 Solution

In polar coordinates $(r,\phi )$, a radial asterand a circulating vortexboth have $q=+1$. The continuous family $\theta =\phi +\chi$, $0\le \chi \le \pi /2$, deforms one into the other without making $\mathbf{p}$ vanish away from the core, proving their topological equivalence. The hyperbolic configuration $\theta =-\phi$ has $q=-1$.