The notion of a knot concordance is a rich subject in low-dimensional topology, see Livingston's survey. More precisely:

For $i=0,1$, let $K_i$ be knots in $S^3$. A *knot concordance* from $K_0$ to $K_1$ is a smooth annulus $A=S^1 \times [0,1]$ in $S^3 \times [0,1]$ such that $\partial A= -(K_0) \cup K_1$ where $-$ denotes the reversed orientation.

Using this relation, we can form a group structure on the set of oriented knots in $S^3$, denoted by $\mathcal{C}$.

Let $K$ be a *slice* knot in $S^3$, that is, $K$ bounds a smooth disk $D$ embedded in $B^4$. We can also show that $K$ is slice if and only if $K$ is concordant to the unknot in $S^3$.

Let $M$ be a closed oriented $3$-manifold. I wonder:

- Can we talk about a knot $M$ behaves like the unknot in $S^3$?
- Can we generalize the notion of knot concordance to the oriented knots in $M$?
- (Extra) Can we define inverses of knots in $M$ as in the case of $S^3$?