Invariants of Legendrian Knots in Thickened Convex Surfaces

Recently, Zijian Rong and I defined a new differential graded algebra associated to a Legendrian knot Λ in a thickened convex surface Σ×ℝ. The algebra is defined combinatorially and in the same spirit as the Chekanov-Eliashberg DGA for Legendrians in ℝ3, but incorporates the data of the dividing set Γ𝑜𝑓Σ. The algebra is generated by (countably many) Reeb chords of Λ, and its differential counts certain immersed polygons in the projection 𝜋: Σ×ℝ→Σ×{0} with boundary on 𝜋(Λ)∪Γ. In this talk, I will outline our construction and sketch the proofs of our main results, which establish well-definedness and invariance of the homology of the DGA under Legendrian isotopy. As an application, I will describe how it can be used to distinguish knots in thickened convex surfaces that are not distinguishable by their classical invariants. Time permitting, I will comment on some future directions.