Grid Diagrams of Fibered Knots

TL;DR

This paper develops an efficient method to identify special grid diagrams of fibered knots, demonstrating its effectiveness by finding such diagrams for most fibered prime knots with up to 13 crossings.

Contribution

The authors introduce a new computational approach to determine the existence of specific grid diagrams for fibered knots, advancing understanding of their combinatorial representations.

Findings

Successfully identified grid diagrams for 5385 of 5397 fibered prime knots with ≤13 crossings.

Developed an efficient algorithm implemented in Python for analyzing grid diagrams.

Provided evidence supporting the conjecture that all fibered knots admit such special grid diagrams.

Abstract

Grid diagrams are special representations of knots in the three-sphere that are used to define a combinatorial version of knot Floer homology. Paolo Ghiggini and Yi Ni showed that knot Floer homology detects fibered knots. Their results imply, in particular, that grid diagrams with a unique grid state whose Alexander grading is maximal only exist for fibered knots. Whether every fibered knot admits such a diagram remains an open question. Here, we investigate the existence of such special grid diagrams for fibered knots. We develop an efficient method for deciding whether a given grid diagram meets the even stricter condition of having a unique grid state that realizes an upper bound for the Alexander function. By implementing this method in a Python package, we find suitable grid diagrams for 5385 of the 5397 fibered prime knots with crossing number at most 13.