Grid Diagrams of Fibered Knots

TL;DR

The paper investigates whether every fibered knot admits a grid diagram with a unique grid state attaining the Alexander-function upper bound. It develops an efficient framework based on winding numbers to bound the Alexander function, introduces the notion of perfect (and unique) grid states, and proves that a unique perfect state implies a local minimal-entry condition in the winding matrix; this yields a practical algorithm for testing uniqueness. A Python package, griddiagrams, implements commutation- and stabilization-based searches to find nice grid diagrams for fibered knots, achieving solutions for 5385 of 5397 fibered prime knots with crossing number $\leq 13$, and leaving only 12 unresolved. The work provides a scalable computational approach to connect fiberedness, grid-homology constraints, and Alexander-genus data, while highlighting open questions about the remaining knots and potential algorithmic enhancements.

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.

Figures (6)

• Figure 1: For this planar realization of a grid diagram, the winding number at the red point $p$ is -2. To see this, one can choose any ray emerging out of $p$ and count its signed intersection with the knot projection, as explained in Definition \ref{['def: winding number']}. Here, two possible rays are drawn, both intersect the knot projection twice in a clockwise direction.
• Figure 2: A planar realization of a grid diagram of the trefoil with a grid state visualized by red dots. Entries of the winding matrix are written in grey.
• Figure 3: A visual explanation of Proposition \ref{['prop: idea']}. Left: An example of a grid diagram where no column of the winding matrix has a unique minimal value. The red dots represent entries of the winding matrix that are minimal for this column. Right: For each column, we choose two minimal entries (blue and red). The blue dots represent a column perfect grid state. Then we draw a curve (red) until a point of the grid state is visited twice. The closed loop, which is created from this curve, tells us which blue dots can be switched to red dots to create a new column perfect grid state.
• Figure 4: An example of a grid diagram with a grid state that is maximal and unique but not perfect.
• Figure 5: A planar grid diagram of the trefoil knot.

Theorems & Definitions (17)

• Proposition : I
• proof
• Definition 1.1
• Definition 1.2
• Definition 1.3
• Definition 1.4
• Definition 1.5
• Definition 1.6
• Example 1.7
• Definition 2.1