A two-crossing closed braid factorization complex

The four resolutions and all signed edges. Write $C_{ab}=C({\mathrm{\Gamma }}_{ab})$ for the resolution with choices $a,b\in \{0,1\}$ at the two positive crossings, including all unchanged arc factors. The outer terms are $C^{-2}=C_{00}\{0,4\},C^{-1}=C_{10}\{0,2\}\oplus C_{01}\{0,2\},C^0=C_{11},$ and zero otherwise. In this summand order the outer differentials are ${\mathrm{\partial }}^{-2}=({\chi }_0\otimes 1,-1\otimes {\chi }_0),{\mathrm{\partial }}^{-1}=(1\otimes {\chi }_0,{\chi }_0\otimes 1)\mathrm{.}$ The minus sign comes from the first source factor having cochain degree $-1$. Each edge has internal bidegree $(0,0)$ after the indicated shifts and raises outer degree by $1$. The edge maps commute before the signs, since they act on different tensor factors, so ${\mathrm{\partial }}^{-1}{\mathrm{\partial }}^{-2}=0$. All four terms retain their inner two-periodic factorization differentials, with the same total potential.

The potential. Before closure, the boundary points of the braid diagram carry labels; every internal label occurs in exactly two local factors with opposite signs, so the sum of the local potentials is $a{\sum }_p{ϵ}_p{x}_p$ over the boundary points, and the square of the inner factorization differential is that element; the outer crossing differential still squares to zero by step 1.1. After closing the braid the closure arcs identify the boundary points in pairs with opposite orientations, so each boundary label occurs once with sign $+1$ and once with sign $-1$; the total potential is $0$ and every outer term of $C(D)$ is a genuine two-periodic complex of bigraded $\mathbb{Q}[a]$-modules.

The induced differential and the cohomology. For each term $C^j(D)$ the termwise cohomology $C{H}^j(D)$ is computed after removing the contractible summands, and $\mathrm{\partial }$ commutes with the internal differentials, so it descends to maps $\mathrm{\partial }\text{\hspace{0.17em}⁣}:C{H}^j(D)\to C{H}^{j+1}(D)$; the four resolutions contribute the four summands of $CH(D)$ and the induced maps are the sums of the four edge maps of the cube with Koszul signs. The resulting cohomology $H(D)$ is the trigraded cohomology of the diagram, and the trigradings of the four summands are inherited from the terms; no invariance statement is made here, and the representative is taken up to contractible summands in $K({\mathrm{h}\mathrm{m}\mathrm{f}}_0)$.