A contact structure on a $3$-manifold refines traditional knot theory by introducing a geometric criterion for knots. A knot whose tangent space sits within the two-plane field of the contact structure is called Legendrian. In recent decades, Legendrian knot theory has proven to be a useful tool in low-dimensional topology and contact geometry and a fruitful area of study in its own right. Chekanov and Eliashberg's independent work in the late 90's associated a differential graded algebra (DGA) to a Legendrian knot in the standard contact structure on $\mathbb{R}^3$. Numerous Legendrian knot invariants have since been derived from this DGA.

I will outline recent work that assigns a new differential graded algebra to a Legendrian knot in $\mathbb{R}^3$. The definition of the DGA is motivated by considering Morse-theoretic data from a generating family. A generating family $f_x$ for a Legendrian knot is a family of functions whose critical values encode a projection of the knot. The DGA is defined combinatorial using an algebraic analogue of a generating family first introduced by Pushkar. We will discuss the motivation and construction of this DGA and relationships between the new DGA and the Chekanov-Eliashberg DGA.