$$ (ii) \quad \left| \overline{\text{WindNum}}(\alpha, \beta) - \overline{\text{WindNum}}(\alpha, \gamma) \right| - \left| \overline{\text{WindNum}}(\beta, \gamma) \right| \le 1. We say that $\mathrm{Ring}(I_{\mathrm{in}}, I_{\mathrm{out}})$ is *rooted* if it is equipped with some fixed path $\eta$ that is traversing it, called the *reference path* of this ring. In a rooted ring $\mathrm{Ring}(I_{\mathrm{in}}, I_{\mathrm{out}})$, we measure all winding numbers with respect to $\eta$, hence we shall use the shorthand $\overline{\mathrm{WindNum}}(\alpha) = \overline{\mathrm{WindNum}}(\alpha, \eta)$ when $\eta$ is implicit or clear from context. Here, we implicitly assume that the walk $\alpha$ is edge disjoint from $\eta$. This requirement will always be met by the following assumptions: (i) $H$ is a plane multigraph where we have $4n + 1$ parallel copies of every edge, and we assume that the reference path $\eta$ consists of only the 0-th copy $e_0$; and (ii) whenever we consider the winding number of a walk $\alpha$, it will edge-disjoint from the reference curve $\eta$ as it will not contain the 0-th copy of any edge. (In particular, the walks of the (weak) linkages that we consider will always satisfy this property.) Note that any visitor walk in $\mathrm{Ring}(I_{\mathrm{in}}, I_{\mathrm{out}})$ with both endpoints in $I_{\mathrm{in}}$ is discretely homotopic to a segment of $I_{\mathrm{in}}$, and similarly for $I_{\mathrm{out}}$. Thus, we derive the following observation. **Observation 7.1.** Let $\alpha$ be a visitor in $\mathrm{Ring}(I_{\mathrm{in}}, I_{\mathrm{out}})$. Then, $|\overline{\mathrm{WindNum}}(\alpha)| \le 1$. Recall the notion of a weak linkage defined in Section 5, which is a collection of edge-disjoint non-crossing walks. When we use the term *weak linkage of order k* in $\mathrm{Ring}(I_{\mathrm{in}}, I_{\mathrm{out}})$, we refer to a weak linkage such that each walk has both endpoints in $V(I_{\mathrm{in}}) \cup V(I_{\mathrm{out}})$. For brevity, we abuse the term ‘weak linkage’ to mean a weak linkage in a ring when it is clear from context. Note that every walk in a weak linkage $\mathcal{P}$ is an inner visitor, or an outer visitor, or a traversing walk. This partitions $\mathcal{P}$ into $P_{\mathrm{in}}, P_{\mathrm{out}}, P_{\mathrm{traverse}}$. A weak linkage is *traversing* if it consists only of traversing walks. Assuming that $\mathrm{Ring}(I_{\mathrm{in}}, I_{\mathrm{out}})$ is rooted, we define the *winding number* of a traversing weak linkage $\mathcal{P}$ as $\overline{\mathrm{WindNum}}(\mathcal{P}) = \overline{\mathrm{WindNum}}(P_1)$. Recall that any two walks in a weak linkage are non-crossing. Then as observed in [14, Observation 4.4],¹² |\overline{\text{WindNum}}(P_i) - \overline{\text{WindNum}}(\mathcal{P})| \le 1 \quad \text{for all } i = 1, \dots, k. $$
The above definition is extended to any weak linkage $\mathcal{P}$ in the ring as follows: if there is no walk in $\mathcal{P}$ that traverses the ring, then $\overline{\text{WindNum}}(\mathcal{P}) = 0$, otherwise $\overline{\text{WindNum}}(\mathcal{P}) = \overline{\text{WindNum}}(\mathcal{P}_{\text{traverse}})$. Note that, two aligned weak linkages $\mathcal{P}$ and $\mathcal{Q}$ in the ring may have different winding numbers (with respect to any reference path). Replacing a linkage $\mathcal{P}$ with an aligned linkage $\mathcal{Q}$ having a “small” winding number will be the main focus of this section.
Lastly, we define a labeling of the edges based on the winding number of a walk (this relation is made explicit in the observation that follows).
Definition 7.4. Let $(G, S, T, g, k)$ be a good instance of Planar Disjoint Paths, and $H$ be the radial completion of $G$. Let $\alpha$ be a (not self-crossing) walk in $H$, and let $\beta$ be a path in $H$ such that $\alpha$ and $\beta$ are edge disjoint. Let us fix (arbitrary) orientations of $\alpha$ and $\beta$, and define the left and right side of the path $\beta$ with respect to its orientation. The labeling $\text{label}_\beta^\alpha$ of each ordered pair of consecutive edges, $(e, e') \in E_H(\alpha) \times E_H(\alpha)$ by ${-1, 0, +1}$ with respect to $\beta$, where $e$ occurs before $e'$ when traversing $\alpha$ according to its orientation is defined as follows.
• The pair $(e, e')$ is labeled $+1$ if $e$ is on the left of $\beta$ while $e'$ is on the right of $\beta$.
• else, $(e, e')$ is labeled $-1$ if $e$ is on the right while $e'$ is on the left of $\beta$;
- otherwise e and e' are on the same side of β and (e, e') is labeled 0.
Note that in the above labeling only pairs of consecutive edges may get a non-zero label, depending on how they cross the reference path. For the ease of notation, we extend the above
12This inequality also follows from the second property of Proposition 7.1 by setting α to be the reference path, β = P1 and γ = Pi and noting that WindNum(β, γ) = 0.