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In the literature, one sometimes sees the claim that finitely presented quandles (in particular, knot quandles) are "hard to deal with". Hence, a great deal of effort has gone into studying finite quandles and counting homomorphisms onto them, and so on. However, I have not yet come across any theorems that state formal undecidability results for finitely presented quandles similar to those for finitely presented groups. In fact, I have yet to see any formulation of such problems. (For instance, a theorem stating that the isomorphism problem is undecidable for finitely presented quandles.)

Do such results exist in the literature and, if so, could someone please provide references?

(Asked previously here on math.stackexchange, without response.)

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up vote 6 down vote accepted

The word problem for quandles was formulated by Rena Levitt and Sam Nelson. In Levitt's research statement the problem was formulated as follows:

Word Problem for Quandles. Given a finitely generated quandle $Q$, is there an algorithm to determine if two words in the generators represent the same element of $Q$?

Levitt and Nelson solved the word problem for some families of quandles. In particular, they proved:

Theorem. Finitely generated free quandles and knot quandles have a solvable word problem.

You can see more information in these slides of a talk given by Levitt in 2012. As far as I know, the paper is not yet published.

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