The universal quandle-coloring separation conjecture

Let S=(Q1,Q2,)\mathcal S=(Q_1,Q_2,\ldots) be a sequence of finite quandles, and define the coloring invariant

C(K)=(ColQ1(K),ColQ2(K),).C(K)=\bigl(\operatorname{Col}_{Q_1}(K),\operatorname{Col}_{Q_2}(K),\ldots\bigr).

Universal quandle-coloring separation conjecture. There exists such a sequence satisfying, for all knots K,KK,K' ,

C(K)=C(K)if and only ifK=K or K=mr(K).C(K')=C(K)\quad\text{if and only if}\quad K'=K\text{ or }K'=mr(K).

This is a stronger global form of the finite-knot separation problem and is presented as an open conjecture; the paper gives finite computational evidence but does not establish the assertion for all knots.

Sources & referencesView supporting material

Primary source

W. Edwin Clark, Mohamed Elhamdadi, Masahico Saito and Timothy Yeatman, “Quandle Colorings of Knots and Applications”, arXiv:1312.3307 (2014).

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