Geometry & Topology

Fibered knots and potential counterexamples to the Property 2R and Slice-Ribbon Conjectures

Robert E Gompf, Martin Scharlemann, and Abigail Thompson

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If there are any 2–component counterexamples to the Generalized Property R Conjecture, a least genus component of all such counterexamples cannot be a fibered knot. Furthermore, the monodromy of a fibered component of any such counterexample has unexpected restrictions.

The simplest plausible counterexample to the Generalized Property R Conjecture could be a 2–component link containing the square knot. We characterize all two-component links that contain the square knot and which surger to #2(S1×S2). We exhibit a family of such links that are probably counterexamples to Generalized Property R. These links can be used to generate slice knots that are not known to be ribbon.

Article information

Geom. Topol., Volume 14, Number 4 (2010), 2305-2347.

Received: 21 January 2010
Revised: 24 August 2010
Accepted: 29 September 2010
First available in Project Euclid: 21 December 2017

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Zentralblatt MATH identifier

Property R Slice-Ribbon Conjecture Andrews–Curtis moves


Gompf, Robert E; Scharlemann, Martin; Thompson, Abigail. Fibered knots and potential counterexamples to the Property 2R and Slice-Ribbon Conjectures. Geom. Topol. 14 (2010), no. 4, 2305--2347. doi:10.2140/gt.2010.14.2305.

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  • S Akbulut, Cappell–Shaneson homotopy spheres are standard, Ann. of Math. (2) 171 (2010) 2171–2175
  • A J Casson, C M Gordon, Reducing Heegaard splittings, Topology Appl. 27 (1987) 275–283
  • T D Cochran, J P Levine, Homology boundary links and the Andrews–Curtis conjecture, Topology 30 (1991) 231–239
  • M H Freedman, The topology of four-dimensional manifolds, J. Differential Geom. 17 (1982) 357–453
  • M H Freedman, R E Gompf, S Morrison, K Walker, Man and machine thinking about the smooth $4$–dimensional Poincaré conjecture, Quantum Topol. 1 (2010) 171–208
  • D Gabai, Foliations and the topology of $3$–manifolds. II, J. Differential Geom. 26 (1987) 461–478
  • D Gabai, Foliations and the topology of $3$–manifolds. III, J. Differential Geom. 26 (1987) 479–536
  • D Gabai, Surgery on knots in solid tori, Topology 28 (1989) 1–6
  • S Gersten, On Rapaport's example in presentations of the trivial group, Preprint (1987)
  • R E Gompf, Killing the Akbulut–Kirby $4$–sphere, with relevance to the Andrews–Curtis and Schoenflies problems, Topology 30 (1991) 97–115
  • R E Gompf, More Cappell–Shaneson spheres are standard, Algebr. Geom. Topol. 10 (2010) 1665–1681
  • R E Gompf, A I Stipsicz, $4$–manifolds and Kirby calculus, Graduate Studies in Math. 20, Amer. Math. Soc. (1999)
  • J A Hillman, Alexander ideals of links, Lecture Notes in Math. 895, Springer, Berlin (1981)
  • R Kirby, A calculus for framed links in $S\sp{3}$, Invent. Math. 45 (1978) 35–56
  • R Kirby (editor), Problems in low-dimensional topology, AMS/IP Stud. Adv. Math. 2, Amer. Math. Soc. (1997)
  • R Kirby, P Melvin, Slice knots and property ${\rm R}$, Invent. Math. 45 (1978) 57–59
  • F Laudenbach, V Poénaru, A note on $4$–dimensional handlebodies, Bull. Soc. Math. France 100 (1972) 337–344
  • D Rolfsen, Knots and links, Math. Lecture Series 7, Publish or Perish, Houston, TX (1990) Corrected reprint of the 1976 original
  • M Scharlemann, A A Thompson, Surgery on a knot in (surface ${\times}\ I$), Algebr. Geom. Topol. 9 (2009) 1825–1835