Invariants of Legendrian knots and decompositions of front diagrams

The authors prove that the sufficient condition for the existence of an augmentation of the Chekanov–Eliashberg differential algebra of a Legendrian knot, which is contained in a recent work of the first author, is also necessary. As a by-product, the authors describe an algorithm for calculating Chekanov–Eliashberg invariants in terms of the front diagram of a Legendrian knot.