Commutation and stabilization invariance of grid complexes

A cyclic permutation is a change of fundamental cut on the same grid torus. It identifies states, rectangles, and marking monomials bijectively. The finite pair-count gradings also agree under this change. To check this directly for a column cut, use the signed set $P=\mathbf{x}-\mathbb{O}$ in $M_O=I(P,P)+1$. The wrapping column contains one state point of sign $+1$ and one $O$ of sign $-1$. Only their comparisons with nonwrapping points change. At an integer state height the signed number of all points below is zero, and at the height of a marking in row $r$ it is one, since the states have heights $0,\dots ,n-1$ and the markings have one height in each open row. Thus the signed difference of the below/above counts for either wrapping point, excluding itself, is one. Multiplying by the two signs and adding gives zero; the comparisons between the two wrapping points cancel from this sum and do not change. Hence $I(P,P)$ is unchanged. Apply the same count to $\mathbf{x}-\mathbb{X}$, and interchange coordinates for a row cut. Both Maslov functions and the Alexander formula are unchanged. The torus-identification map is therefore a filtered chain isomorphism, with inverse the opposite cut change and zero inverse homotopies.

For a commutation, use the maps and homotopies in [F1]. Their weights retain every $O$ variable separately, and their filtration drop is the number of $X$ in the polygon. Thus they are filtered equivalences over the full ring; restricting the ring action to any chosen marking variable preserves these identities. Their associated-graded maps retain precisely polygons avoiding $X$. Setting the variables to zero gives the tilde equivalence. The identical conclusion holds for the switches used below.

For a southwest $X$ stabilization, [F2] supplies the map to the cone of $U_1+U_2$ on the polynomial extension of the smaller complex. By [F3], it has a finite matrix inverse and both filtered inverse homotopies. Compose with the explicit retraction in [F5]. Its inclusion, divided-difference homotopy, and evaluation are linear over every old variable, so the resulting equivalence and homotopies are over any chosen retained old-variable action. If either grid uses another chosen variable, including the newly inserted one, first or last compose with the chosen-variable equivalences of [F4]. Hence the assertion holds for arbitrary choices of the single variable action, without setting any independent variable equal to another.

The other local stabilization types reduce to this one using the following finite operations. An $O$ stabilization introduces a vertical segment joining marks in consecutive rows. Commute this short segment horizontally toward the next $X$ endpoint; while passing another vertical segment, its length-one interval is either disjoint or nested, since no marking height lies strictly between its two endpoints. At a shared endpoint level the exchange is a switch. Stop at the next $X$, where deleting that short segment is an $X$ destabilization back to the original grid. Tracking the empty corner gives the pairs $O:NE$ with $X:SW$, $O:SE$ with $X:NW$, $O:NW$ with $X:SE$, and $O:SW$ with $X:NE$. Among the four $X$ types, exchanging the two new adjacent rows moves the empty corner across a horizontal side, and exchanging the two new adjacent columns moves it across a vertical side; these are switches because the central opposite-type marking shares a column or row with the adjacent same-type marking. One or two such switches carry any empty corner to the southwest corner. These operations concern only the added row, added column, and their endpoint segments; after deleting them the original row and column orders are unchanged. Thus every stabilization is a southwest $X$ stabilization followed by finitely many commutations, switches, and cut changes.

Compose the equivalences for that finite sequence, using step 1.2 for commutations and switches and step 1.3 for its sole stabilization. Explicit inverse homotopies compose: for equivalences $f_1,f_2$ with inverses $g_1,g_2$ and homotopies $g_i{f}_i+1=d{H}_i+H_{i}d$, a homotopy for $g_1{g}_2{f}_2{f}_1+1$ is $H_1+g_1{H}_2{f}_1$. The counterpart on the other complex has the analogous formula. Every summand remains filtered and has degree $+1$. Reversing the roles of the two grids proves the destabilization assertion as well. All map formulas are finite sums of composable grid-domain counts and the explicit polynomial formulas in [F3]–[F5].

The full polynomial equivalence with the stabilization cone in step 1.3 can also be tensored with the simultaneous ordinary coefficient quotient. Its cross map $U_1+U_2$ becomes zero, so [F5] identifies the quotient cone with the collapsed smaller complex tensored with $V$. The first summand's grading and filtration shift is $(-1,-1)$, giving exactly the displayed bidegrees. Taking the associated-graded equivalence first and then setting every variable to zero gives the same direct sum for tilde homology. Commutation and switches preserve tilde homology by step 1.2, and cut changes do by step 1.1. These comparisons prove the tilde and collapsed assertions for every stabilization type.