Eigenvalue varieties of Brunnian links

TL;DR

This paper proves that the eigenvalue variety of certain Brunnian links in 3-sphere contains a significant nontrivial component, extending known results about knot invariants to a broader class of links.

Contribution

It generalizes the nontriviality of the A-polynomial from knots to nontrivial, non-Hopf Brunnian links, showing their eigenvalue varieties have maximal dimension components.

Findings

Eigenvalue variety of nontrivial Brunnian links contains a nontrivial component.

Extension of A-polynomial nontriviality from knots to Brunnian links.

Eigenvalue varieties have maximal dimension components for these links.

Abstract

In this article, it is proved that the eigenvalue variety of the exterior of a nontrivial, non-Hopf, Brunnian link in $\mathbb{S}^{3}$ contains a nontrivial component of maximal dimension. This generalises, for Brunnian links, the nontriviality of the $A$-polynomial of a nontrivial knot in $\mathbb{S}^{3}$.