Towers of baryons of the $\mathcal{N}=2$ quark model band in the large $N_c$ limit

We perform a complete analysis of the spectrum of all the states in the $\mathcal{N}=2$ quark model harmonic oscillator band with $N_f=3$ in the large $N_c$ limit including the often disregarded antisymmetric multiplet $[\mathbf{20},1^{+}]$. We include configuration mixing effects. We find that the states in the $[\mathbf{56},L^{+}]$ and $[\mathbf{70},L^{+}]$ with $L=0$, 2 fall into nine towers of degenerate states. We find that nonstrange antisymmetric states fall into three towers and respond to the same structure as the states in the $[\mathbf{70},1^{-}]$ multiplet. We also show explicitly the compatibility of these results and the scattering resonance picture.