On Floer homology and the Berge conjecture on knots admitting lens space surgeries
HTML articles powered by AMS MathViewer
- by Matthew Hedden PDF
- Trans. Amer. Math. Soc. 363 (2011), 949-968 Request permission
Abstract:
We complete the first step in a two-part program proposed by Baker, Grigsby, and the author to prove that Berge’s construction of knots in the three-sphere which admit lens space surgeries is complete. The first step, which we prove here, is to show that a knot in a lens space with a three-sphere surgery has simple (in the sense of rank) knot Floer homology. The second (conjectured) step involves showing that, for a fixed lens space, the only knots with simple Floer homology belong to a simple finite family. Using results of Baker, we provide evidence for the conjectural part of the program by showing that it holds for a certain family of knots. Coupled with work of Ni, these knots provide the first infinite family of non-trivial knots which are characterized by their knot Floer homology. As another application, we provide a Floer homology proof of a theorem of Berge.References
- Kenneth L. Baker, Small genus knots in lens spaces have small bridge number, Algebr. Geom. Topol. 6 (2006), 1519–1621. MR 2253458, DOI 10.2140/agt.2006.6.1519
- Kenneth L. Baker, Surgery descriptions and volumes of Berge knots. I. Large volume Berge knots, J. Knot Theory Ramifications 17 (2008), no. 9, 1077–1097. MR 2457837, DOI 10.1142/S0218216508006518
- Kenneth L. Baker, J. Elisenda Grigsby, and Matthew Hedden, Grid diagrams for lens spaces and combinatorial knot Floer homology, Int. Math. Res. Not. IMRN 10 (2008), Art. ID rnm024, 39. MR 2429242, DOI 10.1093/imrn/rnn024
- J. Berge. Some knots with surgeries yielding lens spaces. unpublished manuscript
- Steven A. Bleiler and Richard A. Litherland, Lens spaces and Dehn surgery, Proc. Amer. Math. Soc. 107 (1989), no. 4, 1127–1131. MR 984783, DOI 10.1090/S0002-9939-1989-0984783-3
- Doo Ho Choi and Ki Hyoung Ko, Parametrizations of 1-bridge torus knots, J. Knot Theory Ramifications 12 (2003), no. 4, 463–491. MR 1985906, DOI 10.1142/S0218216503002445
- Marc Culler, C. McA. Gordon, J. Luecke, and Peter B. Shalen, Dehn surgery on knots, Bull. Amer. Math. Soc. (N.S.) 13 (1985), no. 1, 43–45. MR 788388, DOI 10.1090/S0273-0979-1985-15357-1
- H. Doll, A generalized bridge number for links in $3$-manifolds, Math. Ann. 294 (1992), no. 4, 701–717. MR 1190452, DOI 10.1007/BF01934349
- Paolo Ghiggini, Knot Floer homology detects genus-one fibred knots, Amer. J. Math. 130 (2008), no. 5, 1151–1169. MR 2450204, DOI 10.1353/ajm.0.0016
- Hiroshi Goda and Masakazu Teragaito, Dehn surgeries on knots which yield lens spaces and genera of knots, Math. Proc. Cambridge Philos. Soc. 129 (2000), no. 3, 501–515. MR 1780501, DOI 10.1017/S0305004100004692
- Hiroshi Goda, Hiroshi Matsuda, and Takayuki Morifuji, Knot Floer homology of $(1,1)$-knots, Geom. Dedicata 112 (2005), 197–214. MR 2163899, DOI 10.1007/s10711-004-5403-2
- Matthew Hedden, Knot Floer homology of Whitehead doubles, Geom. Topol. 11 (2007), 2277–2338. MR 2372849, DOI 10.2140/gt.2007.11.2277
- P. Kronheimer, T. Mrowka, P. Ozsváth, and Z. Szabó, Monopoles and lens space surgeries, Ann. of Math. (2) 165 (2007), no. 2, 457–546. MR 2299739, DOI 10.4007/annals.2007.165.457
- Charles Livingston, Computations of the Ozsváth-Szabó knot concordance invariant, Geom. Topol. 8 (2004), 735–742. MR 2057779, DOI 10.2140/gt.2004.8.735
- Ciprian Manolescu, Peter Ozsváth, and Sucharit Sarkar, A combinatorial description of knot Floer homology, Ann. of Math. (2) 169 (2009), no. 2, 633–660. MR 2480614, DOI 10.4007/annals.2009.169.633
- Louise Moser, Elementary surgery along a torus knot, Pacific J. Math. 38 (1971), 737–745. MR 383406
- Yi Ni, Link Floer homology detects the Thurston norm, Geom. Topol. 13 (2009), no. 5, 2991–3019. MR 2546619, DOI 10.2140/gt.2009.13.2991
- Peter Ozsváth and Zoltán Szabó, Absolutely graded Floer homologies and intersection forms for four-manifolds with boundary, Adv. Math. 173 (2003), no. 2, 179–261. MR 1957829, DOI 10.1016/S0001-8708(02)00030-0
- Peter Ozsváth and Zoltán Szabó, Holomorphic disks and topological invariants for closed three-manifolds, Ann. of Math. (2) 159 (2004), no. 3, 1027–1158. MR 2113019, DOI 10.4007/annals.2004.159.1027
- Peter Ozsváth and Zoltán Szabó, Holomorphic disks and three-manifold invariants: properties and applications, Ann. of Math. (2) 159 (2004), no. 3, 1159–1245. MR 2113020, DOI 10.4007/annals.2004.159.1159
- Peter Ozsváth and Zoltán Szabó, Holomorphic triangles and invariants for smooth four-manifolds, Adv. Math. 202 (2006), no. 2, 326–400. MR 2222356, DOI 10.1016/j.aim.2005.03.014
- Peter Ozsváth and Zoltán Szabó, Knot Floer homology and the four-ball genus, Geom. Topol. 7 (2003), 615–639. MR 2026543, DOI 10.2140/gt.2003.7.615
- Peter Ozsváth and Zoltán Szabó, On knot Floer homology and lens space surgeries, Topology 44 (2005), no. 6, 1281–1300. MR 2168576, DOI 10.1016/j.top.2005.05.001
- Peter Ozsváth and Zoltán Szabó, Holomorphic disks and knot invariants, Adv. Math. 186 (2004), no. 1, 58–116. MR 2065507, DOI 10.1016/j.aim.2003.05.001
- Peter Ozsváth and Zoltán Szabó, Holomorphic disks and genus bounds, Geom. Topol. 8 (2004), 311–334. MR 2023281, DOI 10.2140/gt.2004.8.311
- Peter S. Ozsváth and Zoltán Szabó, Knot Floer homology and integer surgeries, Algebr. Geom. Topol. 8 (2008), no. 1, 101–153. MR 2377279, DOI 10.2140/agt.2008.8.101
- P. S. Ozsváth and Z. Szabó. Knot Floer homology and rational surgeries. math.GT/0504404, 2005.
- J. Rasmussen. Floer homology and knot complements. Ph.D. thesis, Harvard University, 2003.
- Jacob Rasmussen, Lens space surgeries and a conjecture of Goda and Teragaito, Geom. Topol. 8 (2004), 1013–1031. MR 2087076, DOI 10.2140/gt.2004.8.1013
- J. Rasmussen. Lens space surgeries and $L$-space homology spheres. Preprint, available at http://arxiv.org/abs/0710.2531.
Additional Information
- Matthew Hedden
- Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824
- MR Author ID: 769768
- Email: mhedden@math.msu.edu
- Received by editor(s): January 5, 2009
- Received by editor(s) in revised form: May 23, 2009
- Published electronically: September 22, 2010
- Additional Notes: The author was supported in part by NSF Grant #DMS-0706979.
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 363 (2011), 949-968
- MSC (2000): Primary 57M25, 57M27
- DOI: https://doi.org/10.1090/S0002-9947-2010-05117-7
- MathSciNet review: 2728591