Proposition

Show that the integer $n$ of Problem 4-24[Spivak] is unique.

Solution

Let $c = c_{1, n} + \partial c^2$.

\[\begin{align*} \int_{c} d\theta &= \int_{c_{1, n}} d\theta + \int_{\partial c^2} d\theta \\ &= \int_{\partial c_{1, n}} \theta + \int_{\partial^2 c^2} \theta & \text{(Stokes' theorem)} \\ &= \int_{\partial c_{1, n}} \theta + 0 \\ &= 2\pi n. \end{align*}\]

Thus the value $n$ must be unique.