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On the Topology of Hypocycloid Curves

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On the Topology of Hypocycloid Curves Física Teórica, Julio Abad, 1–16 (2008) ON THE TOPOLOGY OF HYPOCYCLOIDS Enrique Artal Bartolo∗ and José Ignacio Cogolludo Agustíny Departamento de Matemáticas, Facultad de Ciencias, IUMA Universidad de Zaragoza, 50009 Zaragoza, Spain Abstract. Algebraic geometry has many connections with physics: string theory, enu- merative geometry, and mirror symmetry, among others. In particular, within the topo- logical study of algebraic varieties physicists focus on aspects involving symmetry and non-commutativity. In this paper, we study a family of classical algebraic curves, the hypocycloids, which have links to physics via the bifurcation theory. The topology of some of these curves plays an important role in string theory [3] and also appears in Zariski’s foundational work [9]. We compute the fundamental groups of some of these curves and show that they are in fact Artin groups. Keywords: hypocycloid curve, cuspidal points, fundamental group. PACS classification: 02.40.-k; 02.40.Xx; 02.40.Re . 1. Introduction Hypocycloid curves have been studied since the Renaissance (apparently Dürer in 1525 de- scribed epitrochoids in general and then Roemer in 1674 and Bernoulli in 1691 focused on some particular hypocycloids, like the astroid, see [5]). Hypocycloids are described as the roulette traced by a point P attached to a circumference S of radius r rolling about the inside r 1 of a fixed circle C of radius R, such that 0 < ρ = R < 2 (see Figure 1). If the ratio ρ is rational, an algebraic curve is obtained. The simplest (non-trivial) hypocycloid is called the deltoid or the Steiner curve and has a history of its own both as a real and complex curve. S C r P R Figure 1: Hypocycloid ∗[email protected] [email protected] 2 E. Artal and J.I. Cogolludo Hypocycloids first appeared as trajectories of motions or integral solutions of vector fields, describing physical phenomena. Modern physics also finds these objects useful. For instance, in the context of superstring compactifications of Calabi-Yau threefolds, certain Picard-Fuchs equations arise naturally. The monodromy group of such equations is the target duality group acting on the moduli of string theory and can be computed as the fundamental group of the complement of the bifurcation locus in a deformation space. Using the celebrated Lefschetz-Zariski theorems of hyperplane sections (in the homo- topy setting), these monodromy groups can be recovered in the context of complements of complex algebraic projective curves. For this reason, braid monodromies and fundamental groups of complements of plane curves have been intensively studied not only by by mathematicians, but also by physicists in the past decades. Our purpose is to investigate the topology of the complement of some of those interesting hypocycloids using their symmetries and the structure of their affine and projective singular- ities in a very effective way. In order to do so, we need to introduce Zariski-van Kampen method [9, 7], braid monodromies and Chebyshev polynomials and exploit their properties. 2. First properties of complex hypocycloids Let us construct a parametrization of a hypocycloid as a real curve. Since ρ is a positive ` rational number, it admits the irreducible form ρ := N , where ` and N are coprime positive 1 integers. Also note that ρ and 1−ρ define the same curve, hence ρ 2 (0; 2 ), i.e., k := N−` > `, R will be assumed. For simplicity, the external circle C can be assumed to have radius 1. If Ck;` denotes the real hypocycloid given by k and `, it is not difficult to prove that ` cos kθ + k cos `θ ` sin kθ − k sin `θ X (θ) = ; Y (θ) = (1) k;` N k;` N R provides a parametrization of Ck;`. This parametrization is useful for drawing the hypocycloid but a rational one is preferred. Given n 2 N we denote by Tn, Un, and Wn the Chebyshev polynomials defined by ! 1 1 cos nθ = T (cos θ); sin (n + 1)θ = sin θ U (cos θ); sin n + θ = sin θ W (cos θ): (2) n n 2 2 n Let us recall that Tn, Un, and Wn have degree n and they have zeroes at: (2r−1) π Tn(x) = 0 , x = cos 2n r = 1; :::; n; r π Un(x) = 0 , x = cos r = 1; :::; n; (3) n+1 r π Wn(x) = 0 , x = cos 1 r = 1; :::; n: n+ 2 The following rational parametrization is obtained: 1−t2 1−t2 Pk;` 2 2t Qk;` 2 x (t):= 1+t ; y (t):= 1+t ; (4) k;` N k;` N(1 + t2) On the topology of hypocycloids 3 where Pk;`(x):= `Tk (x) + kT` (x), and Qk;`(t):= `Uk−1 (x) − kU`−1 (x). p If the parameter t is allowed to run along the complex numbers outside {± −1g one obtains a complex plane curve, which will be called the complex hypocycloid, or simply 2 R 2 hypocycloid for short, and denoted by Ck;` ⊂ C . Note that Ck;` ⊂ Ck;` \ R . Moreover, let us recall that any affine complex curve in C2 defined by rational parametric equations t 7! p1(t) ; p2(t) can be embedded in P2 := CP2, the complex projective plane, p3(t) p3(t) by homogenizing its parametric equations and removing denominators. This way, the pa- rameter space becomes P1 and the new projective parametric equations become [t : s] 7! d−d1 t d−d2 t d−d3 t [s p1 s : s p2 s : s p3 s ), where di := deg pi(t), and d := maxfd1; d2; d3g. The ¯ 2 complexp projective hypocycloid will be denoted by Ck;` ⊂ P . Note that the former parame- ters {± −1g can be interpreted as the points at infinity of the complex hypocycloid. Proposition 2.1. The complex projective hypocycloid C¯k;` is a rational curve of degree 2k with the following properties: (i) The curve Ck;` is invariant by an action of the dihedral group D2N . (ii) The singular points of Ck;` are only ordinary nodes and ordinary cusps arranged as follows: N cusps, N(` − 1) (real) nodes, and N(k − ` − 1) (non-real) nodes. (iii) The intersection with the line at infinity consists of two points with local equations uk − vk−` = 0 tangent to the line at infinity (these points are singular if k − ` > 1). (iv) The parametrization given in (4) is an immersion (outside the N cusps and eventually the points at infinity if k − ` > 1). 2π Proof. First, let us show that the rotation of angle N and the reflection with respect to the horizontal axis given by the equation fy = 0g globally fix Ck;`. These two symmetries generate a dihedral group of order 2N, denoted by D2N . The reflection is an immediate consequence of the fact that the first coordinate of the affine parametrization is even xk;`(−t) = xk;`(t), while the second coordinate is odd yk;`(−t) = −yk;`(t), see (4). In order to visualize the rotation, it is more convenient to use trigonometric notation. One can check that 0 2π 1 0 2πk 2πk 1 0 1 BXk;` θ + C cos − sin X (θ) B N C B N N C B k;` C B C = B C B C ; B C B C B C @B 2π AC @sin 2πk cos 2πk A @Y (θ)A Yk;` θ + N N N k;` 2πk Since this is a rotation of degree N and gcd(k; N) = 1 then part (i) follows. Note that h 2 2 ki '([t : s]:= pk;`(t; s) : 2tsqk;`(t; s): N(s + t ) (5) is a parametrization of the projective hypocycloid C¯k;`, where ! ! s2 − t2 s2 − t2 p (t; s):= (s2 + t2)kP ; q (t; s):= (s2 + t2)k−1Q : k;` k;` s2 + t2 k;` k;` s2 + t2 Note that pk;` is homogeneous of degree k and qk;` is homogeneous of degree k − 1. Sincep the k−1 k−2 leading coefficient of Pk;` is 2 ` and the one of Qk;` is 2 `, see [8], then '([1 : ± −1]) = 4 E. Artal and J.I. Cogolludo p [1 : ± −1 : 0] and hence this parametrization induces a well-defined surjective map from 1 P to C¯k;`. Hence, outside a finite number of points (those where the parametrization is not 1 injective) C¯k;` is isomorphic to P , which implies that C¯k;` is a rational curve. Also, its degree corresponds with the degree of any of its parametric equations, namely 2k. It is a straightforward computation that 2 d'1 4k`ts d'2 2k`s = − (U − + U − ) ; and = (T − T ) : dt N(s2 + t2)2 k 1 ` 1 dt N(s2 + t2) k ` One should consider two different cases: • If N is even, then one can use the following two formulas (see [8]): Uk−1 + U`−1 = 2T k−` U k+` and (Tk(x) − T`(x)) + x(Uk−1(x) + U`−1(x)) = (Uk(x) + U`−2(x)). Therefore: 2 2 −1 2 d'1 8k`ts = − T k−` U k+` −1; dt N(s2 + t2)2 2 2 (6) d'2 2 2 d'1 8k`ts 2ts − (s − t ) = T k−` +1U k+` −1: dt dt N(s2 + t2) 2 2 2 2 0 d'1 0 d'2 s −t Thus the common zeroes to ' := and ' := are given by tsU k+` 2 2 = 0, 1 dt 2 dt 2 −1 s +t that is, ( ! ) s2 − t2 2rπ k + ` (t; s) j t = 0; s = 0; or = cos ; r = 1; :::; − 1 : s2 + t2 k + ` 2 k+` This makes a total of 2 2 − 1 + 2 = N singularities. Using the previous equations 0 0 one can check that the order of t in ('1;'2) is (1,2) and consequently such a singularity 2 3 is an ordinary cusp of equation y − x whose tangent is the line L0 := fy = 0g.
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