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

As pointed out in the above paper (Section 7) there are some differences between the two, as LCH vanishes under stabilizations where as $HC(M,\xi,L)$ does not. Moreover, the constructions seem to count different things as the differential in sutured contact homology counts pseudoholomorphic curves avoiding $\mathbb{R}\times\partial(M\setminus N(L))$ whereas the differential in LCH counts curves with boundary in $\mathbb{R}\times L$.

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

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    $\begingroup$ I am not sure about a known relationship, but you should expect a relation by shrinking the tubular neighborhood $N(L)$ to its core $L$. The analogy one dimension higher is to relate Lagrangian boundary conditions (of J-holomorphic curves in symplectic 4-manifolds) with asymptotic Reeb orbit conditions (of J-holomorphic curves in the complement of a Weinstein neighborhood $N$ of the Lagrangian) -- here there is a correspondence between Reeb orbits on $\partial N$ and geodesics on the Lagrangian, and this is documented in several papers. $\endgroup$ – Chris Gerig Dec 29 '18 at 21:07

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