The Dabkowski-Sahi invariant and 4-moves for links

Dabkowski and Sahi defined an invariant of a link in the 3-sphere, which is preserved under 4-moves. This invariant is a quotient of the fundamental group of the complement of the link. It is generally difficult to distinguish between the Dabkowski-Sahi invariants of given links. In this paper, we give a necessary condition for the existence of an isomorphism between the Dabkowski-Sahi invariant of a link and that of the corresponding trivial link. Using this condition, we provide a practical obstruction to a link to be trivial up to 4-moves.