# Knot cobordisms, bridge index, and torsion in Floer homology

## TL;DR

This paper establishes inequalities relating torsion in knot Floer homology to cobordism properties, leading to new bounds on knot invariants like bridge index, fusion number, and ribbon distance, with sharp results for certain torus knots.

## Contribution

It introduces novel inequalities connecting torsion in knot Floer homology with cobordism features, providing sharp bounds on several knot invariants and insights into ribbon and concordance properties.

## Key findings

- Bounds on bridge index and fusion number are sharp for specific torus knots.
- Torsion order provides lower bounds on band-unlinking and ribbon unknots.
- Knot Floer homology bounds the ribbon distance and cobordism distance between knots.

## Abstract

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. 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. We show 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.