Knots of low knot Floer width

TL;DR

This paper classifies the chain homotopy types of knot Floer complexes for knots with width 2, revealing their structural properties and implications for simplifying Montesinos knots.

Contribution

It provides a complete classification of knot Floer complexes of width 2 and demonstrates their implications for the simplification of Montesinos knots.

Findings

Knot Floer complexes of width 2 have no nontrivial local systems.

All Montesinos knots admit a basis that can be simultaneously simplified horizontally and vertically.

Abstract

This paper classifies the chain homotopy equivalence types of knot Floer complexes $CFK_{\mathbb{F}[U,V]}(K)$ 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.