  We introduce new skein invariants of links based on a procedure where we
first apply the skein relation only to crossings of distinct components, so as
to produce collections of unlinked knots. We then evaluate the resulting knots
using a given invariant. A skein invariant can be computed on each link solely
by the use of skein relations and a set of initial conditions. The new
procedure, remarkably, leads to generalizations of the known skein invariants.
We make skein invariants of classical links, $H[R]$, $K[Q]$ and $D[T]$, based
on the invariants of knots, $R$, $Q$ and $T$, denoting the regular isotopy
version of the Homflypt polynomial, the Kauffman polynomial and the Dubrovnik
polynomial. We provide skein theoretic proofs of the well-definedness of these
invariants. These invariants are also reformulated into summations of the
generating invariants ($R$, $Q$, $T$) on sublinks of a given link $L$, obtained
by partitioning $L$ into collections of sublinks.
