Cocyclic quasoid knot invariants

We describe some method that associates two chain complexes to every $X$ and every mapping $Q: X\times X\times X\to X$ satisfying a few conditions motivated by Reidemeister moves. These complexes differ by boundary homomorphisms: For one complex, the boundary homomorphism is the difference of two operators; and for the other, their sum. We prove that each element of the third cohomology group of these complexes correctly defines an invariant of oriented links. We provide the results of calculations of cohomology groups for all various mappings $Q$ on sets of order at most 4.