A higher-order genus invariant and knot Floer homology

Horn, Peter

doi:10.1090/S0002-9939-10-10263-9

A higher-order genus invariant and knot Floer homology
HTML articles powered by AMS MathViewer

by Peter D. Horn PDF

Proc. Amer. Math. Soc. 138 (2010), 2209-2215 Request permission

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

A. J. Casson and C. McA. Gordon, On slice knots in dimension three, Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976) Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978, pp. 39–53. MR 520521
Jeff Cheeger and Mikhael Gromov, Bounds on the von Neumann dimension of $L^2$-cohomology and the Gauss-Bonnet theorem for open manifolds, J. Differential Geom. 21 (1985), no. 1, 1–34. MR 806699
A. J. Casson and C. McA. Gordon, Cobordism of classical knots, À la recherche de la topologie perdue, Progr. Math., vol. 62, Birkhäuser Boston, Boston, MA, 1986, pp. 181–199. With an appendix by P. M. Gilmer. MR 900252
Tim Cochran, Shelly Harvey, and Constance Leidy, Derivatives of knots and second-order signatures, http://arXiv.org/pdf/0808.1432, 2008. To appear in Algebra & Topology
Jae Choon Cha and Charles Livingston, Knotinfo: Table of knot invariants, http://www.indiana.edu/$\sim$knotinfo, February 2010.
Tim D. Cochran, Kent E. Orr, and Peter Teichner, Knot concordance, Whitney towers and $L^2$-signatures, Ann. of Math. (2) 157 (2003), no. 2, 433–519. MR 1973052, DOI 10.4007/annals.2003.157.433
P. R. Cromwell, Homogeneous links, J. London Math. Soc. (2) 39 (1989), no. 3, 535–552. MR 1002465, DOI 10.1112/jlms/s2-39.3.535
Marc Culler, Gridlink 2.0, software, available at http://www.math.uic.edu/ $\sim$culler/gridlink.
Stefan Friedl, Eta invariants as sliceness obstructions and their relation to Casson-Gordon invariants, Algebr. Geom. Topol. 4 (2004), 893–934. MR 2100685, DOI 10.2140/agt.2004.4.893
Matthew Hedden, Knot Floer homology of Whitehead doubles, Geom. Topol. 11 (2007), 2277–2338. MR 2372849, DOI 10.2140/gt.2007.11.2277
—, in preparation, 2008.
Peter D. Horn, The first-order genus of a knot, Math. Proc. Cambridge Philos. Soc. 146 (2009), no. 1, 135–149. MR 2461873, DOI 10.1017/S0305004108001886
Andras Juhasz, Knot Floer homology and Seifert surfaces, Algebr. Geom. Topol. 8 (2008), no. 1, 603–608. MR 2443240, DOI 10.2140/agt.2008.8.603
Carl F. Letsche, An obstruction to slicing knots using the eta invariant, Math. Proc. Cambridge Philos. Soc. 128 (2000), no. 2, 301–319. MR 1735303, DOI 10.1017/S0305004199004016
Yair N. Minsky, The classification of punctured-torus groups, Ann. of Math. (2) 149 (1999), no. 2, 559–626. MR 1689341, DOI 10.2307/120976
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 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
Jacob Rasmussen, Floer homology and knot complements, Ph.D. thesis, Harvard University, 2003.
Shinichi Suzuki, Introduction to knot theory, Saien-su-sha (in Japanese), 1991.
Wilbur Whitten, Isotopy types of knot spanning surfaces, Topology 12 (1973), 373–380. MR 372842, DOI 10.1016/0040-9383(73)90029-3