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.

已知亚历山大多项式可检测纤维化纽结以及在圆周上纤维化的3 - 流形。在本文中，我们表明当亚历山大多项式无法得出结论时，在[KL]中所研究的“纽结邻接”概念可用于获得纽结和3 - 流形纤维化的阻碍。作为一个应用，给定一个纤维化纽结\(K'\)，我们构造出无穷多个非纤维化纽结，它们与\(K'\)具有相同的亚历山大多项模以及直到某一阶数相同的瓦西里耶夫不变量。我们的构造还为每个\(n\in N\)提供了不可约3 - 流形的例子，这些流形不能通过\(\leq n\)阶的科克伦 - 梅尔文有限型不变量来区分。