| $$ | |
| (ii) \quad \left| \overline{\text{WindNum}}(\alpha, \beta) - \overline{\text{WindNum}}(\alpha, \gamma) \right| - \left| \overline{\text{WindNum}}(\beta, \gamma) \right| \le 1. | |
| $$ | |
| 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 | |
| <sup>12</sup>This inequality also follows from the second property of Proposition 7.1 by setting α to be the reference path, | |
| β = P<sub>1</sub> and γ = P<sub>i</sub> and noting that <span style="text-decoration: overline;">WindNum</span>(β, γ) = 0. |