Hostname: page-component-745bb68f8f-b95js Total loading time: 0 Render date: 2025-01-11T02:11:52.708Z Has data issue: false hasContentIssue false
  • English
  • Français

HEEGAARD FLOER HOMOLOGY AND RATIONAL CUSPIDAL CURVES

Part of: Local theory Curves Differential topology Low-dimensional topology

Published online by Cambridge University Press:  05 December 2014

MACIEJ BORODZIK  and
CHARLES LIVINGSTON
MACIEJ BORODZIK
Affiliation:
Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02-097 Warsaw, Poland; mcboro@mimuw.edu.pl
CHARLES LIVINGSTON
Affiliation:
Department of Mathematics, Indiana University, Bloomington, IN 47405, USA; livingst@indiana.edu

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We apply the methods of Heegaard Floer homology to identify topological properties of complex curves in $\mathbb{C}P^{2}$. As one application, we resolve an open conjecture that constrains the Alexander polynomial of the link of the singular point of the curve in the case that there is exactly one singular point, having connected link, and the curve is of genus zero. Generalizations apply in the case of multiple singular points.

MSC classification

Primary: 14H50: Plane and space curves
Secondary: 14B05: Singularities 57M25: Knots and links in $S^3$ 57R58: Floer homology
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 2 , 01 July 2014 , e28
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/3.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2014

References

Arnold, V. I., Varchenko, A. N. and Gussein-Zade, S. M., Singularities of Differentiable Mappings. II (Nauka, Moscow, 1984).Google Scholar
Bodnár, J. and Némethi, A., ‘Lattice cohomology and rational cuspidal curves’, Preprint, 2014, arXiv:1405.0437.Google Scholar
Borodzik, M. and Némethi, A., ‘Spectrum of plane curves via knot theory’, J. Lond. Math. Soc. 86 (2012), 87–110.Google Scholar
Brieskorn, E. and Knörrer, H., Plane Algebraic Curves (Birkhäuser, Basel–Boston–Stuttgart, 1986).Google Scholar
Coolidge, J., A Treatise on Plane Algebraic Curves (Oxford University Press, Oxford, 1928).Google Scholar
Fernández de Bobadilla, J., Luengo, I., Melle-Hernández, A. and Némethi, A., ‘On rational cuspidal projective plane curves’, Proc. Lond. Math. Soc. 92 (2006), 99–138.Google Scholar
Fernández de Bobadilla, J., Luengo, I., Melle-Hernández, A. and Némethi, A., ‘Classification of rational unicuspidal projective curves whose singularities have one Puiseux pair’, inProceedings of Sao Carlos Workshop 2004 Real and Complex Singularities, Series Trends in Mathematics (Birkhäuser, 2007), 31–46.Google Scholar
Greuel, G. M., Lossen, C. and Shustin, E., Introduction to Singularities and Deformations, Springer Monographs in Mathematics (Springer, Berlin, 2007).Google Scholar
Hancock, S., Hom, J. and Newmann, M., ‘On the knot Floer filtration of the concordance group’, Preprint, 2012, arXiv:1210.4193.Google Scholar
Hedden, M., ‘On knot Floer homology and cabling. II’, Int. Math. Res. Not. 2009(12) 2248–2274.Google Scholar
Hedden, M., Livingston, C. and Ruberman, D., ‘Topologically slice knots with nontrivial Alexander polynomial’, Adv. Math. 231 (2012), 913–939.Google Scholar
Liu, T., ‘On planar rational cuspidal curves’, PhD Thesis, MIT, 2014, Available at http://dspace.mit.edu/bitstream/handle/1721.1/90190/890211671.pdf.Google Scholar
Matsuoka, T. and Sakai, F., ‘The degree of rational cuspidal curves’, Math. Ann. 285 (1989), 233–247.Google Scholar
Milnor, J., Singular Points of Complex Hypersurfaces, Annals of Mathematics Studies, 61 (Princeton University Press and the University of Tokyo Press, Princeton, NJ, 1968).Google Scholar
Moe, T. K., ‘Rational cuspidal curves’, Master Thesis, University of Oslo, 2008, permanent link at University of Oslo: https://www.duo.uio.no/handle/123456789/10759.Google Scholar
Nagata, M., ‘On rational surfaces. I. Irreducible curves of arithmetic genus 0 or 1’, Mem. Coll. Sci., Univ. Kyoto, A 32 (1960), 351–370.Google Scholar
Nayar, P. and Pilat, B., ‘A note on the rational cuspidal curves’, Bull. Pol. Acad. Sci. Math. 62 (2014), 117–123.Google Scholar
Némethi, A., ‘Lattice cohomology of normal surface singularities’, Publ. RIMS. Kyoto Univ. 44 (2008), 507–543.Google Scholar
Némethi, A. and Nicolaescu, L., ‘Seiberg–Witten invariants and surface singularities: splicings and cyclic covers’, Sel. Math. New Ser. 11 (2005), 399–451.Google Scholar
Némethi, A. and Róman, F., ‘The lattice cohomology of S d3(K)’, inZeta Functions in Algebra and Geometry, Contemporary Mathematics, 566 (American Mathematical Society, Providence, RI, 2012), 261–292.Google Scholar
Orevkov, S., ‘On rational cuspidal curves. I. Sharp estimates for degree via multiplicity’, Math. Ann. 324 (2002), 657–673.Google Scholar
Ozsváth, P. and Szabó, Z., ‘Absolutely graded Absolutely graded Floer homologies and intersection forms for four-manifolds with boundary’, Adv. Math. 173 (2003), 179–261.Google Scholar
Ozsváth, P. and Szabó, Z., ‘Holomorphic disks and knot invariants’, Adv. Math. 186 (2004), 58–116.Google Scholar
Ozsváth, P. and Szabó, Z., ‘On knot Floer homology and lens space surgeries’, Topology 44 (2005), 1281–1300.Google Scholar
Piontkowski, J., ‘On the number of cusps of rational cuspidal plane curves’, Exp. Math. 16(2) (2007), 251–255.Google Scholar
Saito, M., ‘Exponents and Newton polyhedra of isolated hypersurface singularities’, Math. Ann. 281 (1988), 411–417.Google Scholar
Tono, K., ‘On the number of cusps of cuspidal plane curves’, Math. Nachr. 278 (2005), 216–221.Google Scholar
Wall, C. T. C., Singular Points of Plane Curves, London Mathematical Society Student Texts, 63 (Cambridge University Press, Cambridge, 2004).Google Scholar