Toroidal grid diagrams and gradings for oriented knots

Let $K\subset S^3$ be a smooth oriented knot, meaning one oriented component of a finite smooth oriented link in the standard oriented three-sphere $S^3={\mathbb{R}}^3\cup \{\mathrm{\infty }\}$. The grid orientation below fixes the sign convention for its Alexander grading.

For an integer $n\ge 2$, put $T_n={\mathbb{R}}^2\mathrm{/}(n\mathbb{Z}{)}^2$, with horizontal circles ${\alpha }_i=\{y=i\}$ and vertical circles ${\beta }_j=\{x=j\}$ for $i,j\in \mathbb{Z}\mathrm{/}n\mathbb{Z}$. A toroidal grid diagram of grid number $n$ is a pair of disjoint sets $\mathbb{O},\mathbb{X}$ of cell centers $(j+\frac{1}{2},i+\frac{1}{2})$, read modulo $n$, such that every row and every column contains exactly one point of each set. In the fundamental square $[0,n)\times [0,n)$, join the two marks in each row and each column by the straight segment between their centers. At each crossing of the projected link, the vertical segment is over and the horizontal segment is under. Orient each horizontal segment from its $O$-marking to its $X$-marking and each vertical segment from its $X$-marking to its $O$-marking. The markings and these segments determine an oriented link. Following each component from an $O$ along its row to an $X$ and then along that column to the next $O$ defines a permutation of the $n$ $O$-markings; its cycles are the link components. The grid represents a knot precisely when this permutation is one $n$-cycle.

A grid state is a subset $\mathbf{x}\subset {\bigcup }_{i,j}({\alpha }_i\cap {\beta }_j)$ containing exactly one intersection on each horizontal and each vertical circle. After numbering the circles, grid states correspond bijectively to permutations of $\{1,\dots ,n\}$, so there are $n!$ states. For states $\mathbf{x},\mathbf{y}$, a rectangle from $\mathbf{x}$ to $\mathbf{y}$ is an embedded torus rectangle whose sides lie in grid circles, whose northeast and southwest corners lie in $\mathbf{x}$, whose northwest and southeast corners lie in $\mathbf{y}$, and whose remaining state points agree. Rectangles are subsets of the torus, so their lifts may cross a chosen fundamental-domain edge. Such a rectangle is empty when its interior contains no point of $\mathbf{x}$; write ${\mathrm{Rect}}^{\circ }(\mathbf{x},\mathbf{y})$ for the empty rectangles from $\mathbf{x}$ to $\mathbf{y}$.

To define the gradings, fix the fundamental square $[0,n)\times [0,n)$. For finite signed point sets $P,Q$ in this square, set $I(P,Q)=\mathrm{\#}\{(p,q):p_1<q_1,p_2<q_2\}$ and $J(P,Q)=(I(P,Q)+I(Q,P))\mathrm{/}2$, extended bilinearly to signed point sets. Regard $\mathbf{x}$, $\mathbb{O}$, and $\mathbb{X}$ as signed point sets and define

$M_{\mathbb{O}}(\mathbf{x})=J(\mathbf{x}-\mathbb{O},\mathbf{x}-\mathbb{O})+1,M_{\mathbb{X}}(\mathbf{x})=J(\mathbf{x}-\mathbb{X},\mathbf{x}-\mathbb{X})+1.$

For a knot grid, define the rational Alexander value

$A_G(\mathbf{x})=\frac{M_{\mathbb{O}}(\mathbf{x})-M_{\mathbb{X}}(\mathbf{x})}{2}-\frac{n-1}{2}\mathrm{.}$

The lemma establishing integrality and the rectangle-change formula is a well-definedness supplier for this grading. For any rectangle $r$, let $O_i(r)$ and $X_i(r)$ be $1$ if the interior of $r$ contains the corresponding marking and $0$ otherwise; let $n_{\mathbb{O}}(r)={\sum }_i{O}_i(r)$ and $n_{\mathbb{X}}(r)={\sum }_i{X}_i(r)$.

For illustration, this $3\times 3$ marking pattern has one marking of each type in every row and column; its cell centers are read modulo three on the torus.

Oriented links, knots, and ambient isotopy in the standard three-sphere

Integrality and rectangle-change formula for the grid Alexander grading ↗

Ozsvath, Stipsicz, and Szabo, Grid Homology for Knots and Links, §§3.1–3.2 and 4.1–4.3

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

Manolescu, An Introduction to Knot Floer Homology, §4.1