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    Hernandes, M. E. Hernandes, M. E. Rodrigues and Ruas, M. A. S. 2007. $${\mathcal{A}}_e$$ -codimension of germs of analytic curves. manuscripta mathematica, Vol. 124, Issue. 2, p. 237.

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  • Mathematical Proceedings of the Cambridge Philosophical Society, Volume 117, Issue 2
  • March 1995, pp. 213-222

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

  • David Mond (a1)
  • DOI:
  • Published online: 24 October 2008

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.

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[1]N. A'Campo . Le groupe de monodromie du déploiement des singularitiés isolées de courbes planes I. Math. Annalen, 213 (1975), 131.

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[3]J. Damon and D. Mond . -codimension and the vanishing topology of discriminants. Invent. Math. 106 (1991), 217242.

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[18]K. Saito . Quasihomogene isolierte Singularitäten von Hyperflächen. Invent. Math. 14 (1971), 123142.

[19]C. T. C. Wall . Finite determinacy of smooth map-germs. Bull. London Math. Soc., 13 (1981) 481539.

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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
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