Multivariate Alexander quandles, III. Sublinks

Abstract

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 $L^{\prime }$ have $Q_A(L)\cong Q_A(L^{\prime })$, then after an appropriate re-indexing of the components of $L$ and $L^{\prime }$, there will be a module isomorphism $M_A(L)\cong M_A(L^{\prime })$ 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.

v1: 15 pages, 4 figures. v2: minor edits. v3: 16 pages, 4 figures. minor edits. Further changes may be made before publication in the Journal of Knot Theory and its Ramifications