  Polyak proved that the set $\{\Omega1a,\Omega1b,\Omega2a,\Omega3a\}$ is a
minimal generating set of oriented Reidemeister moves. One may distinguish
between forward and backward moves, obtaining $32$ different types of moves,
which we call directed oriented Reidemeister moves. In this article we prove
that the set of $8$ directed Polyak moves $\{ \Omega{1a}^\uparrow,
\Omega{1a}^\downarrow, \Omega{1b}^\uparrow, \Omega{1b}^\downarrow,
\Omega{2a}^\uparrow, \Omega{2a}^\downarrow, \Omega{3a}^\uparrow,
\Omega{3a}^\downarrow \}$ is a minimal generating set of directed oriented
Reidemeister moves. We also specialize the problem, introducing the notion of a
$L$-generating set for a link $L$. The same set is proven to be a minimal
$L$-generating set for any link $L$ with at least $2$ components. Finally, we
discuss knot diagram invariants arising in the study of $K$-generating sets for
an arbitrary knot $K$, emphasizing the distinction between ascending and
descending moves of type $\Omega3$.
