Seminar on low-dimensional mathematics
March 31, 2006

Sergei Anisov (Utrecht)
Combinatorics of cut loci in lens manifolds

Consider a triangulation of a regular $p$-gon by its diagonals.
Choose a number $q$, $0<q<p$, coprime with $p$. Rotate a $p$-gon
by $2\pi q/p$ (so that its $i$-th vertex $A_i$ takes the place
of $A_{i+q}$). What is the minimal (over all trianglulations)
number of diagonals that are not mapped to other diagonals of
that triangulation by the rotation? Denote it by $N(p,q)$. For
example, $N(65,8)=13$.

This question arises in the following way. Consider the lens
manifold $L_{p,q}$ with the standard metric inherited from the
covering 3-sphere. Consider the cut locus $C(x)$ for a point
$x\in L_{p,q}$. For some $x$, this cut locus contains a circle,
neighborhood of which locally looks as an open book with $p$
sheets (or like the white matter around the axis of an orange),
but globally has a monodromy determined by $q$. We study small
perturbations of $C(x)$ that are simple polyhedra. Then $N(p,q)$
is the minimal number of vertices of such a polyhedron.

A formula for $N(p,q)$ will be presented. The proof of the fact
that at least so many diagonals are destroyed by the rotation
is based on the study of the Euclid algorithm. An example showing
that the lower estimate for $N(p,q)$ is in fact sharp, is based
on the geometric properties of $L_{p,q}$ and is described in terms
of the so called Farey tesselation of the hyperbolic plane. For
example, $N(34,13)=5$, so there exists a triangulation of a regular
34-gon such that the rotation by $2\pi\frac{13}{34}$ preserves 26
(out of 31) diagonals.

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