Invariants of Legendrian knots in thickened convex surfaces

We define a differential graded algebra associated to Legendrian knots in thickened convex surfaces $\Sigma\times \mathbb{R}$. The algebra is defined in the same spirit as the Chekanov-Eliashberg DGA for Legendrians in $\mathbb{R}^3$, but makes use of the data of the dividing set $\Gamma$ of $\Sigma$. The algebra is generated by countably many Reeb chords of the Legendrian $\Lambda$, and its differential counts certain immersed polygons in the projection $\pi:\Sigma\times \mathbb{R}\to \Sigma\times \{0\}$ with boundary on $\pi(\Lambda)\cup \Gamma$. We show that the differential squares to zero and that the stable tame isomorphism type of the DGA is invariant under Legendrian isotopy. Finally, we compute several examples and use the invariant to distinguish Legendrian knots in thickened convex surfaces that cannot be distinguished by the classical invariants.