Extension of Chekanov-Eliashberg algebra using annuli

TL;DR

This paper introduces a new Legendrian knot invariant extending the Chekanov-Eliashberg algebra by incorporating annuli, providing a richer algebraic structure that captures additional geometric information.

Contribution

It defines a deformation of the Chekanov-Eliashberg algebra using annuli and demonstrates its invariance and computability from knot projections.

Findings

Invariant distinguishes knots with non-vanishing deformation

Invariant can be computed combinatorially from projections

Provides new insights into Legendrian knot invariants

Abstract

We define an SFT-type invariant for Legendrian knots in the standard contact $\mathbb{R}^3$. The invariant is a deformation of the Chekanov-Eliashberg differential graded algebra. The differential consists of a part that counts index zero $J$-holomorphic disks with up to two positive punctures, annuli with one positive puncture, and a string topological part. We describe the invariant and demonstrate its invariance combinatorially from the Lagrangian knot projection, and compute some simple examples where the deformation is non-vanishing.