ScalingStacks

Fix an almost closure $T$ of a braid word $\beta$ for $[\beta]\in\mathrm{Br}_{n+1}$. We call the right-hand closure $L$ and the left-hand closure $L^{\prime}$. We consider the following movies of intermediate diagrams and their associated chain maps between Khovanov–Rozansky complexes.

(3.1)

In the first row, the $\pm$ signs indicate the two versions of this movie, in which the horizontal strand passes in front of ($+$) or behind ($-$) $T$. We denote the composition along the top by $\mathrm{sw}_{+}$ and the composition along the bottom by $\mathrm{sw}_{-}$. In either case we first see a Reidemeister I move (denoted by $R1_{\pm}$), then a composite of Reidemeister II moves (denoted by $R2_{\pm}$), a number of Reidemeister III moves (each denoted by $R3_{\pm}$), a composite of inverse Reidemeister II moves ($R2^{-1}_{\pm}$), and finally an inverse Reidemeister I move ($R1^{-1}_{\pm}$). Our goal is to show that, after making careful use of the freedom, described later, to choose up-to-homotopy representatives of the chain maps for Reidemeister III moves, we have the following: