The Alexander grading of unblocked grid homology

For a grid diagram G with an odd number of link components, the unblocked grid chain module GC⁻ is the internal direct sum of its Alexander pieces over the coefficient ring R, and the unblocked grid differential ∂⁻ preserves the Alexander grading (TauCeti.OddComponentGridDiagram.alexanderChainMinusGrading). The grading therefore descends to the homology ker ∂⁻ ⧸ im ∂⁻, which is the unblocked grid homology GH⁻ (TauCeti.GridDiagram.unblockedHomologyIso): a class has Alexander degree a when it is the class of a cycle all of whose monomials V^e · x have A(x) - |e| = a.

Each variable V_c has Alexander degree -1 on GC⁻, on the concrete homology quotient, and on Mathlib's categorical homology GH⁻. The grading on GH⁻ is transported across the cycle-quotient isomorphism. Its homogeneous pieces are modules over R, since the variables move the Alexander grading. For a knot, every variable acts as U; the resulting degree rule for U is in Homology/KnotAlexander.

Main definitions

• TauCeti.OddComponentGridDiagram.alexanderHomologyGrading: the Alexander grading of ker ∂⁻ ⧸ im ∂⁻.
• TauCeti.OddComponentGridDiagram.alexanderUnblockedHomologyGrading: the corresponding grading of categorical GH⁻.

Main results

• TauCeti.OddComponentGridDiagram.mem_alexanderHomologyGrading_piece_iff: a class has Alexander degree a exactly when it is the class of a cycle of Alexander degree a.
• TauCeti.OddComponentGridDiagram.X_smul_mem_alexanderHomologyGrading_piece: each variable V_c lowers the Alexander grading of grid homology by one.
• TauCeti.OddComponentGridDiagram.X_smul_mem_alexanderUnblockedHomologyGrading_piece: the same degree rule on categorical GH⁻.
• TauCeti.OddComponentGridDiagram.unblockedHomologyEquivOfIntertwining_mem_piece: if an equivalence intertwining two unblocked differentials preserves the Alexander grading of chains, the induced equivalence of GH⁻ preserves the Alexander grading.

References

The Alexander grading of GH⁻ and the action of U in Alexander degree -1 are those of Ozsváth--Stipsicz--Szabó, Grid Homology for Knots and Links, Chapter 4; the definition of τ from them is in Chapter 6.

noncomputable def TauCeti.OddComponentGridDiagram.alexanderHomologyGrading {n : ℕ} (G : OddComponentGridDiagram n) (R : Type u_1) [CommRing R] [CharP R 2] : InternalGrading R (((↑G).unblockedDifferential R).homology ⋯) ℤ

The Alexander grading of unblocked grid homology: the grading of ker ∂⁻ ⧸ im ∂⁻ induced by the Alexander grading of GC⁻, which ∂⁻ preserves. Its degree-a piece consists of the classes of the cycles of Alexander degree a.

theorem TauCeti.OddComponentGridDiagram.mem_alexanderHomologyGrading_piece_iff {n : ℕ} {G : OddComponentGridDiagram n} {R : Type u_1} [CommRing R] [CharP R 2] {a : ℤ} {y : ((↑G).unblockedDifferential R).homology ⋯} : y ∈ (G.alexanderHomologyGrading R).piece ℤ a ↔ ∃ (z : ↥((↑G).unblockedDifferential R).ker), ↑z ∈ G.alexanderChainMinusPiece R a ∧ (((↑G).unblockedDifferential R).homologyπ ⋯) z = y

A class in ker ∂⁻ ⧸ im ∂⁻ has Alexander degree a exactly when it is the class of a cycle of Alexander degree a.

theorem TauCeti.OddComponentGridDiagram.X_smul_mem_alexanderHomologyGrading_piece {n : ℕ} {G : OddComponentGridDiagram n} {R : Type u_1} [CommRing R] [CharP R 2] (i : Fin n) {a : ℤ} {y : ((↑G).unblockedDifferential R).homology ⋯} (hy : y ∈ (G.alexanderHomologyGrading R).piece ℤ a) : MvPolynomial.X i • y ∈ (G.alexanderHomologyGrading R).piece ℤ (a - 1)

The variable V_c lowers the Alexander grading of grid homology by one.

noncomputable def TauCeti.OddComponentGridDiagram.alexanderUnblockedHomologyGrading {n : ℕ} (G : OddComponentGridDiagram n) (R : Type u_1) [CommRing R] [CharP R 2] : InternalGrading R ↑((↑G).unblockedHomology R) ℤ

The Alexander grading on the categorical unblocked grid homology, transported from the cycle quotient. Its degree-a piece consists of the classes of Alexander-homogeneous cycles.

@[simp]

theorem TauCeti.OddComponentGridDiagram.mem_alexanderUnblockedHomologyGrading_piece_iff {n : ℕ} (G : OddComponentGridDiagram n) (R : Type u_1) [CommRing R] [CharP R 2] (a : ℤ) (y : ↑((↑G).unblockedHomology R)) : y ∈ (G.alexanderUnblockedHomologyGrading R).piece ℤ a ↔ (CategoryTheory.ConcreteCategory.hom ((↑G).unblockedHomologyIso R).hom) y ∈ (G.alexanderHomologyGrading R).piece ℤ a

A class of GH⁻ is Alexander-homogeneous of degree a exactly when its image in the cycle quotient is homogeneous of that degree.

theorem TauCeti.OddComponentGridDiagram.X_smul_mem_alexanderUnblockedHomologyGrading_piece {n : ℕ} (G : OddComponentGridDiagram n) (R : Type u_1) [CommRing R] [CharP R 2] (i : Fin n) {a : ℤ} {y : ↑((↑G).unblockedHomology R)} (hy : y ∈ (G.alexanderUnblockedHomologyGrading R).piece ℤ a) : MvPolynomial.X i • y ∈ (G.alexanderUnblockedHomologyGrading R).piece ℤ (a - 1)

A grid variable lowers the Alexander degree of a class of GH⁻ by one.

Equivalences induced by intertwining maps

theorem TauCeti.OddComponentGridDiagram.unblockedHomologyEquivOfIntertwining_mem_piece {n : ℕ} (G : OddComponentGridDiagram n) (R : Type u_1) [CommRing R] [CharP R 2] (G' : OddComponentGridDiagram n) {σ σ' : MvPolynomial (Fin n) R →+* MvPolynomial (Fin n) R} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] (e : GridChainMinus R n ≃ₛₗ[σ] GridChainMinus R n) (he : ∀ (c : GridChainMinus R n), ((↑G').unblockedDifferential R) (e c) = e (((↑G).unblockedDifferential R) c)) (hA : ∀ (a : ℤ), ∀ c ∈ G.alexanderChainMinusPiece R a, e c ∈ G'.alexanderChainMinusPiece R a) {a : ℤ} {y : ↑((↑G).unblockedHomology R)} (hy : y ∈ (G.alexanderUnblockedHomologyGrading R).piece ℤ a) : ((↑G).unblockedHomologyEquivOfIntertwining R (↑G') e he) y ∈ (G'.alexanderUnblockedHomologyGrading R).piece ℤ a

If an equivalence intertwining two unblocked differentials preserves the Alexander grading of chains, the induced equivalence of unblocked homologies preserves the Alexander grading.