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TauCeti.KnotTheory.Grid.Stabilization.Map.Grading

The chain map of an X-stabilization preserves the bigrading #

Let G be a grid diagram of size n with an odd number of link components, let s be a column, and let G' = G.stabilizeX s.castSucc (G.X s).castSucc s be the stabilization splitting the X-marking of column s. Adding the center of the new block to a state of G lowers its doubled Alexander grading by two (GridDiagram.alexanderTwoℤ_stabilizeX_insertPoint), so the parity criterion GridDiagram.even_alexanderTwoℤ_iff shows that G' again has an odd number of components (OddComponentGridDiagram.stabilizeX, in Stabilization/Grading.lean). Both unblocked complexes GC⁻(G') and GC⁻(G) therefore carry the (O-Maslov, Alexander) bigrading of Grading/Unblocked/Chain.lean.

This file proves that the coefficient formula for the stabilization comparison has bidegree (0, 0) over every commutative ring. On a chain c it is H_I^N applied to the off-center part of c, followed by the renaming V_j ↦ V_{s.predAbove j} that merges the two variables of the new block. A monomial V^e · y of c, for y an off-center state, therefore contributes terms V^{e + O(r)} · x with the exponent renamed, one for each rectangle r of the X₂-homotopy from y to the state x' obtained by inserting the center of the new block into x. Renaming variables does not change the total degree, and each such term has the bigrading of V^e · y (GridDiagram.maslovOℤ_stabilizeX_eq_of_isEmpty and GridDiagram.alexander_stabilizeX_eq_of_mem_XHomotopyRectangles).

This formula is the underlying function of GridDiagram.stabilizeXChainMap (GridDiagram.stabilizeXChainMap_apply). Over a commutative ring of characteristic two, it is also the component of GridDiagram.stabilizeXMap (GridDiagram.stabilizeXMap_f_apply), so that chain map has bidegree (0, 0).

Since stabilizeXMap is a quasi-isomorphism (GridDiagram.quasiIso_stabilizeXMap), this is the chain-level input for identifying the bigraded homologies GH⁻(G') and GH⁻(G), and so the invariants τ of a knot grid and of this stabilization.

Main results #

References #

This is the statement that the stabilization quasi-isomorphism is bigraded, in the proof of stabilization invariance in Ozsváth--Stipsicz--Szabó, Grid Homology for Knots and Links, Section 5.2, and Manolescu--Ozsváth--Szabó--Thurston, On combinatorial link Floer homology, Section 3.2.

source
theorem TauCeti.OddComponentGridDiagram.stabilizeXCoefficientFormula_mem_bigradedChainMinusPiece {n : ℕ} (G : OddComponentGridDiagram n) (s : Fin n) (R : Type u_1) [CommRing R] {g : ℤ × ℤ} {c : GridChainMinus R (n + 1)} (hc : c ∈ (G.stabilizeX s).bigradedChainMinusPiece R g) :
Finsupp.mapRange ⇑(MvPolynomial.rename s.predAbove) ⋯ (((↑G).stabilizeXOffCenterToCenter s R) (((↑G).stabilizeXOffCenterProjection s R) c)) ∈ G.bigradedChainMinusPiece R g

The stabilization coefficient formula preserves the bigrading. For the stabilization splitting the X-marking of column s, the displayed formula sends a chain of GC⁻(G') homogeneous of bidegree g to a chain of GC⁻(G) homogeneous of the same bidegree over any commutative ring. In characteristic two, this is the component of stabilizeXMap.

source
theorem TauCeti.OddComponentGridDiagram.stabilizeXMap_mem_alexanderChainMinusPiece {n : ℕ} (G : OddComponentGridDiagram n) (s : Fin n) (R : Type u_1) [CommRing R] [CharP R 2] {a : ℤ} {c : GridChainMinus R (n + 1)} (hc : c ∈ (G.stabilizeX s).alexanderChainMinusPiece R a) :
(ModuleCat.Hom.hom (CategoryTheory.CategoryStruct.comp ((ModuleCat.restrictScalars ↑(MvPolynomial.rename s.castSucc.succAbove)).map (CategoryTheory.eqToHom ⋯)) (CategoryTheory.CategoryStruct.comp (((↑G).stabilizeXMap s R).f ()) (CategoryTheory.eqToHom ⋯)))) c ∈ G.alexanderChainMinusPiece R a

The stabilization chain map preserves the Alexander grading. The chain map GC⁻(G') ⟶ GC⁻(G) of the stabilization splitting the X-marking of column s sends a chain of GC⁻(G') of Alexander degree a to a chain of GC⁻(G) of the same Alexander degree.