Transitions and winding convention

Positive traversal means increasing developed angle. For each oriented overlap component, let $n_{gf}=1$ for an upper-to-lower marked seam crossing, $n_{gf}=-1$ for its reverse, and $n_{gf}=0$ for an ordinary crossing. The fixed presentation prescribes $S_{gf}=J(\mu {)}^{n_{gf}}$. Hence

The last chronological transition acts on the left.

For a closed ordered chart word based in $f_0$,

The integer $N$ is retained as a topological mark, not reconstructed from the matrix. In the angular covering of the punctured cone, a loop of winding $N$ has angular lift increment $N(2\pi -\mu )$.

Transport from the base-face coordinates to the developed rest reference frame is conjugation by ${{\mathcal{A}}_{f_0}}^{-1}$. Transport onward to the declared frame is conjugation by $A$.