A LEGENDRIAN CONSPECTUS

BY ROGER CASALS

Abstract. 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 ofmathematics 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 polynomialequation and Legendrian knots, explored in the article [2]. We will work with the following example:

X = {(x, y, z) : xyz + x+ z = 1} ⊆ C3,

i.e. we define X to be the space of triples of complex numbers (x, y, z) which verify the polynomialequation P (x, y, z) = 1, where P (x, y, z) = xyz + x + z. I would like to explain to you why thesymplectic 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} ⊆ C3.

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 ofthe geometry of X that is robust with respect to small polynomial perturbations. In addition, lots ofthe 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 Figure1 is genuinely useful because just by pictorial manipulation we can solve many questions about thegeometry of the space X. That is, we can translate questions about X to combinatorial questionsabout 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 boththe 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 Painleve I equation

g′′(t) = 6g2(t) + t,1

2 BY ROGER CASALS

which, for instance, features in particle physics via the Yang–Mills functional. Then given a solutionof the differential equation we may want to deform it with some parameters (x, y, z) in a way that itstill has certain properties, in our case, the same monodromy. Then the parameters must satisfy thepolynomial equation

xyz + x+ z = 1.

For reference, this is proven in the article [14]. In consequence, it is crucial to understand the solutionsto this polynomial equation in order to properly understand how solutions of the Painleve I behave.

But wait, algebraic geometers already study polynomials: why would you use symplectic topology ?

Because symplectic topology captures the geometry of polynomial equations even up to perturbations!

Figure 2. The algebraic geometer sees Carolina, Cobalt and Duke blues, the sym-plectic geometer sees one blue: a much more stable viewpoint.

Example 1.1 (Elliptic Curves). Let us for instance consider the family of polynomial equationsy2 = x(x − 1)(x − λ) whose solutions per each value of λ ∈ C \ {0, 1} are depicted in Figure 3. Thealgebraic geometry of the space of solutions depends strongly on the value λ whereas the symplectictopology of all these spaces is the same. �

This robustness is extremely useful: models obtained from experimental data carry a measuringindeterminacy and oftentimes even precise theoretical models include parameters. Choosing differentvalues of these parameters yield different behaviours1, however if we just vary the parameters in asmall interval, the behaviour will generically be the same. This then discretizes the study to finitelymany values of these parameters since we want to think of nearby values as giving equivalent solutions.

In conclusion, we would like to treat a polynomial equation and a small perturbation of it as beingthe same and understand the common geometric properties of their solutions. In our case, we mighthave a differential equation which is a perturbation of Painleve I and then the parameters (x, y, z)would be constrained to a perturbation such as

xyz + x+ z = 1 =⇒ xyz + 1.1x+ 0.9 · z − 0.01 · xy = 1.2.

We then would like to say that the space of solutions are equivalent and this is where algebraicgeometry is excessively demanding: equivalence in algebraic geometry requires a bijection given by apolynomial map which does in general not exist when the coefficients of a polynomial are perturbed.

In contrast, the more flexible viewpoint from symplectic geometry enriches our toolbox with a wealthof new techniques2 in order to solve problems and in particular understand the space X in depth.

2. Legendrian Knots

Let us suppose we find ourselves on an ice rink, once we have our skates on we can model our positionin the rink with three coordinates (x, y; θ) ∈ R× (−π/3, π/3) where (x, y) ∈ R are planar coordinates

1The Rayleigh number ρ in the Lorenz system is such an example: for the value ρ = 28 the system can becomechaotic whereas for ρ = 99.96 the system has knotted periodic orbits (the knot they form is the one in Figure 1)

2Here is an example close to my heart: very recently algebraic geometers [9] have managed to prove that the solutionsof xy2 + z2 + w2 = 1 and xy2 + z2 + w2 = 1 are not equivalent via a polynomial map. This is lots of effort, but with

the pictures as in Figure 1 one can immediately prove that they are not even symplectically equivalent!

A LEGENDRIAN CONSPECTUS 3

on the ice rink R and the coordinate θ ∈ (−π/3, π/3) measures the angle of the skate blade. Now wecome to the fundamental question on ice skating:

Q: Given any two positions P0 = (x0, y0, θ0) and P1 = (x1, y1, θ1) can we move from P0 to P1 ?

The experimental answer is not only fun, but also quite indisputable: it is possible to skate betweenany two points of the rink even if we constrain the direction in which the blade points at the beginningand the end3. However, it is relevant to point out that this is, rigorously speaking, not immediate:the blade imposes a constraint on the possible velocities from a given initial state (x0, y0, θ0).

Figure 3. Snoopy tracing an embedded Legendrian knot in the 3–dimensional con-figuration space Rink× (−π/3, π/3) which connects the point P0 to the point P1.

Indeed, if the blade is at an angle θ = θ0 we can either move forward in the direction

v1 = sin(θ0)∂x − cos(θ0)∂y

of the blade or veer the blade itself, which corresponds to the velocity v2 = ∂θ. In constrast, thereare forbidden directions: if the blade is pointing forward θ = 0 then moving to the right ∂x will oftenresult in an ankle sprain. In short, at each point the velocities are constrained to be on the tangent2–plane defined by the linear equation

cos(θ)dx+ sin(θ)dy = 0.

Near a point, a choice of a 2–plane field can have two behaviours: either they represent the tangentspaces to a set of disjoint embedded surfaces (a foliation) in the 3–space R× (−π/3, π/3) or they donot. The former case is not generic under small C1–perturbations, whereas the latter is. Now, thecrucial observation: if the equation we obtained for the ice skating question had the first behaviourthe answer to the question would be no ! Certainly, if we choose the two positions P0 and P1 to lie intwo different disjoint surfaces then it is impossible to conform to the mechanical constraints of the iceblade – which force us to remain in the same surface – and at the same time reach the other surface.

The conclusion of this discussion is that the equation cos(θ)dx+ sin(θ)dy = 0 defines a 2–plane fieldwhich is not integrable by surfaces, and the experimental evidence implies that between any two pointsthere exists a path in our configuration space R× (−π/3, π/3) which is tangent to those 2–planes allthe time and connects them. Let us give objects their proper names: a 2–plane field which cannotbe integrated is called a contact structure and a path which is tangent to such a 2–plane field iscalled a Legendrian knot. So good news: we have just learnt what contact topology and the theoryof Legendrian knots study.

3. From the Polynomial to the Knot

Let us connect the previous two concepts: the affine variety X and the Legendrian knot in Figure 1.We start with the space of complex solutions

X = {(x, y, z) : xyz + x+ z = 1} ⊆ C3,

3Given that this applicatin will be reviewed in Winter, feel free to go and try.

4 BY ROGER CASALS

which can be described with three degrees of freedom (x, y, z) constrained to one equation, and henceits complex dimension is 2 = 3− 1. In consequence, in real coordinates this is a 4–dimensional spaceand we aim at describing it. For a geometric topologist, a description is often a diagramatic andpictorial notion, let me give a simple example.

Example 3.1 (Thickened 2–sphere). Consider the set I = {(x, y, z) ∈ R3 : 1 ≤ ‖(x, y, z)‖ ≤ 2},which is a 3–dimensional annulus, i.e. an spherical shell. We can construct this shell I by startingwith a solid 3–ball D3 and attaching another 3–ball D2 × D1 along the boundary ∂D2 × D1. Thislooks like building an igloo on top of the first solid 3–ball, as depicted in Figure 4, and thus we candescribe the 3–space I by the (red) circle S1 in the boundary of ∂D3 where the igloo is attached.

Figure 4. The space I, in the middle, constructed by attaching an igloo to a solid3–ball. This is described by the red knot in ∂D3 indicating the attaching region. �

We can consider the 4–dimensional version of this example, we can construct a 4–dimensional spaceby attaching a 3–dimensional igloo D2×D2 along ∂D2×D2 and by following the rightmost picture inFigure 4 we can describe this by drawing the attaching circle ∂D2×{0} ⊆ ∂D2×D2 in the boundary3–sphere S3 = ∂D4 ⊆ D4. Thus, we obtain a knot in the 3–space S3.

Now comes the glorious part: if our initial 4–dimensional real space consists of the complex solutionsof a polynomial equation in three variables, then this knot is a Legendrian knot! Even better, allthe symplectic information of the space is contained in the knot as long as it is Legendrian, i.e. tangentto the 2–plane fields in the 3–sphere4.

Figure 5. The Legendrian knot in the domain of the projection R3 −→ R2 and itsimage into R3. The projection has cusps singularities because of the tangencies ofthe Legendrian knot.

Finally, Figure 5 pictorially explains the reason why we can draw a Legendrian knot as in Figure 1. Incoordinates, we can project our Legendrian knot from the three space (x, y, θ) to the two coordinates(y, θ) and this still recover all the information. Indeed, since the Legendrian knot is forced to betangent to the 2–plane field cos(θ)dx+ sin(θ)dy = 0, we can recover the x–coordinate by integrating:

dx

dy= − tan(θ).

4The contact structure in the 3–sphere is the one described in the ice–skating problem where we compactify itsconfiguration space to the 3–sphere S3, with a minor modification near infinity making the 2–planes match.

A LEGENDRIAN CONSPECTUS 5

The Figure 1 shows a Legendrian knot with many components, and the vertical walls (α, β, δ, γ)represent four worm–holes along which the knots can pass through5.

The recipe that inputs a polynomial equation and outputs a Legendrian knot can be found in [2].This is extremely useful from the research viewpoint because the study of Legendrian knots is quiteadvanced and we have further developed a diagramatic calculus to simplify pictures fast. In conse-quence we can prove new results about affine varieties by studying knots ! Many of these results canbe found in [2, 4], and hopefully many others will come soon.

References

[1] V.I. Arnol’d, Mathematical methods of classical mechanics, GTM 60, Springer-Verlag (1989).

[2] R. Casals, E. Murphy, Legendrian Fronts for Affine Varieties, arXiv:1610.06977.

[3] R. Casals, E. Murphy, Differential algebra of cubic graphs, in preparation.[4] R. Casals, E. Murphy, A frontal view on Lefschetz fibrations II, IAS Video Lectures (2016).

[5] R. Casals, D.M. Pancholi, F. Presas, Almost contact 5-manifolds are contact, Ann. of Math. 182 (2015), 429–490.

[6] R. Casals, J.L. Perez, A. del Pino, F. Presas, Existence h-principle for Engel structures, arXiv:

[7] K. Cieliebak, Y. Eliashberg, The topology of rat. and poly. convex dom. , Invent. Math. 199 (2015) 215–238.[8] S.K. Donaldson, Lefschetz pencils on symplectic manifolds, J. Differential Geom. 53 (1999), no. 2, 205–236.

[9] A. Dubouloz, D.R. Finston, On exotic affine 3-spheres, J. Algebraic Geometry, 23 (2014) 445–469.

[10] Y.M. Eliashberg, W.P. Thurston, Confoliations, University Lecture Series AMS, 13 (1998).[11] Y. Eliashberg, A. Givental, H. Hofer, Introduction to symplectic field theory, GAFA (2000) 560–673.

[12] P.B. Kronheimer, T.S. Mrowka, Witten’s conjecture and property P, Geom. Topol. 8 (2004), 295–310.

[13] P. Seidel, Fukaya categories and Picard-Lefschetz theory, Zurich Lectures Adv. Math. EMS (2008).[14] M. van der Put, M. Saito, Moduli spaces for linear differential equations and the Painleve equations, Ann. Inst.

Fourier (Grenoble) 59 (2009), no. 7, 2611–2667.

Massachusetts Institute of Technology, Department of Mathematics, 77 Mass. Ave. , Cambridge MA, USA,

University College of London, Department of Mathematics, Gordon St, London WC1H, United Kingdom

E-mail address: casals@mit.edu

5The reason for the worm–holes is that we must work in the connected sum #4(S1 × S2) instead of S3.