Eigenvalue varieties of Brunnian 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. Eigenvalue varieties were first introduced to generalize the $A$–polynomial of knots in ${\mathbb{S}}^3$ to manifolds with nonconnected toric boundary. The result presented here generalizes, for Brunnian links, the nontriviality of the $A$–polynomial of nontrivial knots in ${\mathbb{S}}^3$.