A Consistent Set of Empirical Scaling Relations for Spiral Galaxies: The
  $(v_{\rm max},\,M_{\rm DM})$-$(\sigma_0,\,M_{\rm BH},\,\phi)$ Relations
  Using the latest sample of 48 spiral galaxies having a directly measured
supermassive black hole mass, $M_{BH}$, we determine how the maximum disk
rotational velocity, $v_{max}$ (and the implied dark matter halo mass,
$M_{DM}$), correlates with the (i) $M_{BH}$, (ii) central velocity dispersion,
$\sigma_0$, and (iii) spiral-arm pitch angle, $\phi$. We find that
$M_{BH}\propto v_{max}^{10.62\pm1.37}\propto M_{DM}^{4.35\pm0.66}$,
significantly steeper than previously reported, and with a root mean square
scatter ($0.58$ dex) similar to that about the $M_{BH}$-$\sigma_0$ relation for
spiral galaxies - in stark disagreement with claims that $M_{BH}$ does not
correlate with disks. Moreover, this $M_{BH}$-$v_{max}$ relation is consistent
with the unification of the Tully-Fisher relation (involving the total stellar
mass, $M_{*,tot}$) and the steep $M_{BH}\propto M_{*,tot}^{3.05\pm0.53}$
relation observed in spiral galaxies. We also find that $\sigma_0\propto
v_{max}^{1.55\pm0.25}\propto M_{DM}^{0.63\pm0.11}$, consistent with past
studies connecting stellar bulges (with $\sigma_0\gtrsim100\,{\rm
km\,s}^{-1}$), dark matter halos, and a nonconstant $v_{max}/\sigma_0$ ratio.
Finally, we report that
$\tan|\phi|\propto(-1.18\pm0.19)\log{v_{max}}\propto(-0.48\pm0.09)\log M_{DM}$,
providing a novel formulation between the geometry (i.e., the logarithmic
spiral-arm pitch angle) and kinematics of spiral galaxy disks. While the
$v_{max}$-$\phi$ relation may facilitate distance estimations to face-on spiral
galaxies through the Tully-Fisher relation and using $\phi$ as a proxy for
$v_{max}$, the $M_{DM}$-$\phi$ relation provides a path for determining dark
matter halo masses from imaging data alone. Furthermore, based on a spiral
galaxy sample size that is double the size used previously, the self-consistent
relations presented here provide dramatically revised constraints for theory
and simulations.
