The 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 will use extensively. The problem of classifying Legendrian knots in standard contact R3 up to Legendrian isotopy has attracted much recent attention. In this note, we study one particular construction on Legendrian knots, the Legendrian satellite construction, which relates knots in R3 and in S1 × R2. This is the Legendrian analogue of the satellite construction in the smooth category, which glues a link in the solid torus S1 × R2 into a tubular neighborhood of a knot in R3 to produce a link in R3. We examine some consequences of Legendrian satellites for the Legendrian knot classification problem in both R3 and S1×R2; in particular, we recover previously-known results for knots in both spaces, and prove a new result in S1 × R2 (Proposition 2.11). The motivation for this work is that Legendrian satellites may provide nontrivial, nonclassical invariants of stabilized Legendrian knots in R3. Here we recall that there are two stabilization operators S± on Legendrian knots, decreasing tb by 1 and changing r by ±1, which replace a segment of the knot’s front projection by a zigzag, as shown in Figure 1. Understanding stabilized knots is an important open problem in Legendrian knot theory; it also has repercussions for the classification of transverse knots. It seems possible that satellites of stabilized knots may contain interesting information through the Chekanov-Eliashberg differential graded algebra invariant [2, 8], which is derived from contact homology [3]. We will show that, unfortunately, the DGAs of the simplest Legendrian satellites of stabilized knots do not encode any useful information. The computation used in the proof may be of interest as the first involved computation manipulating the DGA invariant directly, rather than using easier invariants such as Poincaré polynomials [2] or the characteristic algebra [8]. In any case, more complicated satellites may well give nonclassical invariants of stabilized knots, as has been suggested by Michatchev [7]. We define the construction in Section 2, and show how it immediately implies facts about solidtorus links, including some that could not be shown using any previously known techniques. In