 
Summary: Intersections of curves on the torus
Ian Agol
May 15, 1999
For a pair of curves ff, fi on a torus, call their intersection number \Delta(ff; fi). We'll use the term
slope for an embedded curve on the torus. Given a number d, let \Delta Ÿ (d) be the maximal number of
a collection of slopes on the torus with pairwise intersection numbers bounded by d. This definition
comes from theorems about Dehn filling in 3manifold topology (for a survey, see Gordon's paper
[2]). In this note, we'll give upper bounds and asymptotics for \Delta Ÿ (d), and compute it for small
values of d.
If we choose a basis for the homology on the torus, such that ff = (a; b); fi = (c; d); then
\Delta(ff; fi) = jad \Gamma bcj. If ff is a slope, then gcd(a; b) = 1. Let R be the ring Z or Zn = Z=nZ.
Let R \Theta be the set of invertible elements in R. Let R 2\Theta be the set of coprime pairs in R 2 , that
is points (a; b) so that there is a pair (m; n) 2 R 2 with ma + nb = 1, and denote the collection
of lines in R 2\Theta by RP 1 . Lines in RP 1 are equivalence classes of points of R 2\Theta under the relation
(a; b) ¸ (c; d) () (a; b) = m(c; d); m 2 R \Theta : ZP 1 = QP 1 is the space of slopes on the torus (which
is the reason they are called slopes). Let /(n) = jZnP 1 j.
Lemma 0.1. /(n) = n
Q
qjn (1 + 1
q
