Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-30T20:15:03.277Z Has data issue: false hasContentIssue false
  • English
  • Français

Looking at bent wires – -codimension and the vanishing topology of parametrized curve singularities

Published online by Cambridge University Press:  24 October 2008

David Mond
David Mond
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7AL
Get access Rights & Permissions [Opens in a new window]

Extract

Projecting a knot onto a plane – or, equivalently, looking at it through one eye – one sees a more or less complicated plane curve with a number of crossings (‘nodes’); viewing it from certain positions, some other more complicated singularities appear. If one spends a little time experimenting, looking at the knot from different points of view, then provided the knot is generic, one can convince oneself that there is only a rather short list of essentially distinct local pictures (singularities) – see Fig. 3 below. All singularities other than nodes are unstable: by moving one's eye slightly, one can make them break up into nodes. For each type X the following two numbers can easily be determined experimentally:

1. the codimension in ℝ3 of the set View(X) of centres of projection (viewpoints) for which a singularity of type X appears, and

2. the maximum number of nodes n into which the singularity X splits when the centre of projection is moved.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 117 , Issue 2 , March 1995 , pp. 213 - 222
Copyright
Copyright © Cambridge Philosophical Society 1995

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]A'Campo, N.. Le groupe de monodromie du déploiement des singularitiés isolées de courbes planes I. Math. Annalen, 213 (1975), 1–31.CrossRefGoogle Scholar
[2]Burghelea, D. and Verona, A.. Local homological properties of analytic sets. Manuscripta Math. 7 (1972), 55–66.CrossRefGoogle Scholar
[3]Damon, J. and Mond, D.. -codimension and the vanishing topology of discriminants. Invent. Math. 106 (1991), 217–242.CrossRefGoogle Scholar
[4]David, J. M. S.. Projection-generic curves. J. London Math. Soc. (2), 27 (1983), 552–562.CrossRefGoogle Scholar
[5]Gaffney, T. and Mond, D.. Weighted homogeneous maps from the plane to the plane. Math. Proc. Camb. Phil. Soc. 109 (1991), 451–470.CrossRefGoogle Scholar
[6]Greuel, G.-M.. Dualität in der lokalen Kohomologie isolierter Singularitäten. Math. Ann. 250 (1980), 157–173.CrossRefGoogle Scholar
[7]Gusein-Zade, S. M.. Dynkin diagrams for certain singularities of functions of two real variables. Functional Analysis and Appl. 8 (1974), 295–300.CrossRefGoogle Scholar
[8]De Jong, T. and van Straten, D.. Disentanglements, Singularity Theory and Applications, Warwick 1989. Springer Lecture Notes in Math. 1462 (1991), 198–211.Google Scholar
[9]Looijenga, E. J. N. and Steenbrink, J. H. M.. Milnor numbers and Tjurina numbers of complete intersections. Math. Ann. 271 (1985), 121–124.CrossRefGoogle Scholar
[10]Marar, W. L.. Mapping fibrations. Manuscripta Math. 80 (1993) 273–281.CrossRefGoogle Scholar
[11]Martinet, J.. Singularités des fonctions et applications différentiables (Pontificia Universidade Católica; Rio de Janeiro, 1974) translated as Singularities of smooth functions and mappings, London Math. Soc. Lecture Notes in Math. 58 (Cambridge University Press, 1982).Google Scholar
[12]Mather, J.. Stability of C ∞ mappings II: Infinitesimal stability implies stability. Ann. Math. 89 (1969), 254–291.CrossRefGoogle Scholar
[13]Mather, J.. Stability of C ∞ mappings VI: the nice dimensions. Proc. Liverpool Singularities Symposium Vol. 1 Springer Lecture Notes in Math. 192 (1971), 207–255.CrossRefGoogle Scholar
[14]Milnor, J.. Singular points of complex hypersurfaces. Ann. of Math. Studies 61 (Princeton University Press, 1968).Google Scholar
[15]Mond, D.. Singularities of the exponential map of the tangent bundle associated with an immersion. Proc. London Math. Soc. (3) 53 (1986), 357–384.CrossRefGoogle Scholar
[16]Mond, D.. Vanishing cycles for analytic maps. Singularity Theory and Applications, Warwick 1989, Springer Lecture Notes in Math. 1462 (1991), 221–234.CrossRefGoogle Scholar
[17]Mond, D. and Pellikaan, R.. Fitting ideals and multiple points of analytic maps. In Algebraic Geometry and Complex Analysis, Patzcuaro 1987, Springer Lecture Notes in Math. 1414 (1989), 107–161.CrossRefGoogle Scholar
[18]Saito, K.. Quasihomogene isolierte Singularitäten von Hyperflächen. Invent. Math. 14 (1971), 123–142.CrossRefGoogle Scholar
[19]Wall, C. T. C.. Finite determinacy of smooth map-germs. Bull. London Math. Soc., 13 (1981) 481–539.CrossRefGoogle Scholar