  Density functional theory methods are applied to investigate the properties
of the new superconductor $\beta$-YbAlB$_4$ and its polymorph
$\alpha$-YbAlB$_4$. We utilize the generalized gradient approximation + Hubbard
U (GGA+U) approach with spin-orbit(SO) coupling to approximate the effects of
the strong correlations due to the open $4f$ shell of Yb. We examine closely
the differences in crystal bonding and symmetry of $\beta$-YbAlB$_4$ and
$\alpha$-YbAlB$_4$. The in-plane bonding structure amongst the dominant
itinerant electrons in the boron sheets is shown to differ significantly. Our
calculations indicate that, in both polymorphs, the localized 4$f$ electrons
hybridize strongly with the conduction sea when compared to the related
materials YbRh$_{2}$Si$_{2}$ and YbB$_{2}$. Comparing $\beta$-YbAlB$_4$ to the
electronic structure of related crystal structures indicates a key role of the
7-member boron coordination of the Yb ion in $\beta$-YbAlB$_4$ in producing its
enhanced Kondo scale and superconductivity. The Kondo scale is shown to depend
strongly on the angle between the B neighbors and the Yb ion, relative to the
$x-y$ plane, which relates some of the physical behavior to structural
characteristics.
