Commensurability of 1-cusped hyperbolic 3-manifolds (Q2782671)

Examples of non-fibered hyperbolic knot complements in homology 3-spheres are given that are not commensurable to fibered knot complements. This is motivated by Thurston's conjecture that every finite volume hyperbolic 3-manifold has a finite cover which fibers over the circle. The main result of the paper is a criterion, stated in terms of non-integral reducible representations in the \(\text{PSL}_2\mathbb C\)-character variety of \(\pi_1(M)\), which guarantees that the complement \(M\) of a knot in a homology 3-sphere is not commensurable to a fibered knot complement. The main idea combines the criterion that the Alexander polynomial of a fibered knot is monic with the fact that the roots of the Alexander polynomial are related to the eigenvalues of reducible \(\text{PSL}_2\mathbb C\)-representations of \(\pi_1(M)\). It is shown that the criterion applies to the complements of many non-fibered 2-bridge knots. Also, examples are given of pairs of commensurable 1-cusped hyperbolic 3-manifolds exactly one of which fibers.