We study nematic liquid crystal textures exhibiting topological defects (TDs) on a two-dimensional (2D) toroidal shell. For the toroidal topology the total topological charge of TDs is equal to zero. We use a mesoscopic Landau-de Gennes approach which features a 2D nematic order tensor Q. We show that fat tori unbind TDs. If no extrinsic free energy couples Q with the Weingarten tensor of the torus, then defects and antidefects are assembled along the innermost and the outermost circles of the torus, respectively. In this case, we estimate the critical condition for the onset of TDs using an electrostatic analogy. If, on the other hand, an extrinsic free energy is present, then defects are repelled from these regions.
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