Product Structures for Legendrian Contact Homology

TL;DR

This paper introduces new algebraic structures, including cup and Massey products and an A_infinity structure, on linearized Legendrian contact homology to capture nonlinear information while maintaining computational feasibility.

Contribution

It develops invariant algebraic structures on linearized LCH, enabling the detection of non-isotopic Legendrian knots and higher-order linearizations.

Findings

Identified infinite families of Legendrian knots not isotopic to their mirrors.

Reinterpreted duality theorem using cup product.

Recovered higher-order linearizations of LCH.

Abstract

Legendrian contact homology (LCH) and its associated differential graded algebra are powerful non-classical invariants of Legendrian knots. Linearization makes the LCH computationally tractable at the expense of discarding nonlinear (and noncommutative) information. To recover some of the nonlinear information while preserving computability, we introduce invariant cup and Massey products - and, more generally, an A_\infty structure - on the linearized LCH. We apply the products and A_\infty structure in three ways: to find infinite families of Legendrian knots that are not isotopic to their Legendrian mirrors, to reinterpret the duality theorem of the fourth author in terms of the cup product, and to recover higher-order linearizations of the LCH.