Clark–Elhamdadi–Saito–Yeatman conjecture on finite quandle coloring invariants

For a knot or link LL, let K(L)\mathcal{K}(L) denote its fundamental quandle, and let χL(X):=Homquandle(K(L),X)\chi_L(X):=|\operatorname{Hom}_{\operatorname{quandle}}(\mathcal{K}(L),X)| be the quandle counting invariant associated with a finite quandle XX. Clark–Elhamdadi–Saito–Yeatman conjecture. There exists a sequence of finite quandles

S=(Q1,Q2,,Qn,)S=(Q_1,Q_2,\ldots,Q_n,\ldots)

such that the invariant

C(L):=(χL(Q1),χL(Q2),,χL(Qn),)C(L):=(\chi_L(Q_1),\chi_L(Q_2),\ldots,\chi_L(Q_n),\ldots)

satisfies, for all knots K,KK,K', C(K)=C(K)C(K')=C(K) if and only if K(K)=K(K)\mathcal{K}(K')=\mathcal{K}(K). Thus, the sequence of coloring invariants would distinguish knots exactly up to isomorphism of their fundamental quandles. The claim is presented as an unproved conjecture and would provide a countable family of finite-quandle invariants detecting the fundamental quandle of every knot.

Sources & referencesView supporting material

Primary source

Eric Ramos, “Asymptotic behaviors in the homology of symmetric group and finite general linear group quandles”, arXiv:1706.02809 (2017).

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