The Chekanov-Eliashberg dga, or Legendrian contact homology, was the first modern invariant of Legendrian knots in $\mathbb{R}^3$. This talk is a progress report on a project to give a stable homotopy refinement of Legendrian contact homology, induing operations like Steenrod squares on linearized Legendrian contact homology. In general, contact homology is defined by counting psuedoholomorphic curves, but in this case those counts reduce to the Riemann Mapping Theorem and the invariant is described purely combinatorially, from an appropriate knot diagram, and our refinement has a similarly combinatorial flavor. The talk will start at the beginning, briefly recalling the basics of Legendrian knot theory and the construction of Legendrian contact homology, and then outline our program to refine it, sketch its status, and describe some examples.

This is joint with Lenhard Ng and Sucharit Sarkar.