Biquandles and virtual links
The axioms of quandle theory may be understood as arising from knot theory by taking arcs as generators and requiring the resulting algebraic structure to be unchanged by Reidemeister moves. If we replace ``arcs'' with ``semi-arcs'' obtained by dividing knot diagrams at both over- and under-crossings, the resulting algebraic structure is known as a {biquandle}. In this paper, biquandles are obtained from {biracks,} which are algebraic structures derived from {switches}, algebraic solutions to the Yang-Baxter equation. Every birack defines a representation of the virtual braid group \(VB_n\), with resulting invariants of virtual knots. Some examples of switches and their resulting biracks are given, along with their applications to virtual knot theory. In particular, the fundamental biquandle is conjectured to be a complete invariant of virtual knots up to mirror image.
scientific article; zbMATH DE number 2136837
scientific article