Functorial LCH for immersed Lagrangian cobordisms

For $1$-dimensional Legendrian submanifolds of $1$-jet spaces, we extend the functorality contact homology DG-algebra (DGA) from embedded exact Lagrangian cobordisms, as in \cite{EHK}, to a class immersed cobordisms by considering their lifts conical cobordisms. To cobordism $\Sigma$ $\Lambda_-$ $\Lambda_+$, associate an DGA map, which is diagram $$\alg(\Lambda_+) \stackrel{f}{\rightarrow} \alg(\Sigma) \stackrel{i}{\hookleftarrow} \alg(\Lambda_-), $$ where $f$ map and $i$ inclusion map. This construction gives functor between suitably defined categories Legendrians with DGAs maps. In algebraic preliminary, consider analog mapping cylinder setting DG-algebras establish several its properties. As application give examples augmentations twist knots that can be induced filling single double point but cannot any orientable filling.