In his famous ASM paper, Kuperberg uses a skein relation to give an algebraic proof of a Yang-Baxter equation where the Boltzmann weights satisfy the field-free condition. In this paper, we use Kuperberg's techniques to give proofs of a few Yang-Baxter equations where the Boltzmann weights satisfy the free-fermionic condition. In particular, we use skein relations to prove the Yang-Baxter equation for Gamma-Gamma ice which is a free-fermionic six-vertex model introduced by Brubaker, Bump and Friedberg.