A Hodge-theoretic study of augmentation varieties associated to Legendrian knots/tangles
- Author(s): Su, Tao
- Advisor(s): Shende, Vivek;
- Borcherds, Richard
- et al.
In this article, we give a tangle approach in the study of Legendrian knots in the standard
contact three-space. On the one hand, we define and construct Legenrian isotopy invariants
including ruling polynomials and Legendrian contact homology differential graded algebras
(LCH DGAs) for Legendrian tangles, generalizing those of Legendrian knots. Ruling polynomials
are the Legendrian analogues of Jones polynomials in topological knot theory, in the
sense that they satisfy the composition axiom.
On the other hand, we study certain aspects of the Hodge theory of the “representation
varieties (of rank 1)” of the LCH DGAs, called augmentation varieties, associated to Legendrian
tangles. The augmentation variety (with fixed boundary conditions), hence its mixed
Hodge structure on the compactly supported cohomology, is a Legendrian isotopy invariant
up to a normalization. This gives a generalization of ruling polynomials in the following
sense: the point-counting/weight (or E-) polynomial of the variety, up to a normalized factor,
is the ruling polynomial. This tangle approach in particular provides a generalization
and a more natural proof to the previous known results of M.Henry and D.Rutherford. It
also leads naturally to a ruling decomposition of this variety, which then induces a spectral
sequence converging to the MHS. As some applications, we show that the variety is of Hodge-Tate
type, show a vanishing result on its cohomology, and provide an example-computation
of the MHSs.