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

## TL;DR

This paper extends Joyce's results on involutory medial quandles from knots to links, providing bounds on their size, showing they are stronger invariants than homology groups, and characterizing when they are infinite.

## Contribution

It generalizes Joyce's knot results to links, establishes bounds on involutory medial quandle size, and demonstrates its superiority as a link invariant over homology.

## Key findings

- IMQ(L) size bounds depend on link components and determinant
- IMQ(L) is infinite iff determinant is zero
- IMQ(L) can distinguish links with identical homology groups

## 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 $\mu \geq 2$ components, the order of the involutory medial quandle is bounded as follows: \[ \frac{\mu | \det L |}{2} \geq |\text{IMQ}(L)| \geq \frac{ \mu | \det L |} {2^{\mu -1}}. \] 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 $L'$ are links with $\text{IMQ}(L) \cong \text{IMQ}(L')$, then $H_1(X_2) \cong H_1(X'_2)$; but it is possible to have $H_1(X_2) \cong H_1(X'_2)$ and $\text{IMQ}(L) \not \cong \text{IMQ}(L')$. In fact, it is possible to have $X_2 \cong X'_2$ and $\text{IMQ}(L) \not \cong \text{IMQ}(L')$.