Given a cobordism between two knots in the 3-sphere, we present an inequality involving torsion orders of the knot Floer homology of the knots, and the number of critical points of index zero, one, and two of the cobordism. In particular, the torsion order gives a lower bound on the number of minima appearing in a slice disk of a knot. This has a number of topological applications: The torsion order gives lower bounds on the bridge index and the band-unlinking number of a knot, and on the fusion number of a ribbon knot.

It also gives a lower bound on the number of bands appearing in a ribbon concordance between two knots. Our bounds on the bridge index and fusion number are sharp for $T_{p,q}$ and $-T_{p,q} \# T_{p,q}$, respectively. We also show that the bridge index of $T_{p,q}$ is minimal within its concordance class.

The torsion order bounds a refinement of the cobordism distance on knots, which is a metric. As a special case, we can bound the number of band moves required to get from one knot to the other. Knot Floer homology also gives a lower bound on Sarkar's ribbon distance, and exhibit examples of ribbon knots with arbitrarily large ribbon distance from the unknot. This is joint work with Miller and Zemke.

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