Multivariate Alexander quandles, III. Sublinks

If $L$ is a classical link then the multivariate Alexander quandle, $Q_A(L)$, is a substructure of the multivariate Alexander module, $M_A(L)$. In the first paper of this series we showed that if two links $L$ and $L^{\mathrm{\prime }}$ have $Q_A(L) \cong Q_A(L’)$,thenafteranappropriatere-indexingofthecomponentsof$L$and$L’$,therewillbeamoduleisomorphism$M_A(L) \cong M_A(L’)$ of a particular type, which we call a"Crowell equivalence." In the present paper we show that $Q_A(L)$ (up to quandle isomorphism) is a strictly stronger link invariant than $M_A(L)$ (up to re-indexing and Crowell equivalence). This result follows from the fact that $Q_A(L)$ determines the $Q_A$ quandles of all the sublinks of $L$, up to quandle isomorphisms.