The Alexander grading of the `X`-marking state of a torus link grid #

In the standard torus link grid torusLink p q of grid number n = (p + 1) + (q + 1), the X-marking in column c sits in row c + (q + 1) taken modulo n: in row c + q + 1 for c ≤ p and in row c - (p + 1) for c > p (torusLink_X_val_of_le, torusLink_X_val_of_lt). This file computes the Alexander grading of the X-marking state G.X and shows that no grid state has a larger one.

Both facts come from the column-sum form of the Alexander grading (GridDiagram.alexanderTwoℤ_eq_sum_southwestCount): twice the Alexander grading of a state x is, up to a constant depending only on the diagram, twice the sum over the columns c of southwestCount 𝕏 c (x c) - min c (x c). Each summand is at most zero (GridState.southwestCount_le_min), and at the X-marking state every summand vanishes (southwestCount_X_torusLink_X): the X-markings southwest of an X-corner are exactly those of the columns to its left in the same block of the shifted diagonal. Hence A(x) ≤ A(G.X) for every grid state x (alexanderTwoℤ_le_torusLink_X), and evaluating the constant gives 2 A(G.X) = p q (alexanderTwoℤ_torusLink_X).

When p + 1 and q + 1 are coprime the diagram is a knot grid and its Alexander grading is integral; then A(G.X) = p q / 2, which is the Seifert genus (p q) / 2 of the (p + 1, q + 1) torus knot (two_mul_alexanderℤ_torusLink_X, alexanderℤ_le_torusLink_X).

Main results #

TauCeti.GridDiagram.southwestCount_X_torusLink_X: at an X-corner of a torus link grid the southwest count of the X-markings is min c (𝕏 c).
TauCeti.GridDiagram.alexanderTwoℤ_torusLink_X: 2 A(G.X) = p q.
TauCeti.GridDiagram.alexanderTwoℤ_le_torusLink_X: the X-marking state has the largest Alexander grading among the grid states.
TauCeti.GridDiagram.two_mul_alexanderℤ_torusLink_X and TauCeti.GridDiagram.alexanderℤ_le_torusLink_X: the integer forms on a torus knot grid.

References #

The Alexander grading of the canonical generator of the torus knot grid is computed in Ozsváth--Stipsicz--Szabó, Grid Homology for Knots and Links, Chapter 6.

Southwest counts at the `X`-corners #

source

theorem TauCeti.GridDiagram.southwestCount_X_torusLink_X (p q : ℕ) (c : Fin (p + 1 + (q + 1))) :

(torusLink p q).X.southwestCount c ((torusLink p q).X.toPerm c) = min ↑c ↑((torusLink p q).X.toPerm c)

At an X-corner (c, 𝕏 c) of a torus link grid, the X-markings strictly southwest are exactly those of the columns to the left of c in the same block of the shifted diagonal, so their number is min c (𝕏 c).

source

theorem TauCeti.GridDiagram.two_mul_sum_min_torusLink_X (p q : ℕ) :

2 * ∑ c : Fin (p + 1 + (q + 1)), min ↑c ↑((torusLink p q).X.toPerm c) = p * (p + 1) + q * (q + 1)

The sum over the columns of a torus link grid of min c (𝕏 c) is p (p + 1) / 2 + q (q + 1) / 2: each block of the shifted diagonal contributes the sum of its column indices within the block.