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We start a systematic analysis of links up to 5-move equivalence. Ourmotivation is to develop tools which later can be used to study skein modulesbased on the skein relation being deformation of a 5-move (in an analogous wayas the Kauffman skein module is a deformation of a 2-move, i.e. a crossingchange). Our main tools are Jones and Kauffman polynomials and the fundamentalgroup of the 2-fold branch cover of S^3 along a link. We use also the fact thata 5-move is a composition of two rational \pm (2,2)-moves (i.e. \pm 5/2-moves)and rational moves can be analyzed using the group of Fox colorings and itsnon-abelian version, the Burnside group of a link. One curious observation isthat links related by one (2,2)-move are not 5-move equivalent. In particular,we partially classify (up to 5-moves)  3-braids, pretzel and Montesinos links, and links up to 9 crossings.

5-MOVE EQUIVALENCE CLASSES OF LINKS AND THEIR ALGEBRAIC INVARIANTS