The bidegrees of the crossing change maps

Exchanging two cyclically adjacent columns a and b = finRotate n a of a grid diagram G whose marking segments interleave is a cross-commutation: G and G' = G.swapColumns a b present links that differ by a crossing change. The pentagon-counting commutation map GridDiagram.commutationMap of a column turn at these columns is a chain map GC⁻(G) → GC⁻(G') at every turn row (GridDiagram.commutationMap_comp_unblockedDifferential), whether or not the segments interleave. Its bidegree GridDiagram.commutationBidegree depends on the turn row. This file finds the turn rows at which it has the bidegrees of the crossing change maps.

For interleaving segments, exactly one of the two O-markings lies on the open arc of the other column (GridDiagram.ColumnsInterleaving.O_mem_columnArc_iff). Going up around the vertical circle, the markings appear in one of two orders.

• If O_a lies on the arc of column b, the order is O_b, O_a, X_b, X_a. The commutation map turning in the row of O_a then has bidegree (0, 0) (GridDiagram.commutationBidegree_eq_zero_of_mem_columnArc).
• Otherwise the order is O_b, X_a, X_b, O_a. The commutation map turning in the row of X_a then has bidegree (-2, -1), that of multiplication by U (GridDiagram.commutationBidegree_eq_of_notMem_columnArc).

A cross-commutation exchanges the two orders (GridDiagram.ColumnsInterleaving.O_swapColumns_mem_columnArc_iff). So of two diagrams related by a cross-commutation, one has a chain map to the other preserving the bigrading, and the other has a chain map back of bidegree (-2, -1). These are the bidegrees of the crossing change maps of Ozsváth--Stipsicz--Szabó, Section 6.1, whose two composites are chain homotopic to multiplication by U. No such homotopy is constructed here. For knot grids the maps induced on GH⁻ are R[U]-linear and homogeneous of Alexander degrees 0 and -1 (GridDiagram.IsKnot.isHomogeneous_unblockedHomologyCommutationLinearMap_of_mem_columnArc and GridDiagram.IsKnot.isHomogeneous_unblockedHomologyCommutationLinearMap_of_notMem_columnArc). This is the shape of the maps in the comparison lemma GridDiagram.IsKnot.tau_le_tau_sub_of_comp_eq_X_pow_smul, which turns them into a bound on the change of τ across a crossing change once the composites are known.

Main results

• TauCeti.GridDiagram.commutationBidegree_eq_zero_of_mem_columnArc and TauCeti.GridDiagram.commutationBidegree_eq_of_notMem_columnArc: turn rows at which the commutation map of a cross-commutation has bidegree (0, 0), respectively (-2, -1).
• isHomogeneous_unblockedHomologyCommutationLinearMap_of_mem_columnArc and isHomogeneous_unblockedHomologyCommutationLinearMap_of_notMem_columnArc, in the namespace TauCeti.GridDiagram.IsKnot: the induced R[U]-linear maps on GH⁻ of a knot grid are homogeneous of Alexander degrees 0 and -1.

References

• P. Ozsváth, A. Stipsicz, Z. Szabó, Grid Homology for Knots and Links, AMS Mathematical Surveys and Monographs 208, 2015, Section 6.1: the crossing change maps and their gradings.
• S. Sarkar, Grid diagrams and the Ozsváth--Szabó tau-invariant, Math. Res. Lett. 18 (2011), arXiv:1011.5265: the crossing change inequality for τ from grid diagrams.

theorem TauCeti.GridDiagram.commutationBidegree_eq_zero_of_mem_columnArc {n : ℕ} {G : GridDiagram n} (C : ColumnTurn n) (h : G.ColumnsInterleaving C.column ((finRotate n) C.column)) (hO : G.O.toPerm C.column ∈ G.columnArc ((finRotate n) C.column)) (hs : C.turnRow = G.O.toPerm C.column) : G.commutationBidegree C = 0

A grading-preserving chain map across a crossing change. If the segments of the two columns of the column turn C interleave and the O-marking of the first column lies on the arc of the second, then the commutation map turning in the row of that O-marking has bidegree (0, 0).

theorem TauCeti.GridDiagram.commutationBidegree_eq_of_notMem_columnArc {n : ℕ} {G : GridDiagram n} (C : ColumnTurn n) (h : G.ColumnsInterleaving C.column ((finRotate n) C.column)) (hO : G.O.toPerm C.column ∉ G.columnArc ((finRotate n) C.column)) (hs : C.turnRow = G.X.toPerm C.column) : G.commutationBidegree C = (-2, -1)

A chain map of bidegree (-2, -1) across a crossing change. If the segments of the two columns of the column turn C interleave and the O-marking of the first column does not lie on the arc of the second, then the commutation map turning in the row of the X-marking of the first column has bidegree (-2, -1), the bidegree of multiplication by U.

theorem TauCeti.GridDiagram.IsKnot.isHomogeneous_unblockedHomologyCommutationLinearMap_of_mem_columnArc {n : ℕ} {G : GridDiagram n} (C : ColumnTurn n) (hG : G.IsKnot) (R : Type u_1) [CommRing R] [CharP R 2] (h : G.ColumnsInterleaving C.column ((finRotate n) C.column)) (hO : G.O.toPerm C.column ∈ G.columnArc ((finRotate n) C.column)) (hs : C.turnRow = G.O.toPerm C.column) : LinearMap.IsHomogeneous (hG.unblockedHomologyCommutationLinearMap R C) ((hG.alexanderUnblockedHomologyGrading R).piece ℤ) ((⋯.alexanderUnblockedHomologyGrading R).piece ℤ) 0

For a knot grid, the chain map of GridDiagram.commutationBidegree_eq_zero_of_mem_columnArc induces an R[U]-linear map on GH⁻ that preserves the Alexander grading.

theorem TauCeti.GridDiagram.IsKnot.isHomogeneous_unblockedHomologyCommutationLinearMap_of_notMem_columnArc {n : ℕ} {G : GridDiagram n} (C : ColumnTurn n) (hG : G.IsKnot) (R : Type u_1) [CommRing R] [CharP R 2] (h : G.ColumnsInterleaving C.column ((finRotate n) C.column)) (hO : G.O.toPerm C.column ∉ G.columnArc ((finRotate n) C.column)) (hs : C.turnRow = G.X.toPerm C.column) : LinearMap.IsHomogeneous (hG.unblockedHomologyCommutationLinearMap R C) ((hG.alexanderUnblockedHomologyGrading R).piece ℤ) ((⋯.alexanderUnblockedHomologyGrading R).piece ℤ) (-1)

For a knot grid, the chain map of GridDiagram.commutationBidegree_eq_of_notMem_columnArc induces an R[U]-linear map on GH⁻ that lowers the Alexander grading by one.