Tokyo Journal of Mathematics

Cobordism of Algebraic Knots Defined by Brieskorn Polynomials

Vincent BLANLŒIL and Osamu SAEKI

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Abstract

In this paper we study the cobordism of algebraic knots associated with weighted homogeneous polynomials, and in particular Brieskorn polynomials. Under some assumptions we prove that the associated algebraic knots are cobordant if and only if the Brieskorn polynomials have the same exponents.

Article information

Source
Tokyo J. Math., Volume 34, Number 2 (2011), 429-443.

Dates
First available in Project Euclid: 30 January 2012

Permanent link to this document
https://projecteuclid.org/euclid.tjm/1327931395

Digital Object Identifier
doi:10.3836/tjm/1327931395

Mathematical Reviews number (MathSciNet)
MR2918915

Zentralblatt MATH identifier
1241.57029

Subjects
Primary: 57Q45: Knots and links (in high dimensions) {For the low-dimensional case, see 57M25}
Secondary: 57Q60: Cobordism and concordance 32S55: Milnor fibration; relations with knot theory [See also 57M25, 57Q45]

Citation

BLANLŒIL, Vincent; SAEKI, Osamu. Cobordism of Algebraic Knots Defined by Brieskorn Polynomials. Tokyo J. Math. 34 (2011), no. 2, 429--443. doi:10.3836/tjm/1327931395. https://projecteuclid.org/euclid.tjm/1327931395


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References

  • V. Blanlœil and F. Michel, A theory of cobordism for non-spherical links, Comment. Math. Helv., 72 (1997), 30–51.
  • V. Blanlœil and O. Saeki, Cobordism of fibered knots and related topics, in “Singularities in geometry and topology 2004”, pp. 1–47, Adv. Stud. Pure Math., 46, Math. Soc. Japan, Tokyo, 2007.
  • P. Du Bois and O. Hunault, Classification des formes de Seifert rationnelles des germes de courbe plane, Ann. Inst. Fourier (Grenoble), 46 (1996), 371–410.
  • A. Durfee, Fibered knots and algebraic singularities, Topology, 13 (1974), 47–59.
  • C. Kang, Analytic types of plane curve singularities defined by weighted homogeneous polynomials, Trans. Amer. Math. Soc., 352 (2000), 3995–4006.
  • M. Kato, A classification of simple spinnable structures on a 1-connected Alexander manifold, J. Math. Soc. Japan, 26 (1974), 454–463.
  • H. C. King, Topological type of isolated critical points, Ann. of Math. (2), 107 (1978), 385–397.
  • D. T. Lê, Sur les nœ uds algébriques, Compositio Math., 25 (1972), 281–321.
  • J. Levine, Knot cobordism groups in codimension two, Comment. Math. Helv., 44 (1969), 229–244.
  • J. Levine, Invariants of knot cobordism, Invent. Math., 8 (1969), 98–110; addendum, ibid. 8 (1969), 355.
  • J. Milnor, Singular points of complex hypersurfaces, Ann. of Math. Stud., Vol. 61, Princeton Univ. Press, Princeton, N.J.; Univ. of Tokyo Press, Tokyo, 1968.
  • J. Milnor and P. Orlik, Isolated singularities defined by weighted homogeneous polynomials, Topology, 9 (1970), 385–393.
  • A. Némethi, The real Seifert form and the spectral pairs of isolated hypersurface singularities, Compositio Math., 98 (1995), 23–41.
  • W. D. Neumann, Invariants of plane curve singularities, in “Nœ uds, tresses et singularité (Plans-sur-Bex, 1982)”, pp. 223–232, Monogr. Enseign. Math., Vol. 31, Enseignement Math., Geneva, 1983.
  • T. Nishimura, Topological invariance of weights for weighted homogeneous singularities, Kodai Math. J., 9 (1986), 188–190.
  • B. Perron, Conjugaison topologique des germes de fonctions holomorphes à singularité isolée en dimension trois, Invent. Math., 82 (1985), 27–35.
  • O. Saeki, Cobordism classification of knotted homology 3-spheres in $S^5$, Osaka J. Math., 25 (1988), 213–222.
  • O. Saeki, Topological types of complex isolated hypersurface singularities, Kodai Math. J., 12 (1989), 23–29.
  • O. Saeki, Real Seifert form determines the spectrum for semiquasihomogeneous hypersurface singularities in $\C^3$, J. Math. Soc. Japan, 52 (2000), 409–431.
  • K. Saito, Quasihomogene isolierte Singularitäten von Hyperflächen, Invent. Math., 14 (1971), 123–142.
  • K. Sakamoto, The Seifert matrices of Milnor fiberings defined by holomorphic functions, J. Math. Soc. Japan, 26 (1974), 714–721.
  • R. Schrauwen, J. Steenbrink, and J. Stevens, Spectral pairs and the topology of curve singularities, in “Complex geometry and Lie theory (Sundance, UT, 1989)”, pp. 305–328, Proc. Sympos. Pure Math., Vol. 53, Amer. Math. Soc., Providence, RI, 1991.
  • J. H. M. Steenbrink, Mixed Hodge structure on the vanishing cohomology, in “Real and complex singularities (P. Holm, ed.)”, pp. 525–563, Stijthoff-Noordhoff, Alphen a/d Rijn, 1977.
  • J. H. M. Steenbrink, Intersection form for quasihomogeneous singularities, Compositio Math., 34 (1977), 211–223.
  • E. Yoshinaga and M. Suzuki, On the topological types of singularities of Brieskorn-Pham type, Sci. Rep. Yokohama Nat. Univ. Sect. I, 25 (1978), 37–43.
  • E. Yoshinaga and M. Suzuki, Topological types of quasihomogeneous singularities in $\C^2$, Topology, 18 (1979), 113–116.
  • O. Zariski, On the topology of algebroid singularities, Amer. J. Math., 54 (1932), 453–465.