Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



A higher-order genus invariant and knot Floer homology

Author: Peter D. Horn
Journal: Proc. Amer. Math. Soc. 138 (2010), 2209-2215
MSC (2010): Primary 57M25
Published electronically: February 9, 2010
MathSciNet review: 2596061
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: It is known that knot Floer homology detects the genus and Alexander polynomial of a knot. We investigate whether knot Floer homology of $ K$ detects more structure of minimal genus Seifert surfaces for $ K$. We define an invariant of algebraically slice, genus one knots and provide examples to show that knot Floer homology does not detect this invariant. Finally, we remark that certain metabelian $ L^2$-signatures bound this invariant from below.

References [Enhancements On Off] (What's this?)

  • [CG78] Andrew J. Casson and Cameron McA. Gordon, On slice knots in dimension three, Algebraic and Geometric Topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, CA, 1976), Proceedings of Symposia in Pure Mathematics, vol. XXXII, American Mathematical Society, Providence, RI, 1978, pp. 39-53. MR 520521 (81g:57003)
  • [CG85] Jeff Cheeger and Mikhael Gromov, Bounds on the von Neumann dimension of $ {L}^2$-cohomology and the Gauss-Bonnet theorem for open manifolds, Journal of Differential Geometry 21 (1985), 1-34. MR 806699 (87d:58136)
  • [CG86] Andrew J. Casson and Cameron McA. Gordon, Cobordism of classical knots, À la recherche de la topologie perdue, Progress in Mathematics, vol. 62, Birkhäuser Boston, Boston, MA, 1986, pp. 181-199. MR 900252
  • [CHL08] Tim Cochran, Shelly Harvey, and Constance Leidy, Derivatives of knots and second-order signatures,, 2008. To appear in Algebra & Topology
  • [CL09] Jae Choon Cha and Charles Livingston, Knotinfo: Table of knot invariants,$ \sim$knotinfo, February 2010.
  • [COT03] Tim Cochran, Kent Orr, and Peter Teichner, Knot concordance, Whitney towers and $ {L}^2$-signatures, Annals of Mathematics 157 (2003), 433-519. MR 1973052 (2004i:57003)
  • [Cro89] Peter Cromwell, Homogeneous links, Journal of the London Mathematical Society 39 (1989), no. 2, 535-552. MR 1002465 (90f:57001)
  • [Cul] Marc Culler, Gridlink 2.0, software, available at $ \sim$culler/gridlink.
  • [Fri04] Stefan Friedl, Eta invariants as sliceness obstructions and their relation to Casson-Gordon invariants, Algebraic & Geometric Topology 4 (2004), 893-934. MR 2100685 (2005j:57016)
  • [Hed07] Matthew Hedden, Knot Floer homology of Whitehead doubles, Geometry and Topology 11 (2007), 2277-2338. MR 2372849 (2008m:57030)
  • [Hed08] -, in preparation, 2008.
  • [Hor09] Peter D. Horn, The first-order genus of a knot, Mathematical Proceedings of the Cambridge Philosophical Society 146 (2009), no. 1, 135-149. MR 2461873 (2009i:57012)
  • [Juh08] András Juhász, Knot Floer homology and Seifert surfaces, Algebraic and Geometric Topology 8 (2008), 603-608. MR 2443240
  • [Let00] Carl F. Letsche, An obstruction to slicing knots using the eta invariant, Mathematical Proceedings of the Cambridge Philosophical Society 128 (2000), no. 2, 301-319. MR 1735303 (2001b:57017)
  • [Min99] Yair Minsky, The classification of punctured-torus groups, Annals of Mathematics 149 (1999), 559-626. MR 1689341 (2000f:30028)
  • [OS04a] Peter Ozsváth and Zoltán Szabó, Holomorphic disks and genus bounds, Geometry and Topology 8 (2004), 311-334. MR 2023281 (2004m:57024)
  • [OS04b] -, Holomorphic disks and knot invariants, Advances in Mathematics 186 (2004), no. 1, 58-116. MR 2065507 (2005e:57044)
  • [Ras03] Jacob Rasmussen, Floer homology and knot complements, Ph.D. thesis, Harvard University, 2003.
  • [Suz91] Shinichi Suzuki, Introduction to knot theory, Saien-su-sha (in Japanese), 1991.
  • [Whi73] Wilbur Whitten, Isotopy types of knot spanning surfaces, Topology 12 (1973), 373-380. MR 0372842 (51:9046)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 57M25

Retrieve articles in all journals with MSC (2010): 57M25

Additional Information

Peter D. Horn
Affiliation: Department of Mathematics, Rice University–MS 136, P.O. Box 1892, Houston, Texas 7725-1892
Address at time of publication: Department of Mathematics, Columbia University, 2990 Broadway, New York, New York 10025

Received by editor(s): April 23, 2009
Received by editor(s) in revised form: July 23, 2009
Published electronically: February 9, 2010
Additional Notes: The author was partially supported by National Science Foundation grant DMS-0706929, the Lodieska Stockbridge Vaughn Fellowhip at Rice University, and the NSF Mathematical Sciences Postdoctoral Research Fellowship DMS-0902786.
Communicated by: Daniel Ruberman
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society