Extension d’un théorème de Louis Antoine

Abstract:

Let $f:[0,1] \times S \to {R^3}$ be a map subject to the conditions: (1) $f|]0,1] \times S$ is $(1,1)$ into; (2) $f|\{ 0\} \times S$ is not $(1,1)$ into; (3) The image ${\Gamma _0} = f(\{ 0\} \times S)$ is a Jordan curve ; (4) $f(\{ 0\} \times S) \cap f(]0,1] \times S) = \emptyset$. Let $\mu$ be the linking number of each of the curves ${\Gamma _t} = f(\{ t\} \times S),t \in ]0,1]$, with ${\Gamma _0}$. Let $v$ be the degree of the mapping $h:S \to {\Gamma _0}$ defined by $h(u) = f(0,u)$. We prove that, if ${\Gamma _0}$ is tame, the integers $\mu$ and $v$ are relatively prime. The question is open in case that ${\Gamma _0}$ is wild.