Grid Diagrams of Fibered Knots

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.

💡 Research Summary

The paper investigates a refined question in knot theory: for which fibered knots does there exist a grid diagram that possesses a unique grid state whose Alexander grading attains the maximal possible value? Building on the foundational result of Ghiggini and Ni—that a knot is fibered if and only if its knot Floer homology in the top Alexander grading is one‑dimensional—the author focuses on the combinatorial incarnation of this statement within grid homology. Proposition I states that a grid diagram with a unique maximal‑grading grid state forces the underlying knot to be fibered, but the converse (whether every fibered knot admits such a diagram) remains open.

To address this, the author introduces an efficient method for testing a given grid diagram against a stricter condition: the existence of a perfect grid state, i.e., a state that realizes the upper bound of the Alexander function as derived from the winding numbers of the diagram. The winding matrix, an n × n array of winding numbers at the lattice points of a planar realization, yields two invariants—the row number r and the column number c—both independent of the chosen realization. The Alexander function A′(x) is bounded above by min{r,c}, and a grid state achieving this bound is called perfect (row‑perfect or column‑perfect when the bound is realized by r or c respectively).

The core of the algorithm is the notion of a loop in the winding matrix. Given a permutation π describing a candidate grid state, for each column one can locate an alternative entry μ(j) that also attains the column minimum. Connecting π(j) to μ(j) vertically and then horizontally to the next π‑entry produces a directed path; if this path closes into a loop, Lemma 2.3 guarantees the existence of another grid state with the same Alexander grading, contradicting uniqueness. Consequently, a unique perfect grid state can exist only if at least one column (or row) contains a unique minimal entry.

The decision procedure proceeds as follows: compute r and c; if they differ, only the direction with the smaller bound needs to be examined. Locate a column with a unique minimal entry, delete that column and its corresponding row, and repeat on the reduced matrix. If at any stage no column (or row) has a unique minimum, the diagram cannot support a unique perfect grid state. The algorithm runs in essentially linear time with respect to the grid size, a dramatic improvement over the naïve factorial‑time search over all n! grid states.

Implementation details are provided in a Python package (named “grid_fiber”), which interfaces with the KnotInfo database to generate grid diagrams for all prime fibered knots with crossing number ≤ 13. Applying the algorithm to the 5 397 such knots, the author finds suitable diagrams for 5 385 of them, i.e., over 99 % coverage. The remaining 12 knots do not admit a unique perfect grid state within the examined grid sizes; they may require larger grids or may only possess a unique maximal (but not perfect) grid state, as illustrated by an explicit example (Figure 4).

The paper concludes with a discussion of the unresolved cases, suggesting that future work could extend the algorithm to detect uniqueness among maximal but non‑perfect states, explore larger grid sizes, or combine grid‑homological criteria with other knot invariants. By providing both a theoretical framework and a practical computational tool, the work makes a substantial contribution to the study of fibered knots via combinatorial Floer homology.