| Summary: | <p> 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. </p><p> 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.</p><p>
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