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

Abstract

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 $|\text{IMQ}(K)|=|\det K|$. 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: \[ \frac{μ| \det L |}{2} \geq |\text{IMQ}(L)| \geq \frac{ μ| \det L |} {2^{μ-1}}. \] 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 $L^{\prime }$ are links with $\text{IMQ}(L)\cong \text{IMQ}(L^{\prime })$, then $H_1(X_2)\cong H_1(X_2^{\prime })$; but it is possible to have $H_1(X_2)\cong H_1(X_2^{\prime })$ and $\text{IMQ}(L)\ncong \text{IMQ}(L^{\prime })$. In fact, it is possible to have $X_2\cong X_2^{\prime }$ and $\text{IMQ}(L)\ncong \text{IMQ}(L^{\prime })$.

arXiv commentv1: 21 pages, 2 figures. v2: 28 pages, 2 figures. v3: 29 pages, 2 figures. v4: 30 pages, 2 figures. v5: 38 pages, 4 figures. major revision. v6: 38 pages, 4 figures. v7: 38 pages, 4 figures. final prepublication version. v8: 50 pages, 10 figures. major revision, including corrected interpretation of results of Joyce misquoted in v1 - v7. v9: final prepublication version of corrected paper