# Geometric topology – Is there a known relationship between sutured contact homology and Legendrian contact homology?

On the one hand, the construction of Colin-Ghiggini-Honda-Hutchings (https://arxiv.org/abs/1004.2942) provides an invariant of a Legendrian $$L$$ in a closed contact collector $$(M, xi)$$ via sutured contact homology of $$M setminus N (L)$$. Moreover, Legendrian contact homology in all its forms (DGA Chekanov-Eliashberg, knot contact homology, …) provides another set of invariants for the Legendrian submanifolds.

As indicated in the document above (section 7), there are some differences between the two, because LCH disappears under stabilization whereas $$HC (M, xi, L)$$ does not. Moreover, the constructions seem to count different things, because the sutured contact homology differential counts pseudoholomorphic curves avoiding $$mathbb {R} times partial (M setminus N (L))$$ while the differential in LCH counts the curves with the limit in $$mathbb {R} times L$$.

Assuming that Legendrian contact homology can be defined for Legendrians in closed varieties (I believe it is currently only defined for Legendrians in spaces of the form $$P times mathbb {R}$$ for $$P$$ an exact symplectic variety, according to the work of Ekholm-Etnyre-Sullivan in https://arxiv.org/abs/math/0505451), is there a reason for it? expect a relationship between sutured contact homology of a Legendrian complement and the Legendrian contact homology?