Knot quandle decomposition along a torus

TL;DR

This paper investigates the structure of the augmented fundamental quandle for knots with incompressible tori, relating satellite knots to their components and exploring algebraic properties of affine quandles.

Contribution

It provides new presentations of fundamental quandles for various links and introduces an algebraic approach to affine quandles, connecting to Alexander modules and colorings.

Findings

Relationship between satellite and component quandles established

Presentations of quandles for links in solid tori and lens spaces derived

Algebraic methods applied to affine quandles and known invariants

Abstract

We study the structure of the augmented fundamental quandle of a knot whose complement contains an incompressible torus. We obtain the relationship between the fundamental quandle of a satellite knot and the fundamental quandles/groups of its companion and pattern knots. General presentations of the fundamental quandles of a link in a solid torus, a link in a lens space and a satellite knot are described. In the last part of the paper, an algebraic approach to the study of affine quandles is presented and some known results about the Alexander module and quandle colorings are obtained.