  We study a two-color loop model known as the $C^{(1)}_2$ loop model. We
define a free-fermionic regime for this model, and show that under this
assumption it can be transformed into a double dimer model. We then compute its
free energy on periodic planar graphs. We also study the star-triangle relation
or Yang-Baxter equations of this model, and show that after a proper
parametrization they can be summed up into a single relation known as Kashaev's
relation. This is enough to identify the solution of Kashaev's relation as the
partition function of a $C^{(1)}_2$ loop model with some boundary conditions,
thus solving an open question of Kenyon and Pemantle about the combinatorics of
Kashaev's relation.
