The diagonal of the X-marking anticommutator

This file computes the diagonal matrix entries of ∂⁻ ∘ H_k + H_k ∘ ∂⁻, where H_k is the X-marking homotopy of TauCeti.KnotTheory.Grid.XHomotopy.Basic.

A two-step decomposition from a grid state x back to x covers a full vertical or horizontal band of the torus, and its two rectangles cover disjoint squares. The decomposition is counted exactly when both rectangles are empty and the band carries a single X-marking, namely X_k. A vertical band carries one X-marking in each column it covers, and a horizontal band one in each row it covers, so it must be the column annulus of column k or the row annulus of the row of X_k, each one square thick. On a grid of size at least two, each thin annulus is cut by x into exactly one pair of rectangles, both automatically empty. The column annulus covers the single O-marking of column k, and the row annulus covers the O-marking in the row of X_k, so the diagonal entry is V_k + V_j, where O_j is the O-marking in the row of X_k (sum_XHomotopyDecompositions_self).

Main results

• TauCeti.GridDiagram.unblockedDecompositionWeight_eq_prod_union: the weight of a returning pair of rectangles is a product over the band it covers.
• TauCeti.GridDiagram.sum_XHomotopyDecompositions_self: on a grid of size at least two, the diagonal entries of ∂⁻ ∘ H_k + H_k ∘ ∂⁻ are V_k + V_j.

References

The annular terms follow Ozsváth--Stipsicz--Szabó, Grid Homology for Knots and Links, Chapter 4.6.

theorem TauCeti.GridDiagram.unblockedDecompositionWeight_eq_prod_union {n : ℕ} (G : GridDiagram n) (R : Type u_1) [CommSemiring R] {x : GridState n} (D : GridRectangleDecomposition x x) : G.unblockedDecompositionWeight R D = ∏ p ∈ D.first.toGridRectangle.coveredSquares ∪ D.second.toGridRectangle.coveredSquares, if p ∈ G.OSet then MvPolynomial.X p.1 else 1

The weight of a returning pair of rectangles is the product, over the band of squares they cover together, of the variable of the column of each O-marked square.

The two thin annuli

The diagonal entries

theorem TauCeti.GridDiagram.sum_XHomotopyDecompositions_self {n : ℕ} (G : GridDiagram n) (R : Type u_1) [CommSemiring R] (hn : 1 < n) (k : Fin n) (x : GridState n) : ∑ D ∈ G.XHomotopyDecompositions k x x, G.unblockedDecompositionWeight R D = MvPolynomial.X k + MvPolynomial.X (G.O.transpose.toPerm (G.X.toPerm k))

On a grid of size at least two, the diagonal matrix entries of ∂⁻ ∘ H_k + H_k ∘ ∂⁻ are V_k + V_j, where O_j is the O-marking in the row of X_k: the column annulus through X_k covers O_k and the row annulus through X_k covers O_j.