Batteries composed of CFx cathodes have high theoretical specific capacities (>860 mA h g-1). Attempts at realizing such batteries coupled with Li anodes have failed to deliver on this promise, however, due to a discharge voltage plateau below the theoretical maximum lowering the realized energy density and difficulties with recharging the system. In this study, we use first-principles calculations to investigate novel carbon allotropes for these battery systems: graphdiyne and "holey" graphene. We first identify stable flourination structures and calculate their band gaps. We demonstrate that the holes in these carbon allotropes can induce the formation of an amorphous LiF network within the carbon and that this formation may, in fact, be kinetically favored. For structures where amorphous LiF forms within the carbon, we predict it is easier to recharge and higher discharge voltages can be achieved. If the LiF forms outside the carbon product, however, it will be crystalline in form and lead to lower discharge voltages and more difficulty in recharging the systems. Finally, we simulate XPS spectra of representative cases, demonstrating an experimental pathway for determining the reaction pathway of these systems. Our work suggests CFx allotropes with holes in them as potential targets for high capacity, rechargeable cathodes for Li batteries, provided they lead to the formation of amorphous LiF within the C structure.