It was conjectured by Gordon and recently proved by Agol that ribbon concordance defined a partial order on the semi group of knots. I know that this question is close related to the slice ribbon conjecture and I am really curious about whether or not some one has construct a non-trivial ribbon concordance from a knot to it self which is not isotopic rel boundary to the cylinder? I heard from a recent lecture given by Maggie Miller that for unknot if we assume some condition I did not remember well, it is equivalent to the classification of ribbon disk. I am really curious about what this condition is and how peopler had done to solve relative questions?
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1$\begingroup$ Take the trivial concordance $K \times I \subset S^3 \times I$ for any knot in the $3$-sphere, and then connect-sum with a non-trivial $2$-knot. That's generally not the trivial concordance. $\endgroup$– Ryan BudneyCommented Mar 13 at 16:36
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1$\begingroup$ sorry，I did not state my question clear yesterday, of course there are a lot of nontrivial concordance from a knot to itself, like the comment above said, adding a 2-knot is a useful construction, but my curiousity lies in is there a ribbon one? $\endgroup$– Judy_xyhCommented Mar 14 at 15:36
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$\begingroup$ This is open in general (as you say, even for the unknot case), even in the topological category (the unknot case might be true in the topological category). The ribbon complement from a knot to itself is an s-cobordism, but I don’t think that anything is known beyond that. $\endgroup$– Ian AgolCommented Mar 21 at 0:55
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$\begingroup$ I am a little bit confuse how to talk about ribbon concordance in topological category when the notion of Morse function and critical points are not well-defined? $\endgroup$– Judy_xyhCommented Mar 23 at 8:04
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In the topological category, a locally flat concordance from knot $K$ to knot $J$ is homotopy-ribbon when the fundamental group of $S^3-K$ injects into the fundamental group of the concordance complement while the fundamental group of $S^3-J$ surjects onto the fundamental group of the concordance complement. I believe that Ian Agol's argument applies in this category, showing that homotopy-ribbon concordance induces a partial ordering on Knots. Many interesting questions about ribbon concordance are interesting in this category, e.g. it's unknown whether $J$ being topologically slice implies there is a homotopy-ribbon concordance from the unknot to $J$.
As Ian points out, we don't know whether a homotopy-ribbon concordance from $K$ to itself must be topologically isotopic to a trivial concordance (although yes for $K$ the unknot we know that this is true -- if we were to cap off the concordance on one end with a trivial disk, we would obtain a $\mathbb{Z}$-disk for the unknot, which we know is unique by either Freedman or Conway--Powell. That tells us our concordance is isotopic to the trivial concordance via an isotopy that possibly moves one boundary component. Up to isotopy of isotopies, there are two oriented self-isotopies of the unknot: the trivial one and rotation about an axis [follows from the Smale conjecture by Hatcher, but I believe there is a simpler argument for this specific case, see e.g. Bob Edwards' writeup of the light bulb theorem in which he deducts the $\pi_1$ case of the Smale conjecture]; we can take the axis such that rotation fixes the unknot setwise, and then change our isotopy to add a final cancelling rotation and contract to obtain an isotopy rel. boundary to the trivial concordance.) However, I don't see how to extend this argument for any nontrivial knot.
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$\begingroup$ The argument in my paper shows that strong homotopy ribbon concordance (in the sense of Miller-Zemke) is a partial order, but I don’t know how to prove the homotopy ribbon case; one needs a relative presentation with an equal number of generators and relators. arxiv.org/abs/1903.05772 $\endgroup$– Ian AgolCommented Mar 25 at 1:26