TITLE: Decidability of knot equivalence in general 3-manifolds? Surface equivalence?
QUESTION [6 upvotes]: Given a closed orientable 3-manifold $M^3$ and two knots $K_1$ and $K_2$ in $M$, is there an algorithm to decide if $K_1$ and $K_2$ are isotopic?  Is there an algorithm to decide if there is a homeomorphism $f: M \to M$ with $f(K_1) = K_2$?
I would expect these questions to be involved (at least as difficult as the recognition problem for 3-manifolds) but I thought that the answer is possibly a folklore consequence of geometrization.  The special case of the first question for $K_1$ unknotted follows from a result of Hass and Lagarias (see Theorem 1.2).  The case where $M = S^3$ is a result of Waldhausen using normal surface theory.
As an aside, I don't know anything about the corresponding questions where instead of considering knots we consider closed embedded surfaces $F_1$ and $F_2$.  Even in the case where $M = S^3$ (so only the isotopy question is relevant) the genus of $F_1$ and $F_2$ is greater than one, this is a total mystery to me.  By looking at tori bounding regular neighborhoods of knots, this is strictly more difficult than the corresponding questions for knots.  Are there results in this direction?

REPLY [4 votes]: Regarding the second question.  This reduces to the homeomorphism problem for three manifolds, which is known due to geometrisation.  Here is a sketch.  Fix $M$ as well as the knots $K$ and $L$.  Let $X_K$ and $X_L$ be the knot complements.  We mark the boundary of each with the meridional slope of $K$ and $L$, respectively. We ask our solution to the homeomorphism problem if $X_K$ and $X_L$ are homeomorphic.  If the answer is "no" then we are done.  If the answer is "yes" we ask for all homeomorphisms from $X_K$ to $X_L$.  For each of these we restrict to the boundary and see if the resulting map takes the meridian of $K$ to that of $L$. If any do, the answer is "yes".  If none do, the answer is "no".
[Edit - as pointed out in the comments, there may be infinitely many homoeomorphisms between $X_K$ and $X_L$.  This happens, for example, when $K$ and $L$ are both unknots in the three-sphere.  In any case, the self-homeomorphisms of $X_K$ restrict to give a virtually cyclic group of homeomorphisms of $\partial X_K$.  When this group is infinite, the cyclic subgroup is generated by a Dehn twist about the slope that dies in homology.]
Regarding the first question.  If $M = S^3$ then again this reduces to the homeomorphism problem for three-manifolds, by appealing to Property P (due to Gordon-Luecke).  Some fiddling about is needed to deal with mirror images.
I don't know the status of the first question for other manifolds off the top of my head.  I'll report back if anything occurs to me. :)