# When do two knots have isomorphic fundamental bikeis?

A kei, also known as an involutive (or involutory) quandle, is a quandle $$(Q,*)$$ 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 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.)

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, pdf)

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,

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? Note that the answers to both can't be no since isomorphic fundamental keis don't imply isomorphic fundamental quandles. (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?