space. This invariant is defined combinatorially by using ideas from Legendrian contact homology. The construction is distinguished from other versions of Legendrian contact algebra by the vertices of Legendrian graphs. A set of countably many generators and a generalized notion of equivalence are introduced for invariance. We show a van Kampen type theorem for the differential graded algebras under the tangle replacement. Our construction recovers many known algebraic constructions of Legendrian links via suitable operations at the vertices.