Grid Diagrams for Lens Spaces and Combinatorial Knot Floer Homology

TL;DR

This paper develops a combinatorial framework using grid diagrams to compute knot Floer homology invariants for knots in lens spaces, aiming to advance understanding of knot classification and the Berge conjecture.

Contribution

It provides explicit combinatorial descriptions of generators, differentials, and gradings for knot Floer homology in lens spaces, extending methods from S^3.

Findings

Explicit combinatorial formulas for generators and differentials.

Conjecture linking Floer homology to knot characterization.

Potential proof of the Berge conjecture with further work.

Abstract

Similar to knots in S^3, any knot in a lens space has a grid diagram from which one can combinatorially compute all of its knot Floer homology invariants. We give an explicit description of the generators, differentials, and rational Maslov and Alexander gradings in terms of combinatorial data on the grid diagram. Motivated by existing results for the Floer homology of knots in S^3 and the similarity of the combinatorics presented here, we conjecture that a certain family of knots is characterized by their Floer homology. Coupled with work of the third author, an affirmative answer to this would prove the Berge conjecture, which catalogs the knots in S^3 admitting lens space surgeries.