  This work is dedicated to the consideration of the construction of a
representation of braid group generators from vertex models with $N$-states,
which provides a great way to study the knot invariant. An algebraic formula is
proposed for the knot invariant when different spins $(N-1)/2$ are located on
all components of the knot. The work summarizes procedure outputting braid
generator representations from three-partite vertex model. This representation
made it possible to study the invariant of a knot with multi-colored links,
where the components of the knot have different spins. The formula for the
invariant of knot with a multi-colored link is studied from the point of view
of the braid generators obtained from the $R$-matrices of three-partite vertex
models. The resulting knot invariant $5_2$ corresponds to the Jones polynomial
and HOMFLY-PT.
