  For the moduli space of Higgs bundles on a Riemann surface of positive genus,
critical points of the natural Morse-Bott function lie along the nilpotent cone
of the Hitchin fibration and are representations of $\mbox{A}$-type quivers in
a twisted category of holomorphic bundles. The critical points that globally
minimize the function are representations of $\mbox{A}_1$. For twisted Higgs
bundles on the projective line, the quiver describing the bottom of the cone is
more complicated. We determine it here. We show that the moduli space is
topologically connected whenever the rank and degree are coprime, thereby
verifying conjectural lowest Betti numbers coming from high-energy physics.
