Seminar

Chekanov-Eliashberg or Legendrian Contact Homology (LCH) DGAs are well-known and powerful Legendrian isotopy invariants for Legendrian knots in the three-space, which also admit a combinatorial description. There's a similar story for Legendrian tangles, which are morally the pieces obtained by cutting the front diagrams of Legendrian knots along vertical lines. Just like 'fundamental groups', the LCH DGAs satisfy a van-Kampen/co-sheaf property, suggesting that the study of their representation varieties (augmentation varieties) behave like character varieties. 

In this talk, we study some aspects of the Hodge theory of the (rank 1) augmentation varieties, which up to a normalization, give rise to Legendrian isotopy invariants. Closed formulas for the point-counting/weight polynomials, are given by the ruling polynomials, the Legendrian analogue of Jones polynomials. The tangle approach also allow us to derive some information about the mixed Hodge structure (MHS) on the compactly supported cohomology. In particular, there's a spectral sequence converging to MHS, and the augmentation varieties are of Hodge-Tate type. Time permitting, I will also try to compute one example.