The orbits of the up-down action of classical and virtual braid diagrams on Z^n are completely determined: trace alone for virtual braids, trace plus a parity-type count for classical braids.
Dehornoy, Diagram colourings, Proceedings of the East Asian School of Knots, Links, and Related Topics February 16–20 , 2004
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Orbits by the up-down action of braid diagrams
The orbits of the up-down action of classical and virtual braid diagrams on Z^n are completely determined: trace alone for virtual braids, trace plus a parity-type count for classical braids.