Now, apply Lemma 7.1 to $\mathcal{P}(u, v)$, $\mathcal{C}(u, v)$ and $\text{Flow}_R(u, v)$ in $\text{Ring}(S_u, S_v)$. We thus obtain a linkage $\mathcal{P}'_{\text{traverse}}(u, v)$ disjoint from $\mathcal{P}_{\text{visitor}}(u, v)$ that is aligned with $\mathcal{P}_{\text{traverse}}(u, v)$. Hence, $\mathcal{P}'(u, v) = \mathcal{P}_{\text{visitor}}(u, v) \cup \mathcal{P}'_{\text{traverse}}(u, v)$ is a linkage in $G$ aligned with $\mathcal{P}(u, v)$. Assume w.l.o.g. that $\mathcal{P}'(u, v)$ uses the 3-rd copy of each edge in $H$. Let us now consider the winding number of $\mathcal{P}'_{\text{traverse}}(u, v)$ with respect to $\text{path}_R(u', v')$. By Lemma 7.1(c), $|\text{WindNum}(\mathcal{P}'_{\text{traverse}}(u, v)) - \text{WindNum}(\text{Flow}_R(u, v))| \le 60\ell + 6$. Now, note that for any path $P' \in \mathcal{P}'_{\text{traverse}}(u, v)$, $|\text{WindNum}(P') - \text{WindNum}(\mathcal{P}'_{\text{traverse}}(u, v))| \le 1$, (recall that the winding number of paths in a linkage differ by at most 1). Similarly, for any path $Q \in \text{Flow}_R(u, v)$, $|\text{WindNum}(\text{Flow}_R(u, v)) - \text{WindNum}(Q)| \le 1$. Therefore, it follows that for any two paths $P' \in \mathcal{P}'_{\text{traverse}}(u, v)$ and $Q \in \text{Flow}_R(u, v)$ $|\text{WindNum}(P') - \text{WindNum}(Q)| \le 60\ell + 8$. Recall that we chose $\eta(u, v)$ as the reference path of $\text{Ring}(S_u, s_v)$ in the above expression. Hence, we may rewrite it as $|\text{WindNum}(P', \eta(u, v)) - \text{WindNum}(Q, \eta(u, v))| \le 60\ell + 8$. Note that $P'$, $Q$ and $\eta(u, v)$ are three edge-disjoint paths traversing $\text{Ring}(S_u, S_v)$. Hence $\text{WindNum}$ is well defined in $\text{Ring}(S_u, S_v)$ for any pair of them. By Proposition 7.1, $|\text{WindNum}(P', Q)| \le |\text{WindNum}(P', \eta(u, v)) - \text{WindNum}(Q, \eta(u, v))| + 1$. We have so far established that for any $P' \in \mathcal{P}'_{\text{traverse}}(u, v)$ and $Q \in \text{Flow}_R(u, v)$, $|\text{WindNum}(P', Q)| \le 60\ell + 9$. Now, consider $\text{path}_R(u', v')$ and recall that for any $Q \in \text{Flow}_R(u, v)$, $|\text{WindNum}(\text{path}_R(u', v'), Q)| \le 1$ by Observation 7.4. Furthermore $\text{path}_R(u', v')$ uses the 0-th copies of edges in $H$. Hence, for each $P' \in \mathcal{P}'_{\text{traverse}}(u, v)$ $\text{WindNum}(P', \text{path}_R(u', v'))$ is well defined in $\text{Ring}(S_u, S_v)$, and by Proposition 7.1, $|\text{WindNum}(P', \text{path}_R(u', v'))| \le |\text{WindNum}(P', Q) - \text{WindNum}(\text{path}_R(u', v'), Q)| + 1 \le 60\ell + 11$.

Finally, consider the paths in $\mathcal{P}'_{\text{visitor}} = \mathcal{P}_{\text{visitor}}$. For each $P \in \mathcal{P}_{\text{visitor}}$ the absolute value of the winding number of $\text{WindNum}(P, \text{path}_R(u', v'))$ is bounded by 1 (by Observation 7.1). Hence, we conclude that $\text{WindNum}(\mathcal{P}', \text{Ring}(S_u, S_v)) \le 60\alpha_{\text{sep}}(k) + 11 < 300 \cdot 2^{ck}$. $\square$

# 8 Pushing a Solution onto the Backbone Steiner Tree

In this section, we push a linkage that is a solution of small winding number onto the backbone Steiner tree to construct a “pushed weak linkage” with several properties that will make its reconstruction (in Section 9) possible. Let us recall that we have an instance $(G, S, T, g, k)$ and $H$ is the radial completion of $G$ enriched with $4n + 1$ parallel copies of each edge. Then we construct a backbone Steiner tree $R$ (Section 6), which uses the 0-th copy of each edge. Formally, a pushed weak linkage is defined as follows.

**Definition 8.1 (Pushed Weak Linkage).** Let $(G, S, T, g, k)$ be an instance of Planar Disjoint Paths, and $R$ be a Steiner tree. A weak linkage $W$ in $H$ is pushed (onto $R$) if $E(W) \cap E(R) = \emptyset$ and every edge in $E(W)$ is parallel to some edge of $R$.

In what follows, we first define the properties we would like the pushed weak linkage to satisfy. Additionally, we partition any weak linkage into segments that give rise to a potential function whose maintenance will be crucial while pushing a solution of small winding number onto $R$. Afterwards, we show how to push the solution, simplify it, and analyze the result. We remark that the simplification (after having pushed the solution) will be done in two stages.

## 8.1 Desired Properties and Potential of Weak Linkages

The main property we would like a pushed weak linkage to satisfy is to use only “few” parallel copies of any edge. This quantity is captured by the following definition of multiplicity, which we will eventually like to upper bound by a small function of $k$.

**Definition 8.2 (Multiplicity of Weak Linkage).** Let $H$ be a plane graph. Let $W$ be a weak linkage in $H$. Then, the multiplicity of $W$ is the maximum, across all edges $e$ in $H$, of the number of edges parallel to $e$ that belong to $E(W)$.