  Every link in the 3-sphere has a projection to the plane where the only
singularities are pairwise transverse triple points. The associated diagram,
with height information at each triple point, is a triple-crossing diagram of
the link. We give a set of diagrammatic moves on triple-crossing diagrams
analogous to the Reidemeister moves on ordinary diagrams. The existence of
n-crossing diagrams for every n>1 allows the definition of the n-crossing
number. We prove that for any nontrivial, nonsplit link, other than the Hopf
link, its triple-crossing number is strictly greater than its
quintuple-crossing number.
