I am trying to learn about the effects of knot mutation on the hyperbolic manifolds obtained via hyperbolic Dehn surgery, and I'm currently reading Ruberman's paper "Mutations and Volumes in $S^3$" (in particular, see p.190-191, and p.212-213) and Christian Millichap's paper (available here: https://arxiv.org/abs/1209.1042 ; in particular, p. 8).
In these papers, we have the following:
Definition 1: Let $K\subset S^3$ be a knot/link. A Conway sphere for $K$ is an embedded $2-$sphere meeting $K$ transversally in four points.
Note that we get a 4-punctured sphere when we examine the Conway sphere in the knot complement, which in turn gives us two symmetries/involution (see p. 190-191 for illustration). Suppose we pick some specific involution $\tau$. Then, as Ruberman remarks, any specific choice of $\tau$ gives a pair of $S^0$'s such that each $S^0$ is preserved by $\tau$. What I don't understand is the subsequent definition:
Definition 2: The Conway sphere $S$ and the mutation $\tau$ are unlinked if these $S^0$'s are unlinked on $K$.
I understand how two (knot) components of a link might be unlinked, but I don't understand how two pair of points can be unlinked. What does this mean, and what would be an example of a Conway sphere $S$ and a mutation $\tau$ which is not unlinked? Does it just mean that that the pairs of lines defined by the $S^0$'s don't overlap so we can construct a tube that joins them together?
(The Ruberman paper is available here: http://gdz.sub.uni-goettingen.de/pdfcache/PPN356556735_0090/PPN356556735_0090___LOG_0015.pdf)