Satellites of Legendrian Knots and Representations of the Chekanov–eliashberg Algebra
نویسندگان
چکیده
We study satellites of Legendrian knots in R and their relation to the Chekanov–Eliashberg differential graded algebra of the knot. In particular, we generalize the well-known correspondence between rulings of a Legendrian knot in R and augmentations of its DGA by showing that the DGA has finite-dimensional representations if and only if there exist certain rulings of satellites of the knot. We derive several consequences of this result, notably that the question of existence of ungraded finite-dimensional representations for the DGA of a Legendrian knot depends only on the topological type and Thurston– Bennequin number of the knot.
منابع مشابه
Invariants of Legendrian Knots and Decompositions of Front Diagrams
The authors prove that the sufficient condition for the existence of an augmentation of the Chekanov–Eliashberg differential algebra of a Legendrian knot, which is contained in a recent work of the first author, is also necessary. As a by-product, the authors describe an algorithm for calculating Chekanov–Eliashberg invariants in terms of the front diagram of a Legendrian knot. 2000 Math. Subj....
متن کاملThe Legendrian Satellite Construction
The spaces R3 and S1 ×R2 both have a standard contact structure given by the kernel of the 1form dz−y dx, where we view the solid torus S1×R2 as R3 modulo the relation (x, y, z) ∼ (x+1, y, z). We will assume that the reader is familiar with some basic concepts in Legendrian knot theory, such as front projections, the Thurston-Bennequin number, and the rotation number; see, e.g., [8], which we w...
متن کاملStudying Legendrian Invariants with FrontLeg
The Chekanov-Eliashberg DGA is a powerful tool for helping us to decide if two Legendrian knots are Legendrian isotopic. However, computing it can be an extremely tedious task. FrontLeg is a program created to automate these calculations. In this paper, we examine the algorithm for constructing the DGA and its Poincaré polynomials. We also describe the design, structure and functionalities of F...
متن کاملClassification of Legendrian Knots and Links
The aim of this paper is to use computer program to generate and classify Legendrian knots and links. Modifying the algorithm in [11], we write a program in Java to generate all grid diagrams of up to size 10. Since classification of Legendrian links up to Legendrian isotopy is equivalent to grid diagrams modulo a set of Cromwell moves including translation, commutation and X:NE,X:SW (de)stabil...
متن کاملChekanov-eliashberg Invariants and Transverse Approximations of Legendrian Knots
In this article, we consider Legendrian and transverse knots in the standard contact space, that is in R3 with the contact structure globally given by the 1-form α = ydx− dz. It is well-known that a little push of an oriented Legendrian knot Γ in the direction of its positive normal within the contact structure changes it into a transverse knot Γ+, whose natural orientation, as determined by α,...
متن کامل