Why is the unknotting number of Borromean rings 1?

Question

Wikipedia claims that the unknotting number of the Borromean rings is 1, which I believe means that they can be totally separated if we are allowed to pass the rings through themselves in a single place. However it seems that making a single crossing switch in the Borromean rings would always leave two of the rings more linked than they were to start with? What have I misunderstood?

Answer 1

You're right, the unknotting (unlinking, really) number is 2.

To see that it is at most 2, simply notice that changing the two crossings shared by a pair of components will unlink the link.

To see that it is at least 2, we follow this paper which provides an overview of several ways of bounding the unlinking number from below. In particular, if we take the chessboard shading in which the unbounded region is unshaded and choose a labeling of the four shaded regions, then following the notation of the paper, $\iota(c) = -1$ for every crossing, and so the Goeritz matrix becomes

$$G = \begin{pmatrix} -3 & 1 & 1 \\ 1 & -3 & 1 \\ 1 & 1 & -3 \end{pmatrix}.$$

This matrix has full rank so by Proposition 4, page 6, the unlinking number $u(L)$ satisfies

$$u(L) \geq 3 - 1 - 0 = 2.$$