Multivariate Alexander quandles, VI. Metabelian groups and 2-component links

TL;DR

This paper explores the algebraic structures associated with links, establishing isomorphisms between Alexander quandles and metabelian quotients, and linking medial quandles to the reduced Alexander invariant for 2-component links.

Contribution

It proves that the multivariate Alexander quandle is isomorphic to a natural image in the metabelian quotient of the link group and relates the medial quandle to the reduced Alexander invariant.

Findings

The fundamental multivariate Alexander quandle is isomorphic to the image in the metabelian quotient of the link group.

The medial quandle of a 2-component link is determined by its reduced Alexander invariant.

Provides new algebraic characterizations of link invariants.

Abstract

We prove two properties of the modules and quandles discussed in this series. First, the fundamental multivariate Alexander quandle $Q_A(L)$ is isomorphic to the natural image of the fundamental quandle in the metabelian quotient $G(L)/G(L)''$ of the link group. Second, the medial quandle of a classical 2-component link $L$ is determined by the reduced Alexander invariant of $L$.