Take a closed $n$-manifold $M$ and fix a nice $n$-ball $B$ in $M$. How much information about $M$ does the set of knotted $(n-2)$-spheres of $B$ which are unknotted *in* $M$ give?



Knots that unknot in a manifold
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. 


