  The Gopakumar-Vafa conjecture is defined and studied for the local geometry
of a curve in a Calabi-Yau 3-fold. The integrality predicted in Gromov-Witten
theory by the Gopakumar-Vafa BPS count is verified in a natural series of cases
in this local geometry. The method involves Gromov-Witten computations, Mobius
inversion, and a combinatorial analysis of the numbers of etale covers of a
curve.
