Grid diagrams and shellability

TL;DR

This paper establishes a novel link between knot Floer homology and shellable posets through grid diagrams, enabling combinatorial constructions of flow categories related to knots.

Contribution

It introduces a new poset construction from grid diagrams that connects knot Floer homology with shellability, providing a combinatorial approach to flow categories.

Findings

The poset derived from a grid diagram has shellable intervals.

Homology of the associated chain complex equals knot Floer homology.

A PL flow category can be combinatorially associated to a grid diagram.

Abstract

We explore a somewhat unexpected connection between knot Floer homology and shellable posets, via grid diagrams. Given a grid presentation of a knot K inside S^3, we define a poset which has an associated chain complex whose homology is the knot Floer homology of K. We then prove that the closed intervals of this poset are shellable. This allows us to combinatorially associate a PL flow category to a grid diagram.