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Sorry for taking so long to respond. Theo's answer is fairly complete. I mentioned the question to Masahico who reminded me that there are results of Kanenobu and others about finite type invariants for ribbon knots. The general case is known to be difficult if not impossible.

I don't agree that a framed braided monoidal 2-category with duals is going to be difficult. But in general finding braided monoidal 2-categories with duals that give non-trivial invariants is a difficult enterprise. I asked one expert about this. He said there were lots, so I said, "Ok name one." See Kevin Walker's response to this question.

Also, before looking at our stuff, someone who wants to learn about knotted surfaces should look at Roseman's paper.

To find non-trivial invariants, whether or not they be finite-type, the best bet so far is the quandle-cocycle invariant.

Now, to return to an answer that the questioner may appreciate: the crossing points of a classical knot are isolated. The crossing points for surfaces are arcs and circles. Crossing changes are not local things. We don't even know if we can unknot by switching crossings between sheets, and sometimes it is hard to imagine how to push a crossing change through a triple point.

There is a lot to learn about knotted surfaces. Yeah sure, you can look at our books, but one should also read the elegant papers by Kanenobu, Satoh, and Hillman.

1

Sorry for taking so long to respond. Theo's answer is fairly complete. I mentioned the question to Masahico who reminded me that there are results of Kanenobu and others about finite type invariants for ribbon knots. The general case is known to be difficult if not impossible.

I don't agree that a framed braided monoidal 2-category with duals is going to be difficult. But in general finding braided monoidal 2-categories with duals that give non-trivial invariants is a difficult enterprise. I asked one expert about this. He said there were lots, so I said, "Ok name one." See Kevin Walker's response to this question .

Also, before looking at our stuff, someone who wants to learn about knotted surfaces should look at Roseman's paper.

To find non-trivial invariants, whether or not they be finite-type, the best bet so far is the quandle-cocycle invariant.

Now, to return to an answer that the questioner may appreciate: the crossing points of a classical knot are isolated. The crossing points for surfaces are arcs and circles. Crossing changes are not local things. We don't even know if we can unknot by switching crossings between sheets, and sometimes it is hard to imagine how to push a crossing change through a triple point.

There is a lot to learn about knotted surfaces. Yeah sure, you can look at our books, but one should also read the elegant papers by Kanenobu, Satoh, and Hillman.