Fig. 12: Classification of three-component links.
The bicoloring is ignored for simplicity. a The linking number between L1 and L2 is −2, and the loop L3 corresponds to an element of form \({\alpha }_{1}{\alpha }_{2}^{3}{\alpha }_{1}^{-1}{\alpha }_{2}^{-3}{\alpha }_{2}^{-2}{\alpha }_{1}^{-4}\) in \({\pi }_{1}({{\mathbb{R}}}^{3}\backslash ({L}_{1}\cup {L}_{2}))\). b The same link after applying the surgery operation depicted in Fig. 13a several times. For the resulting link, the pairwise linking numbers are 0, and the triple linking number μ(123) is 1, so the link is homotopic to the Borromean rings.
