Simple invariants to detect concordance in general 3-manifolds

Question

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.

Answer 1

If you allow the relaxation of the closed condition so that $ \partial Y $ is permitted to be a surface, there is a very simple concordance obstruction for knots in a thickening of the boundary, due to Kauffman.

Let $ \partial Y = \Sigma_g $, a closed orientable surface of genus g. Given a knot $ K \hookrightarrow \Sigma_g \times I $, take its regular projection to $ \Sigma_g $: the result is a knot diagram on $ \Sigma_g $, denoted $ D $.

Pick a crossing, $ c $, of $ D $. Leave $ c $ from any outgoing arc, and traverse $ D $ until you return to $ c $, counting the number of other crossing you passed through. If you passed through an even number of crossings, declare $ c $ to be even, otherwise delcare it odd. Repeat this for the other crossings of $ D $.

Denote by $ J ( D ) $ the sum of the signs of the odd crossings. Kauffman shows this is an invariant of $ K $, and we may define $ J ( K ) = J ( D ) $ [1]. It is shown in [2,3] that if $ J ( K ) \neq 0 $ then $ K $ is not concordant in $ Y \times I $ to the trivial knot. Equivalently, $ K $ does not bound a disc in $ Y \times I $ (it does not bound a disc in any $3$-manifold with boundary $ \Sigma_g $, in fact).

This extends automatically to obstructing concordances between two non-trivial knots $ K_1 $ and $ K_2 $, in the case when $ \partial Y = \Sigma_g \sqcup \Sigma_{g'} $, $ K_1 \hookrightarrow \Sigma_g $, $ K_2 \hookrightarrow \Sigma_{g'} $. If $ J ( K_1 ) \neq J ( K_2 ) $ then $ K_1 $ is not concordant in $ Y \times I $ to $ K_2 $.