András Juhász, Maggie Miller, Ian Zemke

Given a connected cobordism between two knots in the 3‐sphere, our main result is an inequality involving torsion orders of the knot Floer homology of the knots, and the number of local maxima and the genus of the cobordism. This has several topological applications: The torsion order gives lower bounds on the bridge index and the band‐unlinking number of a knot, the fusion number of a ribbon knot, and the number of minima appearing in a slice disk of a 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 ${T}_{p,q}\#{\overline{T}}_{p,q}$, respectively. We also show that the bridge index of ${T}_{p,q}$ is minimal within its concordance class.