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In relativistic heavy ion collisions, top quarks can be produced byphoton-gluon fusion when a photon from the Weizs\"acker-Williams virtual photonfield of one nucleus interacts with a gluon in the other nucleus.Photoproduction with heavy ions at the Large Hadron Collider (LHC) will be thefirst accessible non-hadronic top production channel. We calculate the $t \bart$ photoproduction cross sections, pair mass and top quark rapiditydistributions in peripheral heavy ion collisions. The cross sections aresensitive to the top quark charge and the large-$Q^2$ gluon distribution in thenucleus. 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 $p$Pband $p$Ca cross sections of 5.8 and 3.4 pb respectively, resulting in 6 and 34$t\bar t$ pairs per month.Comment: kinematics clarified (results unchanged); to appear in Eur. Phys. J.  C 13 pages including 2 figures kinematics section clarified (results  unchanged

Photoproduction of top quarks in peripheral heavy ion collisions