Finite grid complexes for knot Floer theory

The grid-complex algorithm takes a finite encoded grid and a mode from $\{\mathrm{t}\mathrm{i}\mathrm{l}\mathrm{d}\mathrm{e},\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{u}\mathrm{s}\text{-}\mathrm{a}\mathrm{s}\mathrm{s}\mathrm{o}\mathrm{c}\mathrm{i}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{d}\text{-}\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}\mathrm{e}\mathrm{d},\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{u}\mathrm{s}\text{-}\mathrm{f}\mathrm{i}\mathrm{l}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{d}\}$. It requires an integer grid number $n\ge 2$, disjoint $O$- and $X$-markings with exactly one of each in every row and column, and an induced component permutation that is one $n$-cycle. If $n<2$ or the marking conditions fail, return invalid-grid; if the component permutation has more than one cycle, return link-input-not-supported. No chain complex is returned on either branch.

For a valid knot grid, number the horizontal and vertical circles $1,\dots ,n$ and enumerate all permutations of $\{1,\dots ,n\}$ in lexicographic order. The state for a permutation $\sigma$ consists of the $n$ intersections $(i,\sigma (i))$. For each ordered pair of states $\mathbf{x},\mathbf{y}$, if they agree in other than exactly $n-2$ points, add no rectangle terms. Otherwise enumerate the at most two torus rectangles with the four nonshared state points as corners, retain those whose interiors contain no point of $\mathbf{x}$, and apply the selected mode. In the full filtered mode, each term drops filtration by $n_{\mathbb{X}}(r)$; the associated-graded mode retains precisely the terms with zero drop, which are the rectangles with no $X$-marking:

1. Tilde mode. Work over ${\mathbb{F}}_2$. Count a retained rectangle with coefficient $1$ exactly when its interior contains no $O$- or $X$-marking.
2. Associated-graded minus mode. Work over ${\mathbb{F}}_2[U_1,\dots ,U_n]$, with each $U_i$ of Maslov degree $-2$ and Alexander filtration degree $-1$. Count a retained rectangle exactly when its interior contains no $X$-marking, with coefficient ${\prod }_{i=1}^n{U}_i^{O_i(r)}$.
3. Full filtered minus mode. Work over ${\mathbb{F}}_2[U_1,\dots ,U_n]$, with each $U_i$ of Maslov degree $-2$ and Alexander filtration degree $-1$. Count every retained rectangle, regardless of its $X$-markings, with coefficient ${\prod }_{i=1}^n{U}_i^{O_i(r)}$. Use the grid Alexander value for each state and assign filtration drop $n_{\mathbb{X}}(r)$ to this term.

Return the ordered state list, the mode and coefficient ring, the exact differential matrix obtained by summing coefficients modulo two, the state Maslov and Alexander values, and the filtration data for the full filtered mode. The algorithm returns this finite presentation only; its identification with knot Floer homology and its independence of the chosen grid are separate theorems. The factor $V^{\otimes (n-1)}$ is not multiplied into the output.

Toroidal knot grids, states, rectangles, and gradings

Integrality of the grid Alexander grading and its rectangle change

Ozsvath, Stipsicz, and Szabo, Grid Homology for Knots and Links, §§4.1–4.6 and 13.2

Manolescu, Ozsvath, and Sarkar, “A combinatorial description of knot Floer homology,” §§1–3

Manolescu, An Introduction to Knot Floer Homology, §4.3