  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.
