The commutation map on the X-marking state

The X-marking state G.X of a grid diagram, whose point in each column is the lower-left corner of the X-marked square, is the grid state x⁻(G) of Ozsváth--Szabó--Thurston. It is a cycle of the unblocked grid complex GC⁻, and its class XMarkingClass in GH⁻ is their class λ⁻(G), which they show to be an invariant of the Legendrian knot the diagram presents. This file shows that the pentagon-counting commutation map Φ of a column commutation carries x⁻(G) to x⁻(G'), where G' is the commuted diagram, so that the induced map on GH⁻ carries λ⁻(G) to λ⁻(G'). Nothing about Legendrian knots is formalised here.

Let a and b = finRotate n a be the commuted columns, and let the turn point lie in square row s. A pentagon turning on its initial side starts at the point of G.X on the line b, the lower-left corner of the X-marked square of column b, and carries that marking unless s is its row; so if the turn row is not the row of X_b, no such pentagon from G.X is counted (initialPentagons_X_eq_empty). A pentagon turning on its terminal side with initial side j starts at the lower-left corner of the X-marked square of column j, and carries that marking unless j = a, where the marking lies below the turn point; so every counted one ends at G.X.swapColumns a b, which is G'.X (pentagons_X_eq_empty). For validated commutation data the pentagon with initial side a is one column wide, hence empty, and it carries neither an X-marking nor an O-marking; for the O-marking of column b this uses that the two columns do not interleave (pentagonCoefficient_X_swapColumns_X).

The condition on the turn row cannot be dropped: when the turn row is the row of X_b, the pentagon with initial side a is not counted, and the pentagons turning on their initial side that are counted instead may carry an O-marking or end at another grid state. The data that ColumnCommutationData.ofNoninterleaving attaches to every column commutation satisfies the condition, so its commutation map always sends x⁻(G) to x⁻(G') (commutationMap_ofNoninterleaving_single_X).

Main results

• TauCeti.GridDiagram.pentagonCoefficient_X_swapColumns_X: if the turn row is not the row of X_b, the pentagon map has coefficient 1 from G.X to G'.X.
• TauCeti.GridDiagram.commutationMap_single_X: if the turn row is not the row of X_b, the commutation map sends the X-marking state of G to that of the commuted diagram.
• TauCeti.GridDiagram.unblockedHomologyCommutationMap_XMarkingClass: under the same condition, the induced map on GH⁻ sends the class of the X-marking state to the class of the X-marking state.
• TauCeti.GridDiagram.commutationMap_ofNoninterleaving_single_X: the commutation map of the data attached to every column commutation sends the X-marking state of G to that of the commuted diagram.

References

• P. Ozsváth, Z. Szabó, D. Thurston, Legendrian knots, transverse knots and combinatorial Floer homology, Geom. Topol. 12 (2008), arXiv:math/0611841: the grid states x⁺ and x⁻, their classes λ⁺ and λ⁻, and their invariance under commutation.
• P. Ozsváth, A. Stipsicz, Z. Szabó, Grid Homology for Knots and Links, AMS Mathematical Surveys and Monographs 208, 2015, Chapter 12: the same material in the conventions of the grid book.

The counted pentagons leaving the X-marking state

theorem TauCeti.GridDiagram.initialPentagons_X_eq_empty {n : ℕ} (G : GridDiagram n) (C : ColumnTurn n) (hs : C.turnRow ≠ G.X.toPerm ((finRotate n) C.column)) (y : GridState n) : G.initialPentagons C G.X y = ∅

If the turn row is not the row of the X-marking of the column after the replaced line, no counted pentagon turning on its initial side leaves the X-marking state: such a pentagon starts at the lower-left corner of that marking's square and covers it.

theorem TauCeti.GridDiagram.pentagons_X_eq_empty {n : ℕ} (G : GridDiagram n) (C : ColumnTurn n) {y : GridState n} (hy : y ≠ (swapColumns C.column ((finRotate n) C.column) G).X) : G.pentagons C G.X y = ∅

No counted pentagon turning on its terminal side leaves the X-marking state for a grid state other than the X-marking state of the commuted diagram.

The commutation map on the X-marking state

theorem TauCeti.GridDiagram.pentagonCoefficient_X_swapColumns_X {n : ℕ} (G : GridDiagram n) (R : Type u_1) [CommSemiring R] (C : G.ColumnCommutationData) (hs : C.turnRow ≠ G.X.toPerm ((finRotate n) C.column)) : G.pentagonCoefficient R C.toColumnTurn G.X (swapColumns C.column ((finRotate n) C.column) G).X = 1

For validated column-commutation data whose turn row is not the row of the X-marking of the second commuted column, the pentagon map has coefficient 1 from the X-marking state of G to that of the commuted diagram: the one counted pentagon is one column wide and carries no O-marking.

theorem TauCeti.GridDiagram.commutationMap_single_X {n : ℕ} (G : GridDiagram n) (R : Type u_1) [CommSemiring R] (C : G.ColumnCommutationData) (hs : C.turnRow ≠ G.X.toPerm ((finRotate n) C.column)) : (G.commutationMap R C.toColumnTurn) (Finsupp.single G.X 1) = Finsupp.single (swapColumns C.column ((finRotate n) C.column) G).X 1

The commutation map preserves the X-marking state. For validated column-commutation data whose turn row is not the row of the X-marking of the second commuted column, the commutation map sends the X-marking state of G to the X-marking state of the commuted diagram.

theorem TauCeti.GridDiagram.commutationMap_ofNoninterleaving_single_X {n : ℕ} (G : GridDiagram n) (R : Type u_1) [CommSemiring R] {a : Fin n} (ha : a ≠ (finRotate n) a) (hG : G.ColumnsNoninterleaving a ((finRotate n) a)) : (G.commutationMap R (ColumnCommutationData.ofNoninterleaving a ha hG).toColumnTurn) (Finsupp.single G.X 1) = Finsupp.single (swapColumns a ((finRotate n) a) G).X 1

The commutation map of the data attached to a column commutation sends the X-marking state of G to the X-marking state of the commuted diagram.

The class of the X-marking state in GH⁻

theorem TauCeti.GridDiagram.unblockedHomologyCommutationMap_XMarkingClass {n : ℕ} (G : GridDiagram n) (R : Type u_1) [CommRing R] [CharP R 2] (C : G.ColumnCommutationData) (hs : C.turnRow ≠ G.X.toPerm ((finRotate n) C.column)) : (G.unblockedHomologyCommutationMap R C.toColumnTurn) (G.XMarkingClass R) = (swapColumns C.column ((finRotate n) C.column) G).XMarkingClass R

The commutation map on GH⁻ preserves the class of the X-marking state. Under the condition of commutationMap_single_X, the induced map on unblocked grid homology sends the class of the X-marking state of G to that of the commuted diagram.