Generating families and Legendrian contact homology in the standard contact space

TL;DR

This paper establishes a connection between generating families and Legendrian contact homology in standard contact space, providing new insights and explicit constructions that relate augmentations, normal rulings, and homology theories.

Contribution

It demonstrates that generating families induce augmentations of the Chekanov-Eliashberg DGA, linking linearized contact homology to singular homology, and offers a new approach to relate augmentations with normal rulings.

Findings

Linearized contact homology is isomorphic to singular homology from generating families.

Sabloff's duality for LCH can be interpreted as Alexander duality.

Explicit construction of generating families for front diagrams with graded normal rulings.

Abstract

We show that if a Legendrian knot in standard contact ${\bb R}^3$ possesses a generating family then there exists an augmentation of the Chekanov-Eliashberg DGA so that the associated linearized contact homology (LCH) is isomorphic to singular homology groups arising from the generating family. In this setting we show Sabloff's duality result for LCH may be viewed as Alexander duality. In addition, we provide an explicit construction of a generating family for a front diagram with graded normal ruling and give a new approach to augmentation $\Rightarrow$ normal ruling.