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.