  Given a saddle fixed point of a surface diffeomorphism, its stable and
unstable curves $W^S$ and $W^U$ often form a homoclinic tangle. Given such a
tangle, we use topological methods to find periodic points of the
diffeomorphism, using only a subset of the tangle with finitely many points of
intersection, which we call a trellis. We typically obtain exponential growth
of periodic orbits, symbolic dynamics and a strictly positive lower bound for
topological entropy. For a simple example occurring in the Henon family, we
show that the topological entropy is at least 0.527.
