A **kei**, also known as an involutive (or involutory) quandle, is a [quandle](https://en.wikipedia.org/wiki/Racks_and_quandles) $(Q,*)$ satisfying the involution condition that $(x*y)*y=x$ for all $x$ and $y$.  Just like we can define a fundamental quandle of a knot using arcs of an oriented knot diagram, we can define the fundamental kei of a knot using arcs of an unoriented knot diagram.  A **bikei**, also known as an involutive biquandle, is a [biquandle](https://en.wikipedia.org/wiki/Biquandle) satisfying similar involution conditions.  Just as we can define the fundamental biquandle of a knot using semiarcs of an oriented knot diagram, we can define the fundamental bikei of a knot using the semiarcs of an unoriented knot diagram.  (These are all defined precisely in Elhamdadi and Nelson’s book _[Quandles: An Introduction to the Algebra of Knots](https://bookstore.ams.org/stml-74)_.)

My question is, under what circumstances are the fundamental bikeis of two knots isomorphic? Now this thesis

>Martina Vaváčková, _Algebraic Structures for Knot Coloring_, Masters thesis, Charles University (2018) ([abstract page](https://is.cuni.cz/webapps/zzp/detail/181764/?lang=en), [pdf](https://is.cuni.cz/webapps/zzp/download/120309226))

shows that the fundamental keis of the granny knot and square knot are isomorphic, whereas their fundamental quandles are not isomorphic.  The recent journal paper

>Katsumi Ishikawa, _Knot Quandles vs. Knot Biquandles_, International Journal of Mathematics (accepted to appear) doi:[10.1142/S0129167X20500159](https://doi.org/10.1142/S0129167X20500159),

on the other hand, shows that the fundamental biquandles of two knots are isomorphic if and only if their fundamental quandles are isomorphic.  So this raises two natural questions: Can two knots have isomorphic fundamental bikeis without having isomorphic fundamental biquandles?  And can two knots have isomorphic fundamental keis without having isomorphic fundamental bikeis? (The converses are clear: isomorphic biquandles imply isomorphic bikeis, and isomorphic bikeis imply isomorphic keis.)

Do the two results I mentioned shed any light on these questions?  Are either of these questions open problems?