We show that the only way of changing the framing of a link by ambient isotopy in an oriented $3$ -manifold is when the manifold has a properly embedded non-separating $S^{2}$ . This change of framing is given by the Dirac trick, also known as the light bulb trick. The main tool we use is based on McCullough’s work on the mapping class groups of $3$ -manifolds. We also relate our results to the theory of skein modules.

中文翻译：

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关于三流形中链接的框架

我们表明，通过定向 $3$ 流形中的环境同位素改变链接框架的唯一方法 是当流形具有正确嵌入的非分离 $S^{2}$ 时 。这种框架的变化是由狄拉克技巧给出的，也称为灯泡技巧。我们使用的主要工具基于 McCullough 在 $3$ -manifolds的映射类组上的工作 。我们还将我们的结果与绞线模块理论联系起来。