Kevin Walker and a member of the forum (by e-mail) let me know about a flaw in the argument in my previous post.  I'll erase it after putting up this post. 

One of my claims was the if $K$ unknots in $M$ then it unknots in $B$, where $K$ is a knot in a ball $B \subset M$.  If $K$ is the unknot in $M$ it bounds a disk $D \subset M$, $K = \partial D$.  In dimension $3$ we perturb the disc so that $D \cap \partial B$ is a collection of circles, this allows us to cap off $D \cap B$ with disc(s) parallel to $\partial B$ (an embedded surgery construction), constructing a disc in $B$ that bounds $K$.  In higher dimensions this argument runs into trouble as it's not clear if you can perform the required embedded surgeries.  

I also claimed that the only new isotopy relation in general is that knots in $B$ can be isotopic (in $M$) to their mirror inverse if $M$ is non-orientable.  I still suspect this is the case and I think I have an idea for an argument but I should think about it some more before posting.  Here is an argument that works in dimension 3:

Say $M$ is a compact boundaryless $3$-manifold, $K$ and $K'$ contained in a ball $B$ are isotopic in $M$. Perform the connect-sum decomposition on $M \setminus K$ and $M \setminus K'$.  Since this is a unique decomposition (modulo issues in the non-orientable case), in both cases you get a list of prime factors that are identical except in the decomposition of $M \setminus K$ you get a summand that's $B \setminus K$ capped off with a disc, and in the other case you get a summand $B \setminus K'$ capped off with a disc. By the Gordon-Luecke theorem $K$ and $K'$ are either isotopic in $B$ or mirror images of each other. 

I'm still hopeful I have an argument that works in general.  I'll edit this post again tomorrow once I'm certain. 

edit: final exams and deadlines are eating up my time at the moment but I'll get around to another revision... eventually. 

Knots that unknot in a manifold
Can anyone direct me to a good book with lots of molarity exercises, i really want to practice this and would like to find as many types of exercises involving molarity. I of course did google this and there are some, but usually they are three or four exercises per resource.  Thanks in advance.
