This is joint work with Y. Pan that applies previous joint work with M. Sullivan. Let Λ ⊂ ℝ3 be a Legendrian knot with respect to the standard contact structure. The Legendrian contact homology (LCH) DG-algebra, 𝓐(Λ), of Λ is functorial for exact Lagrangian cobordisms in the symplectization of ℝ3, i.e., a cobordism L ⊂ Symp(ℝ3) from Λ - to Λ+ induces a DG-algebra map, fL : 𝓐(Λ+) → 𝓐(Λ -). In particular, if L is an exact Lagrangian filling (Λ - = ∅) the induced map is an augmentation ϵL : 𝓐(Λ+) → ℤ/2.
In this talk, I will discuss an extension of this construction to the case of immersed, exact Lagrangian cobordisms based on considering the Legendrian lift Σ of L. When L is an immersed, exact Lagrangian filling a choice of augmentation α for Σ produces an induced augmentation ϵ(L, α) for Λ+. Using the cellular formulation of LCH, we are able to show that any augmentation of Λ may be induced by such a filling.