  A complete surface of constant mean curvature 1 (CMC-1) in hyperbolic 3-space
with constant curvature -1 has two natural notions of "total curvature"-- one
is the total absolute curvature which is the integral over the surface of the
absolute value of the Gaussian curvature, and the other is the dual total
absolute curvature which is the total absolute curvature of the dual CMC-1
surface. In this paper, we completely classify CMC-1 surfaces with dual total
absolute curvature at most 4\pi. Moreover, we give new examples and partially
classify CMC-1 surfaces with dual total absolute curvature at most 8\pi.
