The center complex of a stabilized grid is a polynomial extension #
Let G be a grid diagram of size n, 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. Write A = R[V₀, …, V_{n-1}] and S = R[V₀, …, V_n] for the coefficient rings of
GC⁻(G) and GC⁻(G'). The columns of G become the columns s.castSucc.succAbove c of G';
the remaining column s.castSucc carries the new O-marking, and the old O-marking of column
s now sits in column s.succ. Renaming the variables accordingly makes S an A-algebra, and
singling out the new variable identifies S with the polynomial ring A[X], with
X = V_{s.castSucc} (MvPolynomial.finSuccEquiv').
By TauCeti.KnotTheory.Grid.Stabilization.Cone, GC⁻(G') is the mapping cone of the connecting
map from the center complex I, spanned by the grid states of G, whose differential is that
of GC⁻(G) with the variables renamed into S. This file shows that, after restricting scalars
from S to A, I is the polynomial extension A[X] ⊗[A] GC⁻(G)
(polynomialExtensionIsoStabilizeXCenter), with multiplication by a polynomial q on
the extension corresponding to multiplication by its image in S.
Under this identification multiplication by V_{s.succ} + V_{s.castSucc} on I is
multiplication by X - V_s on A[X] ⊗[A] GC⁻(G), because the ring has characteristic two. The
mapping cone of the latter is homotopy equivalent to GC⁻(G)
(HomologicalComplex.polynomialExtensionMulXSubCHomotopyEquiv), so the mapping cone of
multiplication by V_{s.succ} + V_{s.castSucc} on I is homotopy equivalent to GC⁻(G) as a
complex of A-modules (stabilizeXCenterConeHomotopyEquiv). On the summand I of that cone the
equivalence is evaluation at X = V_s, which merges the variables V_{s.castSucc} and V_{s.succ}
of G' into the variable V_s of G.
This is the cone to which the pair (𝟙, H_I^N) of TauCeti.KnotTheory.Grid.Stabilization.XHomotopy
maps GC⁻(G') = Cone(∂_I^N), so the equivalence here is the step comparing that cone with GC⁻(G)
in the proof of stabilization invariance of GH⁻.
Main definitions #
TauCeti.GridDiagram.polynomialExtensionIsoStabilizeXCenter: the isomorphism of complexes ofA-modules betweenA[X] ⊗[A] GC⁻(G)and the center complex.TauCeti.GridDiagram.stabilizeXCenterConeHomotopyEquiv: the homotopy equivalence between the mapping cone of multiplication byV_{s.succ} + V_{s.castSucc}on the center complex andGC⁻(G).
Main results #
TauCeti.GridDiagram.polynomialExtensionIsoStabilizeXCenter_hom_f_tmul: the isomorphism sendsp ⊗ fto the image ofpinStimesfwith its coefficients renamed intoS.TauCeti.GridDiagram.polynomialExtensionMul_comp_polynomialExtensionIsoStabilizeXCenter_hom: the isomorphism intertwines multiplication byq : A[X]with multiplication by its image inS.TauCeti.GridDiagram.map_inr_comp_stabilizeXCenterConeHomotopyEquiv_homandTauCeti.GridDiagram.stabilizeXCenterConeHomotopyEquiv_inv: the equivalence is evaluation atV_son the summandIof the cone, and its inverse includesGC⁻(G)into that summand as the constant polynomials.
References #
The identification of the center complex with the polynomial extension GC⁻(G)[V₁], and of the
cone of V₁ - V₂ on it with GC⁻(G), are steps of the stabilization argument 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. The
comparison of the whole stabilized complex with that cone is not carried out here.
The underlying modules #
The center complex #
The center complex is the polynomial extension of GC⁻(G). After restricting scalars
from S = R[V₀, …, V_n] to A = R[V₀, …, V_{n-1}] along the renaming of the columns of G into
those of the stabilization, the complex of center states is isomorphic to A[X] ⊗[A] GC⁻(G).
The isomorphism sends p ⊗ x, for x a grid state of G, to the image of p in S under
X ↦ V_{s.castSucc} times the center state x.
Instances For
On a pure tensor p ⊗ f, the identification of A[X] ⊗[A] GC⁻(G) with the center complex
is the image of p in S times f with its coefficients renamed into S. The eqToHoms
identify the chain modules of GC⁻(G) and of the center complex with their underlying free
modules.
The identification of the center complex with the polynomial extension of GC⁻(G)
intertwines multiplication by a polynomial q : A[X] with multiplication by its image in S
under X ↦ V_{s.castSucc}.
The identification of the center complex with the polynomial extension of GC⁻(G)
intertwines multiplication by a polynomial q : A[X] with multiplication by its image in S
under X ↦ V_{s.castSucc}.
The mapping cone of V_{s.succ} + V_{s.castSucc} #
The cone of V_{s.succ} + V_{s.castSucc} on the center complex is GC⁻(G). After
restricting scalars to A = R[V₀, …, V_{n-1}], the mapping cone of multiplication by
V_{s.succ} + V_{s.castSucc} on the complex of center states of the stabilization is homotopy
equivalent to GC⁻(G). In characteristic two this multiplication is multiplication by
X - V_s on the polynomial extension A[X] ⊗[A] GC⁻(G), whose cone is homotopy equivalent to
GC⁻(G) by evaluation at V_s.
Instances For
On the summand of the cone given by the center complex, the homotopy equivalence
stabilizeXCenterConeHomotopyEquiv is evaluation at X = V_s on the polynomial extension of
GC⁻(G).
On the summand of the cone given by the center complex, the homotopy equivalence
stabilizeXCenterConeHomotopyEquiv is evaluation at X = V_s on the polynomial extension of
GC⁻(G).
Evaluation from the center cone kills its shifted summand.
The homotopy inverse of stabilizeXCenterConeHomotopyEquiv includes GC⁻(G) into the
summand of the cone given by the center complex, as the constant polynomials of its polynomial
extension.