  For an arbitrary identity L=R between compositions of maps L and R on tensors
of vector spaces V, a general construction of a 2-cocycle condition is given.
These 2-cocycles correspond to those obtained in deformation theories of
algebras. The construction is applied to a canceling pairings and copairings,
with explicit examples with calculations. Relations to the Kauffman bracket and
knot invariants are discussed.
