Multivariate Alexander quandles, II. The involutory medial quandle of a link (corrected)

Joyce showed that for a classical knot $K$, the involutory medial quandle $\text{IMQ}(K)$ is isomorphic to the core quandle of the homology group $H_1(X_2)$, where $X_2$ is the cyclic double cover of ${\mathbb{S}}^3$, branched over $K$. It follows that $\mathrm{\mid }\text{IMQ}(K)\mathrm{\mid }=\mathrm{\mid }\det K\mathrm{\mid }$. In the present paper, the extension of Joyce’s result to classical links is discussed. Among other things, we show that for a classical link $L$ of $\mu \ge 2$ components, the order of the involutory medial quandle is bounded as follows: $$\frac{\mu \mathrm{\mid }\det L\mathrm{\mid }}{2}\ge \mathrm{\mid }\text{IMQ}(L)\mathrm{\mid }\ge \frac{\mu \mathrm{\mid }\det L\mathrm{\mid }}{2^{\mu -1}}\mathrm{.}$$ In particular, $\text{IMQ}(L)$ is infinite if and only if $\det L=0$. We also show that in general, $\text{IMQ}(L)$ is a strictly stronger invariant than $H_1(X_2)$. That is, if $L$ and $L^{\mathrm{\prime }}$ are links with $$\text{IMQ}$(L) \cong $\text{IMQ}$(L’)$,then$H_1(X_2) \cong H_1(X’_2)$;butitispossibletohave$H_1(X_2) \cong H_1(X’_2)$and$$\text{IMQ}$(L) \not \cong $\text{IMQ}$(L’)$. In fact, it is possible to have $X_2\cong X_2^{\mathrm{\prime }}$ and $$\text{IMQ}$(L) \not \cong $\text{IMQ}$(L’)$.