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?
(The Ruberman paper is available here: http://gdz.sub.uni-goettingen.de/pdfcache/PPN356556735_0090/PPN356556735_0090___LOG_0015.pdf)
