Finite residuality of involutory quandles of knots

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Let DD be a knot diagram, and let IQ(D)IQ(D) denote its involutory quandle. An involutory quandle is trivial if it consists of one element.

Involutory quandles are finitely residual. If IQ(D)IQ(D) is non-trivial, then there is a finite non-trivial involutory quandle QQ and a surjective homomorphism

IQ(D)↠Q.IQ(D) \twoheadrightarrow Q.

This is presented as equivalent to the finite countermodel property needed for the proposed unknot-detection programme to become a decision procedure; the source gives no resolution of the conjecture.

References

Primary source

Andrew Fish and Alexei Lisitsa, “Detecting unknots via equational reasoning, I: Exploration”, arXiv:1405.4211 (2014).

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