International Mathematics Research Notices

International Mathematics Research Notices (2008) Vol. 2008 : article ID rnn024, 39 pages, doi:10.1093/imrn/rnn024 published on May 6, 2008

This Article Abstract Full Text (PDF) Alert me when this article is cited Alert me if a correction is posted Services Email this article to a friend Similar articles in this journal Alert me to new issues of the journal Add to My Personal Archive Download to citation manager Request Permissions How to cite this article Google Scholar Articles by Baker, K. L. Articles by Hedden, M. Social Bookmarking          What's this?

© The Author 2008. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oxfordjournals.org

Grid Diagrams for Lens Spaces and Combinatorial Knot Floer Homology

Kenneth L. Baker¹, J. Elisenda Grigsby² and Matthew Hedden³

¹ School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332, USA
² Department of Mathematics, Columbia University, 2990 Broadway MC4406 NY, NY 10027, USA
³ Department of Mathematics, Massachusetts Institute of Technology, Building 2, Room 236, 77 Massachusetts Avenue, Cambridge, MA 02139-4307, USA

Correspondence: Correspondence to be sent to: mhedden{at}math.mit.edu

Similar to knots in S³, any knot in a lens space has a grid diagram from which one can combinatorially compute all of its knot Floer homology invariants. We give an explicit description of the generators, differentials, and rational Maslov and Alexander gradings in terms of combinatorial data on the grid diagram. Motivated by existing results for the Floer homology of knots in S³ and the similarity of the resulting combinatorics presented here, we conjecture that a certain family of knots is characterized by their Floer homology. Coupled with the work of the third author, an affirmative answer to this would prove the Berge conjecture, which catalogs the knots in S³ admitting lens space surgeries.

References

1. Berge J. Some knots with surgeries yielding lens spaces. Unpublished manuscript.
2. Ghiggini P. Knot Floer homology detects genus-one fibered knots. American Journal of Mathematics. (forthcoming).
3. Greene J., Jabuka S. The slice-ribbon conjecture for 3-stranded pretzel knots. (2007) preprint arXiv:0706.3398v2 [math.GT].
4. Grigsby J. E., Ruberman D., Strle S. Knot concordance and Heegaard Floer homology invariants in branched covers. (2007) preprint arXiv:math/0701460v2 [math.GT].
5. Hedden M. On Floer homology and the Berge conjecture on knots admitting lens space surgeries. (2007) preprint arXiv:0710.0357v2 [math.GT].
6. Jabuka S., Naik S. Order in the concordance group and Heegaard Floer homology. Geometry & Topology (2007) 11:979–94.[CrossRef][ISI]
7. Lee D. A., Lipshitz R. Covering spaces and gradings on Heegaard Floer homology. (2006) preprint arXiv:math/0608001v1 [math.GT].
8. Lipshitz R. A cylindrical reformulation of Heegaard Floer homology. Geometry and Topology (2006) 10:955–1097.[CrossRef]
9. Manolescu C., Owens B. A concordance invariant from the Floer homology of double branched covers. International Mathematics Research Notices (2007) 2007(20):1–22.[Medline]
10. Manolescu C., Ozsváth P., Sarkar S. A combinatorial description of knot Floer homology. In: Geometry & Topology (2007) 11:2339–412.[CrossRef][ISI]
11. Manolescu C., Ozsváth P., Szabó Z., Thurston D. On combinatorial link Floer homology. (2006) preprint arXiv:math/0610559v2 [math.GT].
12. Ni Y. Link Floer homology detects the Thurston norm. (2006) preprint arXiv:math/ 0604360v1 [math.GT].
13. Ni Y. Knot Floer homology detects fibered knots. Inventiones Mathematicae (2007) 170(3):577–608.[CrossRef][ISI]
14. Ozsváth P., Szabó Z. Absolutely graded Floer homologies and intersection forms for four-manifolds with boundary. In: Advances in Mathematics (2003) 173(2):179–261.[CrossRef][ISI]
15. Ozsváth P., Szabó Z. Holomorphic disks and genus bounds. Geometry and Topology (2004) 8:311–34.[CrossRef]
16. Ozsváth P., Szabó Z. Holomorphic disks and knot invariants. In: Advances in Mathematics (2004) 186(1):58–116.[CrossRef][ISI]
17. Ozsváth P., Szabó Z. Holomorphic disks and topological invariants for closed three-manifolds. Annals of Mathematics (2004) 159(3):1027–158.[ISI]
18. Ozsváth P., Szabó Z. Knot Floer homology and rational surgeries. (2005) preprint arXiv:math/0504404v1 [math.GT].
19. Ozsváth P., Szabó Z. Holomorphic disks and link invariants. (2005) preprint arXiv:math/0512286v2 [math.GT].
20. Ozsváth P., Szabó Z. Link Floer homology and the Thurston norm. (2006) preprint arXiv:math/0601618v3 [math.GT].
21. Ozsváth P., Szabó Z. Holomorphic triangles and invariants for smooth four-manifolds. Advances in Mathematics (2006) 202(2):326–400.[CrossRef][ISI]
22. Ozsváth P., Szabó Z., Thurston D. Legendrian knots, transverse knots and combinatorial Floer homology. (2006) preprint math.GT/0611841 [math.GT].
23. Rasmussen J. Floer homology and knot complements. (2003) PhD thesis, Harvard University.
24. Rasmussen J. Lens space surgeries and L-space homology spheres. (2007) preprint arXiv:0710.2531v1 [math.GT].
25. Sarkar S., Wang J. An algorithm for computing some Heegaard Floer homologies. (2006) preprint arXiv:math/0607777v3 [math.GT].
26. Turaev V. Torsion invariants of -structures on 3-manifolds. Mathematical Research Letters (1997) 4(5):679–95.[ISI]
27. Turaev V. Torsions of 3-Dimensional Manifolds (2002) Basel, Switzerland: Birkhäuser. Progress in Mathematics 208.

CiteULike    Connotea    Del.icio.us    What's this?

This Article Abstract Full Text (PDF) Alert me when this article is cited Alert me if a correction is posted Services Email this article to a friend Similar articles in this journal Alert me to new issues of the journal Add to My Personal Archive Download to citation manager Request Permissions How to cite this article Google Scholar Articles by Baker, K. L. Articles by Hedden, M. Social Bookmarking          What's this?