Curve complexes intertwine the braid generators

Decomposition into $k$-strings. Let $\tilde{c}$ be an admissible bigraded curve in normal form and fix $k$. The direct sum decomposition of the graded module $L(\tilde{c})$ into its summands $P(x)$ over crossings restricts, over the subsets of crossings lying in a single $k$-string $\tilde{g}$, to a direct sum decomposition $L(\tilde{c})={⨁}_{\tilde{g}\in \mathrm{st}(\tilde{c},k)}L(\tilde{g})\oplus (\text{crossings with }\mathrm{\mid }{x}_0-k\mathrm{\mid }>1)$. Applying $U_k$, all summands $P(x)$ with $\mathrm{\mid }{x}_0-k\mathrm{\mid }>1$ die by [L2], and for a composable pair of crossings in different $k$-strings the induced map $U_k{\mathrm{\partial }}_{yx}$ is zero: either one of the two crossings has $\mathrm{\mid }{x}_0-k\mathrm{\mid }>1$, or both lie on $d_{k\pm 1}$ and the differential is right multiplication by $(x_0\mathrm{\mid }k\mathrm{\mid }{x}_0)$, which $U_k$ kills [L2]; hence $U_{k}L(\tilde{c})\cong {⨁}_{\tilde{g}}{U}_{k}L(\tilde{g})$ as complexes.

The local intertwiners. For each of the finitely many types of bigraded $k$-strings the source's case-by-case computation (the local lemmas and Cases 1-5 of the proof) writes $R_{k}L(\tilde{g})$ as $L({\tilde{\tau }}_k\tilde{g})$ plus contractible two-term summands with explicit contracting homotopies: for a string $\tilde{g}$ of type $\mathrm{V}\mathrm{I}$ this is the computation $R_k{P}_k\simeq P_k[1]\{1\}$; for the types $\mathrm{I}\mathrm{V},{\mathrm{I}\mathrm{V}}^{\mathrm{\prime }},\mathrm{V},{\mathrm{V}}^{\mathrm{\prime }}$ the summand $U_{k}L(\tilde{g})$ is acyclic; for the types $\mathrm{I},\mathrm{I}\mathrm{I},{\mathrm{I}\mathrm{I}}^{\mathrm{\prime }},\mathrm{I}\mathrm{I}\mathrm{I},{\mathrm{I}\mathrm{I}\mathrm{I}}^{\mathrm{\prime }}$ the complex $U_{k}L(\tilde{g})$ splits off acyclic complexes and the central folding is isomorphic to $L({\tilde{\tau }}_k\tilde{g})$ after the indicated sign isomorphisms, as displayed in the source comparisons [F1]. Those comparisons are relative to the outside complement $\mathrm{\nabla }$. In particular, the type-II${}_0$ comparison sends $(x,y,z,w)$ to $(x,-y-z(k-1\mathrm{\mid }k),z,-w)$ and negates the entire attached right tail (printed p. 43); it is not simply an identity on all outside summands. Composing these relative homotopy equivalences string by string, and step 1.1 gives $R_{k}L(\tilde{c})\cong L({\tilde{\tau }}_k\tilde{c})$.

The inverse-generator analogue. Apply step 2.1 to ${\tilde{\tau }}_k^{-1}\tilde{c}$: it gives $R_{k}L({\tilde{\tau }}_k^{-1}\tilde{c})\cong L(\tilde{c})$. Tensor with $R_k^{-1}$ and use $R_k^{-1}{R}_k\cong \mathrm{Id}$ from [L7]. Thus $L({\tilde{\tau }}_k^{-1}\tilde{c})\cong R_k^{-1}L(\tilde{c})$. No additional inverse case computation is needed.

Induction over braid words. Let $\sigma$ be a word. By [L6] the functor $R_{\sigma }$ is the composite of the factors; inducting over the length of the word using steps 2.1 and 3.1 (and the composition of the induced natural isomorphisms) gives $R_{\sigma }L(\tilde{c})\cong L(\sigma \tilde{c})$ for the action of the braid on bigraded curves through the preferred lifts, which is well defined by the braid/mapping-class dictionary and uses AC exactly there. Applying this to $\tilde{c}={\tilde{b}}_j$ and using that the basic arc has no essential segments, so that $L({\tilde{b}}_j)$ is a single summand $P_{b_j}=P_j$ for the normalized bigradings, gives $R_{\sigma }{P}_j\cong L(\sigma {\tilde{b}}_j)$.

Conclusion. The generator complexes intertwine the half-twist action on bigraded curves, and consequently the braid-word complexes intertwine the braid action; the last isomorphism identifies $R_{\sigma }{P}_j$ with the complex of the twisted basic arc. The local computations are finite and AC is used only in the final word induction.