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Motivated by recent experiments probing anomalous surface states of Diracsemimetals (DSMs) Na$_3$Bi and Cd$_3$As$_2$, we raise the question posed in thetitle. We find that, in marked contrast to Weyl semimetals, the gapless surfacestates of DSMs are not topologically protected in general, except ontime-reversal-invariant planes of surface Brillouin zone. We first demonstratethis in a minimal $4$-band model with a pair of Dirac nodes at ${\bfk}=(0,0,\pm Q)$, where gapless states on the side surfaces are protected onlynear $k_z=0$. We then validate our conclusions about the absence of atopological invariant protecting double Fermi arcs in DSMs using a K-theoryanalysis for space groups of Na$_3$Bi and Cd$_3$As$_2$. Generically, the arcsdeform into a Fermi pocket, similar to {{the surface states of a topologicalinsulator (TI), and this can merge into the}} projection of bulk Dirac Fermisurfaces as the chemical potential is varied. We make sharp predictions for thedoping-dependence of the surface states of a DSM that can be tested by ARPESand quantum oscillation experiments.

Are the double Fermi arcs of Dirac semimetals topologically protected?