A Legendrian Conspectus

This is a light-hearted friendly introduction to the theory of Legendrian knots intended for non-experts in the field. This includes ice-skating and an application to complex geometry. In this note I would like to give a view on contact and symplectic topology. This area of mathematics lies at the intersection of several different branches: complex algebraic geometry [7, 8], geometric topology [10, 12] and differential equations [1, 11] amongst them. To exemplify this I will focus on one of these relations, between the complex solutions of a polynomial equation and Legendrian knots, explored in the article [2]. We will work with the following example: X = {(x, y, z) : xyz + x+ z = 1} ⊆ C, i.e. we define X to be the space of triples of complex numbers (x, y, z) which verify the polynomial equation P (x, y, z) = 1, where P (x, y, z) = xyz + x + z. I would like to explain to you why the symplectic geometry of this space is completely contained in the Figure 1. Figure 1. The contact topologist take on X = {(x, y, z) : xyz + x+ z = 1} ⊆ C. But first, there are two main questions I would like to address: A. Why would we ever care about the symplectic geometry of X ? B. What does the colourful Figure 1 represent, and why is it useful ? In short, the former question can be answered by saying that symplectic topology is the aspect of the geometry of X that is robust with respect to small polynomial perturbations. In addition, lots of the geometric properties of X are already captured by symplectic invariants [11, 13]. The answer to the latter question is even more exciting: Figure 1 represents a Legendrian knot, that is, a knot in 3–space which is constrained to be tangent to certain 2–planes. Having such Figure 1 is genuinely useful because just by pictorial manipulation we can solve many questions about the geometry of the space X. That is, we can translate questions about X to combinatorial questions about a drawing such as Figure 1, establish a set of combinatorial rules and play the game. The following two sections treat each of these questions in more detail and will hopefully convey both the flavour and a small part of the many benefits of the study of Legendrian knots. 1. Polynomial Equations From graph theory to differential equations, the ubiquity of polynomials is manifest in mathematics. For instance, let us suppose that we need to understand the solutions of the Painlevé I equation g′′(t) = 6g(t) + t,