Let $Y$ be a closed, connected, orientable 3-manifold. We call to oriented knots $K_1, K_2$ in $Y$ (smoothly) concordant if there is a smoothly, properly embedded annulus in $Y \times I$ such that the boundary components of the annulus are in $Y \times \{0\}$ and $Y \times \{1\}$ and are $K_1$ and $K_2$ (with there respective orientations).

If you are walking along and explicitly given a 3-manifold (say by a framed link) and a pair of knots $K_1$ and $K_2$ and you are asked if they are concordant, after checking that they are the same class in $\pi_1$, what do you do? Specifically what if $K_2$ is just the unknot - what sorts of simple invariants can I try and compute to see if $K_1$ is not null concordant?

I've seen a couple of papers floating around but I was wondering what kinds of "classical" invariants there are for this sort of problem. For example if $Y = S^3$, the first things I might try would be the Arf invariant and the signature.

As an example, maybe take $Y = S^1 \times S^2$ and take $K_1$ as one of the components of the Whitehead link (where the other component is given 0-framing thus giving $S^1 \times S^2$), and take $K_2$ to be the unknot.

Algebraic linking numbers of knots in 3–manifolds, AG&T, 2003). $\endgroup$3more comments