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.
Determinant of the OU matrix of a braid diagram
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abstract
In this paper, we define the OU matrix of a braid diagram and discuss how the OU matrix reflects the warping degree or the layeredness of the braid diagram, and show that the determinant of the OU matrix of a layered braid diagram is the product of the determinants of the layers. We also introduce invariants of positive braids which are derived from the OU matrix.
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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.