Generating family invariants for Legendrian links of unknots

Abstract

Theory is developed for linear-quadratic at infinity generating families for Legendrian knots in ‫ޒ‬ 3 . It is shown that the unknot with maximal Thurston-Bennequin invariant of 1 has a unique linear-quadratic at infinity generating family, up to fiber-preserving diffeomorphism and stabilization. From this, invariant generating family polynomials are constructed for 2-component Legendrian links where each component is a maximal unknot. Techniques are developed to compute these polynomials, and computations are done for two families of Legendrian links: rational links and twist links. The polynomials allow one to show that some topologically equivalent links with the same classical invariants are not Legendrian equivalent. It is also shown that for these families of links the generating family polynomials agree with the polynomials arising from a linearization of the differential graded algebra associated to the links. 53D10; 57M25

Generating families and Legendrian contact homology in the standard contact space

“…The main results of this paper give an interpretation of a linearized version of the Chekanov and Eliashberg invariant in terms of generating families. This strengthens previously known links between the two classes of invariants [6,7,9,10,17,19]. The Chekanov-Eliashberg DGA of a Legendrian knot and even its homology may be infinite dimensional.…”

“…The differential is defined by counting immersed (or holomorphic) disks. Generating families have provided a second source of non-classical invariants in the work of Traynor and her collaborators [9,17,22] and Chekanov and Pushkar [3]. The main results of this paper give an interpretation of a linearized version of the Chekanov and Eliashberg invariant in terms of generating families.…”

“…Given a 1-parameter family of functions F : R n × R → R and f t = F (·, t), the fiberwise critical set is immersed (under a transversality assumption) into the standard contact space as a Legendrian submanifold , and F is called a generating family for . Traynor [22] introduced a homological invariant obtained from generating families for a class of 2-component links in the solid torus and (with Jordan [9]) in R 3 . A variation † of the Traynor-Jordan invariant, which applies without additional assumptions on the knot type, is the following.…”

Contact Homology, Capacity and Non-Squeezing in \mathbb{R}^{2n}\times S^{1} via Generating Functions

“…They provide a powerful tool in symplectic and contact topology, with important applications also to many of the central problems of these subjects (see for instance Chaperon [4,5], Laudenbach and Sikorav [29], Sikorav [32,33], Givental [22,23], Viterbo [41,39], Traynor [37,38], Théret [34,36,35], Chekanov [6], Eliashberg and Gromov [15], Bhupal [1, 2], Milinković [31], Chekanov and Pushkar [8], Ferrand and Pushkar [18], Jordan and Traynor [28], Colin, Ferrand and Pushkar [12], Chernov and Nemirovski [9,10], Eiseman, Lima, Sabloff and Traynor [13], Fuchs and Rutherford [19]). In particular, Viterbo [41] applied Morse-theoretical methods to the generating function of a Lagrangian submanifold L of the cotangent bundle of a closed manifold B to define invariants c(u, L) ∈ R for any u ∈ H * (B).…”

Obstructions to the Existence and Squeezing of Lagrangian Cobordisms

“…Based on the definitions of the capacities, we define: [10] for more information on generating family homology for Legendrian knots and links. In fact, since L is exact, it lifts to a Legendrian cobordism between the lifts of π(L a ) and π(L b ).…”