Linking number

Remarks

Equivalently, take a regular diagram of the link $K_1 \cup K_2$ and give each crossing between $K_1$ and $K_2$ the sign $+1$ or $-1$ by the right-hand rule. Then $\operatorname{lk}(K_1, K_2)$ is half the sum of these signs. In this form one sees that $\operatorname{lk}(K_1, K_2) = \operatorname{lk}(K_2, K_1)$.

Example: in the Hopf link, the two circles cross twice in a standard diagram, and both crossings have the same sign. So the linking number is $\pm 1$, the sign depending on the orientations.

Non-example of linking: in the unlink of two circles, the diagram has no crossings between the components, so the linking number is $0$. The converse fails: the Whitehead link has linking number $0$, but its two components cannot be pulled apart.

For smooth curves $\gamma_1, \gamma_2$ parametrising $K_1, K_2$, the Gauss integral gives $$\operatorname{lk}(K_1, K_2) = \frac{1}{4\pi} \oint_{\gamma_1} \oint_{\gamma_2} \frac{(\gamma_1 - \gamma_2) \cdot (d\gamma_1 \times d\gamma_2)}{\abs{\gamma_1 - \gamma_2}^3}.$$