Reidemeister moves for triple-crossing link diagrams

Knots and links are classically represented by diagrams: immersed 1-manifolds in the plane where all crossings consist of exactly two strands intersecting transversely (together with over-under information). A diagram of a given link is unique up to ambient isotopy and the three classical Reidemeister moves. In 2013, Colin Adams introduced the concept of "n-diagrams" for any integer n (at least 2), which are immersed 1-manifolds in the plane where all crossings consist of exactly n strands intersecting transversely (together with over-under information). A natural question arises: are there "higher" Reidemeister moves for n-diagrams, in the sense that any two n-diagrams representing the same link are connected by a finite sequence of these moves? I will present a positive answer for n=3, describing a complete set of (five) moves for 3-diagrams. This represents joint work with Colin Adams and Jim Hoste.