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 | References | Similar Articles | Additional Information

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 , we construct infinitely many non-fibered knots that share the same Alexander module with . Our construction also provides, for every , examples of irreducible 3-manifolds that cannot be distinguished by the Cochran-Melvin finite type invariants of order .

**[BZ]**Gerhard Burde and Heiner Zieschang,*Knots*, de Gruyter Studies in Mathematics, vol. 5, Walter de Gruyter & Co., Berlin, 1985. MR**808776****[Ch]**Jae Choon Cha,*Fibred knots and twisted Alexander invariants*, Trans. Amer. Math. Soc.**355**(2003), no. 10, 4187–4200 (electronic). MR**1990582**, 10.1090/S0002-9947-03-03348-8**[C]**Tim D. Cochran,*Noncommutative knot theory*, Algebr. Geom. Topol.**4**(2004), 347–398. MR**2077670**, 10.2140/agt.2004.4.347**[CM]**Tim D. Cochran and Paul Melvin,*Finite type invariants of 3-manifolds*, Invent. Math.**140**(2000), no. 1, 45–100. MR**1779798**, 10.1007/s002220000045**[CGLS]**Marc Culler, C. McA. Gordon, J. Luecke, and Peter B. Shalen,*Dehn surgery on knots*, Ann. of Math. (2)**125**(1987), no. 2, 237–300. MR**881270**, 10.2307/1971311**[Ga1]**David Gabai,*Foliations and the topology of 3-manifolds. III*, J. Differential Geom.**26**(1987), no. 3, 479–536. MR**910018****[FKi]**Stefan Friedl and Taehee Kim,*The Thurston norm, fibered manifolds and twisted Alexander polynomials*, Topology**45**(2006), no. 6, 929–953. MR**2263219**, 10.1016/j.top.2006.06.003**[Ga2]**David Gabai,*Detecting fibred links in 𝑆³*, Comment. Math. Helv.**61**(1986), no. 4, 519–555. MR**870705**, 10.1007/BF02621931**[GKM]**Hiroshi Goda, Teruaki Kitano, and Takayuki Morifuji,*Reidemeister torsion, twisted Alexander polynomial and fibered knots*, Comment. Math. Helv.**80**(2005), no. 1, 51–61. MR**2130565**, 10.4171/CMH/3**[K]**Efstratia Kalfagianni,*Homology spheres with the same finite type invariants of bounded orders*, Math. Res. Lett.**4**(1997), no. 2-3, 341–347. MR**1453064**, 10.4310/MRL.1997.v4.n3.a3**[K1]**E. Kalfagianni,*Crossing changes of fibered knots.*math.GT/0610440**[KL]**E. Kalfagianni and X.-S. Lin,*Knot adjacency and satellites*, Topology Appl.**138**(2004), no. 1-3, 207–217. MR**2035481**, 10.1016/j.topol.2003.08.001**[KL1]**Efstratia Kalfagianni and Xiao-Song Lin,*Knot adjacency, genus and essential tori*, Pacific J. Math.**228**(2006), no. 2, 251–275. MR**2274520**, 10.2140/pjm.2006.228.251**[Kr1]**P. B. Kronheimer,*Embedded surfaces and gauge theory in three and four dimensions*, Surveys in differential geometry, Vol. III (Cambridge, MA, 1996) Int. Press, Boston, MA, 1998, pp. 243–298. MR**1677890****[Kr2]**P. B. Kronheimer,*Minimal genus in 𝑆¹×𝑀³*, Invent. Math.**135**(1999), no. 1, 45–61. MR**1664695**, 10.1007/s002220050279**[Mo]**H. R. Morton,*Fibred knots with a given Alexander polynomial*, Knots, braids and singularities (Plans-sur-Bex, 1982) Monogr. Enseign. Math., vol. 31, Enseignement Math., Geneva, 1983, pp. 205–222. MR**728587****[O]**Tomotada Ohtsuki,*Finite type invariants of integral homology 3-spheres*, J. Knot Theory Ramifications**5**(1996), no. 1, 101–115. MR**1373813**, 10.1142/S0218216596000084**[ST]**Martin Scharlemann and Abigail Thompson,*Unknotting number, genus, and companion tori*, Math. Ann.**280**(1988), no. 2, 191–205. MR**929535**, 10.1007/BF01456051**[Tu]**Vladimir Turaev,*Torsions of 3-dimensional manifolds*, Progress in Mathematics, vol. 208, Birkhäuser Verlag, Basel, 2002. MR**1958479**

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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

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.