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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Knot adjacency and fibering


Authors: Efstratia Kalfagianni and Xiao-Song Lin
Journal: Trans. Amer. Math. Soc. 360 (2008), 3249-3261
MSC (2000): Primary 57M25, 57M27, 57M50
Published electronically: January 30, 2008
MathSciNet review: 2379795
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Abstract: 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 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$.


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Additional Information

Efstratia Kalfagianni
Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824
Email: kalfagia@math.msu.edu

Xiao-Song Lin
Affiliation: Department of Mathematics, University of California, Riverside, California 92521

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04358-4
PII: S 0002-9947(08)04358-4
Keywords: Alexander polynomial, knot adjacency, fibered knots and 3-manifolds, finite type invariants, symplectic structures
Received by editor(s): July 15, 2005
Received by editor(s) in revised form: June 27, 2006
Published electronically: January 30, 2008
Additional Notes: The research of the authors was partially supported by the NSF
Xiao-Song Lin passed away on January 14, 2007
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.