Knots of low knot Floer width

This paper classifies the chain homotopy equivalence types of knot Floer complexes $CF{K}_{\mathbb{F}[U,V]}(K)$ of knot Floer width 2. They have no nontrivial local systems. As an application, this shows that all Montesinos knots admit a basis that can be simultaneously horizontally and vertically simplified.