I'm trying to learn more about hyperbolic 3-manifolds, in particular the geometric implications of doing hyperbolic Dehn surgery to suitable knot complements.

Following this paper by Christian Millichap ( https://arxiv.org/abs/1209.1042 ), we can construct a hyperbolic Montesinos knot $K$ with several tangle regions (see p. 8). We then mutate these tangle regions along a suitable Conway sphere, before doing hyperbolic Dehn surgery to fill up the knot complements. Because of how we constructed the hyperbolic Montesinos knot $K$ (in particular, the relevant pair of $S^0$'s are unlinked on $K$, see What does it mean exactly for a pair of $S^0$'s to be unlinked on a knot $K$?), it can be shown that $K$ and all its mutations yield a hyperbolic 3-manifold with the same volume. Further, by an argument in this paper I don't fully understand (for details, see the proof of Theorem 3 on pg. 11), we can pick appropriate coefficients for the Dehn surgery so that the hyperbolic manifolds resulting from $K$ and its mutations are all non-homeomorphic.

I want to have a more precise (and concrete) understanding of how "different" these hyperbolic 3-manifolds are, beyond the fact that they are non-homeomorphic. In particular, suppose we have a hyperbolic knot $K$ and its mutant $K'$ and we know that they yield non-homeomorphic hyperbolic 3-manifolds (call them $M_{K}$ and $M_{K'}$) with the same finite volume via Dehn surgery.

My question is: since mutations of knots involve "cutting up" a certain region of a knot and rotating it appropriately to get a different knot, is it possible for us to cut out an analogous region of $M_{K}$ and (via some isometry in $H^3$) "rotate" the region to obtain $M_{K'}$?

In particular, I don't understand the geometric aspects of hyperbolic Dehn surgery well enough to know if filling in the knot complement with a solid torus presents an obstruction to this. All I know is that hyperbolic Dehn surgery involves picking a suitable pair of integers $(p,q)$ and choosing a basis $(m,l)$ for the fundamental group of the torus boundary of the knot complement (where m and l corresponds to the meridional and longitudinal side of the torus respectively). We may interpret the knot complement as a hyperbolic 3-manifold with a cusp corresponding to the torus boundary; via hyperbolic Dehn surgery, we can cut off the cusp and glue in a solid torus by mapping the boundary of the meridian disc of the torus to $s=pm+ql$ where $s$ is a slope on the torus boundary of the knot complement. (I am following p. 7 of the Millichap paper here). From a geometric standpoint, all I take from this is that there is a very specific way of gluing the solid torus along the torus boundary of the knot complement, and this might give us problems if we were to "cut and paste" the region in order to get a new hyperbolic 3-manifold. Am I right about this?

If it helps to have something more visual, consider the following two knots, which are the prime Kinoshita-Teraska knot (LH) and the prime Conway knot (RH). This diagram shows how the two are related by a mutation along the Conway sphere (marked in red) by rotation about the y-axis.

As Ian Agol mentions below, it can be verified that this example of knot mutation satisfies the unlinking condition mentioned above, hence Dehn surgery on the knot complements of these two knots yield hyperbolic 3-manifolds with equal volume. Now, suppose we have filled the knot complements out via Dehn surgery and gotten two non-homeomorphic 3-manifolds with the same volume - can we then cut and rotate whatever is analogous to the red region in the hyperbolic 3-manifold (derived from the LH knot) to get the hyperbolic 3-manifold derived from the RH knot?

Montesinos.$\endgroup$