# Multivariate Alexander quandles, III. Sublinks

## TL;DR

This paper demonstrates that the multivariate Alexander quandle $Q_A(L)$ is a strictly stronger invariant than the multivariate Alexander module $M_A(L)$ for classical links, as it determines all sublink quandles.

## Contribution

It proves that $Q_A(L)$ uniquely determines the quandles of all sublinks, making it a more powerful invariant than $M_A(L)$ alone.

## Key findings

- $Q_A(L)$ determines all sublink quandles up to isomorphism.
- $Q_A(L)$ is strictly stronger than $M_A(L)$ as a link invariant.
- The result extends the understanding of link invariants in knot theory.

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