At a crossing $x$, the sign is the orientation of the ordered pair of tangent vectors $(\dot\alpha,\dot\beta)$. Exchanging the two vectors reverses orientation, so
$$
\sigma_{\beta,\alpha}(x)=-\sigma_{\alpha,\beta}(x).
$$
Summing over the same finite set of crossings gives
$$
\langle\alpha,\beta\rangle=-\langle\beta,\alpha\rangle.
$$


Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 156 / 3 / c / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-156/3/c/solution