# What does it mean exactly for a pair of $S^0$'s to be unlinked on a knot $K$?

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)

• My guess is that the two $S^0$ are unlinked if they bound two disjoint intervals on $K$. (This is consistent with the general definition of linking number.) – Marco Golla Jan 3 '18 at 13:13
• Yeah, that makes sense, especially given the fact that we are asked to construct a tube between them (which wouldn't make sense if they bounded intervals on $K$ that were not disjoint). Could you say more about how this relates to the linking number though? – asldjk Jan 3 '18 at 13:15
• Call the two $S^0$s $L$ and $L'$. $L$ bounds an interval $I$ on $K$; if $L$ and $L'$ are linked, the intersection number between $I$ and $L'$ is $1$ (mod 2); if they're not, it's $0$. – Marco Golla Jan 3 '18 at 13:28

Think of $S^1$ as the ideal boundary of the hyperbolic plane, then every embedded $S^0\subset S^1$ determines a unique geodesic in the hyperbolic plane. The linking number of two embedded $S^0$s is defined as the intersection number of the corresponding geodesics. See Section 2 of https://arxiv.org/pdf/1308.3022.pdf