Higher Degree Knot Adjacency as Obstruction to Fibering

It is know that the Alexander polynomial detects fibered knots and 3-manifolds that fiber over the circle. In this note, we show that when the Alexander polynomial becomes inconclusive, the notion of “knot adjacency”, studied in [KL], can be used to obtain obstructions to fibering of knots and of 3-manifolds. As an application, given a fibered knot K′, we construct infinitely many non-fibered knots that share the same Alexander module and the same Vassiliev invariants up to certain orders with K′. Our construction also provides, for every n ∈ N , examples of irreducible 3-manifolds that cannot be distinguished by the Cochran-Melvin finite type invariants of order ≤ n.