  In relativistic heavy ion collisions, top quarks can be produced by
photon-gluon fusion when a photon from the Weizs\"acker-Williams virtual photon
field of one nucleus interacts with a gluon in the other nucleus.
Photoproduction with heavy ions at the Large Hadron Collider (LHC) will be the
first accessible non-hadronic top production channel. We calculate the $t \bar
t$ photoproduction cross sections, pair mass and top quark rapidity
distributions in peripheral heavy ion collisions. The cross sections are
sensitive to the top quark charge and the large-$Q^2$ gluon distribution in the
nucleus. We find a cross section of 94 pb in calcium-calcium collisions,
leading to 190 pairs in a one month ($10^6$ sec) LHC run. We also find $p$Pb
and $p$Ca cross sections of 5.8 and 3.4 pb respectively, resulting in 6 and 34
$t\bar t$ pairs per month.
