Are the double Fermi arcs of Dirac semimetals topologically protected?

Motivated by recent experiments probing anomalous surface states of Dirac semimetals (DSMs) Na$_3$Bi and Cd$_3$As$_2$, we raise the question posed in the title. We find that, in marked contrast to Weyl semimetals, the gapless surface states of DSMs are not topologically protected in general, except on time-reversal-invariant planes of surface Brillouin zone. We first demonstrate this in a minimal $4$-band model with a pair of Dirac nodes at ${\bf k}=(0,0,\pm Q)$, where gapless states on the side surfaces are protected only near $k_z=0$. We then validate our conclusions about the absence of a topological invariant protecting double Fermi arcs in DSMs using a K-theory analysis for space groups of Na$_3$Bi and Cd$_3$As$_2$. Generically, the arcs deform into a Fermi pocket, similar to {{the surface states of a topological insulator (TI), and this can merge into the}} projection of bulk Dirac Fermi surfaces as the chemical potential is varied. We make sharp predictions for the doping-dependence of the surface states of a DSM that can be tested by ARPES and quantum oscillation experiments.