Lemma: Ordered-product identity

For a closed chart word $f_0,f_1,\dots ,f_m=f_0$,

For a loop on a slice avoiding the particle, $N$ is its oriented winding number.

Proof. Substitution of (eq:cocycle) cancels every adjacent pair ${A_f}^{-1}{A}_f$ and gives the displayed product.

To identify $N$, lift the spatial projection of the loop to the angular cover with $\theta \in \mathbb{R}$. Each positive marked-seam crossing increases the sheet index by one, and each negative crossing decreases it by one. The terminal angular coordinate differs from the initial one by $NL$. Since the graph projection is a homeomorphism, this is also the winding number of the loop in the punctured slice.