  Inspiraling and merging binary neutron stars are not only important source of
gravitational waves, but also promising candidates for coincident
electromagnetic counterparts. These systems are thought to be progenitors of
short gamma-ray bursts (sGRBs). We have shown previously that binary neutron
star mergers that undergo {\it delayed} collapse to a black hole surrounded by
a {\it weighty} magnetized accretion disk can drive magnetically-powered jets.
We now perform magnetohydrodynamic simulations in full general relativity of
binary neutron stars mergers that undergo {\it prompt} collapse to explore the
possibility of jet formation from black hole-{\it light} accretion disk
remnants. We find that after $t-t_{\rm BH}\sim 26(M_{\rm NS}/1.8M_\odot)$ms
[$M_{\rm NS}$ is the ADM mass] following prompt black hole formation, there is
no evidence of mass outflow or magnetic field collimation. The rapid formation
of the black hole following merger prevents magnetic energy from approaching
force-free values above the magnetic poles, which is required for the launching
of a jet by the usual Blandford--Znajek mechanism. Detection of gravitational
waves in coincidence with sGRBs may provide constraints on the nuclear equation
of state (EOS): the fate of an NSNS merger--delayed or prompt collapse, and
hence the appearance or nonappearance of an sGRB--depends on a critical value
of the total mass of the binary, and this value is sensitive to the EOS.
