  A $4$-move is a local operation for links consisting in replacing two
parallel arcs by four half twists. At the present time, it is not known if this
induces an unkotting operation for knots. Studying the Dabkowski-Sahi
invariant, we prove that any invariant of knots based on the fundamental group
$\pi_1(S^3\setminus K)$ and preserved by $4$-moves is constant among the
isotopy classes of knots.
