Knot adjacency and fibering

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 {\em knot adjacency} can be used to obtain obstructions to the fibering of knots and of 3-manifolds...

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Bibliographic Details
Published in:Transactions of the American Mathematical Society 2008-06, Vol.360 (6), p.3249-3261
Main Authors: Kalfagianni, Efstratia, Lin, Xiao-Song
Format: Article
Language:English
Subjects:
Algebra
Exact sciences and technology
Integers
Knots
Manifolds and cell complexes
Mathematical manifolds
Mathematical theorems
Mathematics
Nugatory laws
Polynomials
Sciences and techniques of general use
String
Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds
Zero
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Summary: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 {\em knot adjacency} can be used to obtain obstructions to the 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 with K'. Our construction also provides, for every n\in N, examples of irreducible 3-manifolds that cannot be distinguished by the Cochran-Melvin finite type invariants of order \leq n.
ISSN:0002-9947
1088-6850
DOI:10.1090/S0002-9947-08-04358-4