Given two oriented links $K_1, K_2$ in $S^3$, it is an interesting problem to figure out whether there exists an oriented surface with boundary which is embedded (locally flatly) in $S\times [0,1]$ whose boundaries are exactly $K_1\times 0$ and $K_2\times 1$, considered as in- and out-boundary.
I found this notion very similar to the notion of cobordism. Of course in this formulation every link are "cobordant", but there is a restriction on the minimal genus of "cobordism" which depends on how the two links are "similar". (This notion is of course different from link concordance because we're allowing any surface, not just $K_1 \times [0,1]$.)
Just as with cobordisms between manifolds, I think it would be interesting if there is a topological quantum field theory (TQFT) for links and "cobordisms" as described above.
Is there any such TQFT known?