  We study the algebras underlying solvable lattice models of the type fusion
interaction round the face (IRF). We propose that the algebras are universal,
depending only on the number of blocks, which is the degree of polynomial
equation obeyed by the Boltzmann weights. Using the Yang--Baxter equation and
the ansatz for the Baxterization of the models, we show that the three blocks
models obey a version of Birman--Murakami--Wenzl (BMW) algebra. For four
blocks, we conjecture that the algebra is the BMW algebra with a different
skein relation, along with one additional relation, and we provide evidence for
this conjecture. We connect these algebras to knot theory by conjecturing new
link invariants. The link invariants, in the case of four blocks, depend on
three arbitrary parameters. We check our result for $G_2$ model with the seven
dimensional representation and for $SU(2)$ with the isospin $3/2$
representation, which are both four blocks theories.
