  We present the first flexion-focused gravitational lensing analysis of the
first of the strong-lensing "cosmic telescope" galaxy clusters, observed as
part of the Hubble Frontier Fields initiative. Using HST observations of Abell
2744 (z = 0.308), we apply a modified Analytic Image Model (AIM) technique to
measure source galaxy flexion and shear values at a final number density of 82
arcmin$^{-2}$. By using flexion data alone we are able to identify the primary
mass structure aligned along the heart of the cluster in addition to a major
substructure peak offset 1.43' from the cluster core. We generate two types of
nonparametric reconstructions: a flexion aperture mass map, which identifies
the central potential and substructure peak with mass signal-to-noise of
3.5$\sigma$ and 2.3$\sigma$ respectively; and a convergence map derived
directly from the smoothed flexion field. For the primary peak we find a mass
of $1.93\times10^{14}\,h^{-1}\,M_{\odot}$ within a 45" (145h$^{-1}$ kpc)
aperture, and for the western substructure we find a mass of
$7.12\times10^{13}\,h^{-1}\,M_{\odot}$ within a 25" (80h$^{-1}$ kpc) aperture.
The associated peak velocity dispersions were determined to be $\sigma_v$ =
1630 km/s and $\sigma_v$ = 766 km/s, respectively, by fitting nonsingular
isothermal sphere profiles to the flexion data. Additionally, we use
simultaneous shear measurements to independently reconstruct the broader
cluster mass structure, and find that it is unable to reproduce the small-scale
structure associated with the flexion reconstructions. Finally, we perform the
same analysis on the Abell 2744 parallel sky field, and find no strong phantom
signals in the noise reconstructions.
