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A Hodge-theoretic study of augmentation varieties associated to Legendrian knots/tangles


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.

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