Persistent Legendrian contact homology in $\mathbb{R}^3$

Abstract

This work applies the ideas of persistent homology to the problem of distinguishing Legendrian knots. We develop a persistent version of Legendrian contact homology by filtering the Chekanov-Eliashberg DGA using the action (height) functional. We present an algorithm for assigning heights to a Lagrangian diagram of a Legendrian knot, and we explain how each Legendrian Reidemeister move changes the height of generators of the DGA in a way that is predictable on the level of homology. More precisely, a Reidemeister move that changes an area patch of a Lagrangian diagram by {\delta} will induce a 2{\delta}-interleaving on the persistent Legendrian contact homology, computed before and after the Reidemeister move. Finally, we develop strong Morse inequalities for our persistent Legendrian contact homology.