The Hopf invariant of a Haefliger knot

• Autores: Masamichi Takase
• Localización: Mathematische zeitschrift, ISSN 0025-5874, Vol. 256, Nº. 1, 2007, págs. 35-44
• Idioma: inglés
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• Resumen

  • We show that for any given differentiable embedding of the three-sphere in six-space there exists a Seifert surface (in six-space) with arbitrarily prescribed signature. This implies, according to our previous paper, that given such a (6,3)-knot endowed with normal one-field, we can construct a Seifert surface so that the outward normal field along its boundary coincides with the given normal one-field. This aspect enables us to understand the resemblance between Ekholm¿Szucs¿ formula for the Smale invariant and a formula in our previous paper for differentiable (6,3)-knots. As a consequence, we show that an immersion of the three-sphere in five-space can be regularly homotoped to the projection of an embedding in six-space if and only if its Smale invariant is even. We also correct a sign error in our previous paper: ¿A geometric formula for Haefliger knots¿ [Topology 43: 1425¿1447 2004].