invariants of knots that are invariants under band move.

Question

I am asking whether there are known knot invariants which are invariants under band move. Note that band move operation is similar to a connected sum of two knots except that the projections of two knots not necessarily to be disjoint in the band move. They can be linked indeed.

Answer 1

One useful interpretation of a band (or ribbon) move is an elementary cobordism between two links with a single index one critical point.

In more detail: Let $f:S^3 \times I\to\Bbb R$ with $(x,t) \mapsto t$. Then two links $L_1$ and $L_2$ are equivalent by a single band move if and only if there is a cobordism $C\subset S^3 \times I$ between $L_0 \times 0$ and $L_1 \times 1$ where $f$ restricts to a Morse function on $C$ with a single index one critical point (you should prove this yourself as an exercise). To deal with multiple band moves, we can

Note that these cobordisms can have high genus. For instance if I do a band move on a knot I get a 2 component link. If I band these together, I get back to a knot and I have a Morse function where the regular values go from a circle to a pair of circles back to a circle, so the cobordism is a twice-punctured torus.

If one does not put any genus restrictions on the cobordism, the problem is essentially trivial. Any link bounds a Seifert surface and this can be arranged (essentially by Seifert's algorithm) so that such a surface is a disk with some bands glued to the boundary of this disk. It's easy to see the band moves and the cobordism is just a "pushed-in" version of the Seifert surface minus an open disk.

The case that is intensely studied is the case where the genus of such a cobordism is 0. With the most famous question being the following:

Slice-Ribbon Conjecture: If $K$ a knot is smoothly slice ($K\times 0 \cup U\times 1$ bounds a smooth cylinder in $S^3 \times I$) then there is a sequence of band (or ribbon) moves from $K$ to an unlink with the induced cobordism being genus $0$.

One can google or search arXiv to see some of the things people have done relating to this. Still, this conjecture is wide-open, and settling it would likely guarantee you a tenured job for the rest of your life.