Knot adjacency and fibering

Details Knot adjacency and fibering Knot adjacency and fibering

2004-03-01

arxiv.org/abs/math/0403026v7

Transactions of the American Math. Soc., Vol. 360(2008), 3249-3261.

Mathematics

Geometric Topology

18 pages. To appear in Transactions of the AMS

Scientific paper

It is known 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 the paper "Knot adjacency, genus and essential tori" by the authors, 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 natural number n, examples of irreducible 3-manifolds that cannot be distinguished by the Cochran-Melvin finite type invariants of order < n.

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