You are currently browsing the tag archive for the ‘circle packing’ tag.

I am spending a few months in Göttingen as a Courant Distinguished Visiting Professor, and talking a bit to Laurent Bartholdi about rational functions — i.e. holomorphic maps from the Riemann sphere \widehat{\mathbb C} to itself. A rational function is determined (up to multiplication by a constant) by its zeroes and poles, and can therefore generically be put in the form f:z \to P(z)/Q(z) where P and Q are polynomials of degree d. If d=1 then f is invertible, and is called a fractional linear transformation (or, sometimes, a Mobius transformation). The critical points are the zeroes of P'Q-Q'P; note that this is a polynomial of degree \le 2d-2 (not 2d-1) and the images of these points under f are the critical values. Again, generically, there will be 2d-2 critical values; let’s call them V. Precomposing f with a fractional linear transformation will not change the set of critical values.

The map f cannot usually be recovered from V (even up to precomposition with a fractional linear transformation); one needs to specify some extra global topological information. If we let \overline{C} denote the preimage of V under f, and let C denote the subset consisting of critical points, then the restriction f:\widehat{\mathbb C} - \overline{C} \to \widehat{\mathbb C} - V is a covering map of degree d, and to specify the rational map we must specify both V and the topological data of this covering. Let’s assume for convenience that 0 is not a critical value. To specify the rational map is to give both V and a representation \rho:\pi_1(\widehat{\mathbb C} - V,0) \to S_d (here S_d denotes the group of permutations of the set \lbrace 1,2,\cdots,d\rbrace) which describes how the branches of f^{-1} are permuted by monodromy about V. Such a representation is not arbitrary, of course; first of all it must be irreducible (i.e. not conjugate into S_e \times S_{d-e} for any 1\le e \le d-1) so that the cover is connected. Second of all, the cover must be topologically a sphere. Let’s call the (branched) cover \Sigma for the moment, before we know what it is. The Riemann-Hurwitz formula lets one compute the Euler characteristic of \Sigma from the representation \rho. A nice presentation for \pi_1(\widehat{\mathbb C}-V,0) has generators e_i represented by small loops around the points v_i \in V, and the relation \prod_{i=1}^{|V|} e_i = 1. For each e_i define o_i to be the number of orbits of \rho(e_i) on the set \lbrace 1,2,\cdots,d\rbrace. Then

\chi(\Sigma) = d\chi(S^2) - \sum_i (d-o_i)

If each \rho(e_i) is a transposition (i.e. in the generic case), then o_i=d-1 and we recover the fact that |V|=2d-2.

This raises the following natural question:

Basic Question: Given a set of points V in the Riemann sphere, and an irreducible representation \rho:\pi_1(\widehat{\mathbb C} - V,0) \to S_d satisfying \sum_i (d-o_i) = 2d-2, what are the coefficients of the rational function z \to P(z)/Q(z) that they determine (up to precomposition by a fractional linear transformation)?

Read the rest of this entry »

Follow

Get every new post delivered to your Inbox.

Join 176 other followers