Characterizing the Umbral Magnetic Knots of $\delta$-Sunspots

Delta ($\delta$)-spots are active regions (ARs) in which positive and negative umbrae share a penumbra. They are known to be the source of strong flares. We introduce a new quantity, the degree of $\delta$ (Do$\delta$), to measure the fraction of umbral flux participating in the $\delta$-configuration and to isolate the dynamics of the magnetic knot, i.e. adjacent umbrae in the $\delta$-configuration. Using Helioseismic and Magnetic Imager data, we analyze 19 $\delta$-spots and 11 $\beta$-spots in detail, and 120 $\delta$-spots in less detail. We find that $\delta$-regions are not in a $\delta$-configuration for the entire time but spend 55$\%$ of their observed time as $\delta$-spots with an average, maximum Do$\delta$ of 72$\%$. Compared to $\beta$-spots, $\delta$-spots have 2.6$\times$ the maximum umbral flux, 1.9$\times$ the flux emergence rate, 2.6$\times$ the rotation, and 72$\times$ the flare energy. On average, the magnetic knots rotate 17$^{\circ}$ day$^{-1}$ while the $\beta$-spots rotate 2$^{\circ}$ day$^{-1}$. Approximately 72$\%$ of the magnetic knots present anti-Hale or anti-Joy tilts, contrasting starkly with only 9$\%$ of the $\beta$-spots. A positive correlation exists between $\phi_{Do\delta}$ and the flare energy emitted by that region. The $\delta$-spots obey the hemispheric current helicity rule 64$\%$ of the time. 84$\%$ of the $\delta$-spots are formed by single flux emergence events and 58$\%$ have a quadrupolar magnetic configuration. The $\delta$-spot characteristics are consistent with the formation mechanism signatures as follows: 42$\%$ with the kink instability or Sigma effect, 32$\%$ with multi-segment buoyancy, 16$\%$ with collisions and two active regions that are unclassified but consistent with a rising O-ring.