Completeness conjecture for the knot biquandle of virtual links

A knotlike biquandle is a biquandle admitting a presentation in which every explicit relation is short, every generator appears exactly once as input, operator, and output, and the relations occur in the paired switch forms specified above. For a virtual link, its knot biquandle is the biquandle obtained from a diagram by assigning generators to semiarcs and imposing the crossing relations; Tietze moves are the standard presentation moves, while virtual isotopy moves are the Reidemeister-type moves for virtual link diagrams.

Knot-biquandle completeness conjecture. The knot biquandle is a complete invariant of virtual link type: if two knotlike biquandle presentations are related by Tietze moves, then the resulting virtual link diagrams are related by virtual isotopy moves.

Every virtual knot or link biquandle is knotlike, and the biquandle determines the virtual link up to strictly virtual moves. The conjecture asserts that the remaining ambiguity is exactly virtual isotopy, rather than merely strict virtual equivalence.