  We present a gravitational lensing analysis of the galaxy cluster Abell 1689,
incorporating measurements of the weak shear, flexion, and strong lensing
induced in background galaxies. This is the first time that a shapelet
technique has been used to reconstruct the distribution of mass in this
cluster, and the first time that a flexion signal has been measured using
cluster members as lenses. From weak shear measurements alone, we generate a
non-parametric mass reconstruction, which shows significant substructure
corresponding to groups of galaxies within the cluster. Additionally, our
galaxy-galaxy flexion signal demonstrates that the cluster galaxies can be
well-fit by a singular isothermal sphere model with a characteristic velocity
dispersion of $\sigma_v = 295\pm 40 km/s $. We identify a major, distinct dark
matter clump, offset by 40$h^{-1}$kpc from the central cluster members, which
was not apparent from shear measurements alone. This secondary clump is present
in a parametric mass reconstruction using flexion data alone, and its existence
is suggested in a non-parametric reconstruction of the cluster using a
combination of strong and weak lensing. As found in previous studies, the mass
profile obtained by combining weak and strong lensing data shows a much steeper
profile than that obtained from only weak lensing data.
