upper bounded by the number of crossings of $S$ with $P_S$. Thus, if $P_S$ is short, it follows that $\text{labPot}(S) < \alpha_{\text{long}}(k) = 10^4 \cdot 2^{ck}$.
Now, suppose that $P_S$ is long, and let $\text{Ring}(S_u, S_v)$ be the ring that corresponds to $P_S$. By the choice of $\mathcal{Q}$, $|\text{WindNum}(\mathcal{Q}, \text{Ring}(S_u, S_v))| \le \alpha_{\text{winding}}(k) < 300 \cdot 2^{ck}$. Let $\tilde{\mathcal{A}}$ be a collection of maximal subpaths of $S$ that are fully contained within $\text{Ring}(S_u, S_v)$, and let $\hat{P}_S$ be the maximal subpath of $P_S$ that is fully contained within $\text{Ring}(S_u, S_v)$. Then, by Observation 7.2,
By the definition of WindNum, we have that $|\text{WindNum}(\hat{\mathcal{A}}, \hat{P}S)| \le |\text{WindNum}(\mathcal{Q}, \text{Ring}(S_u, S_v))|$ for each $\hat{\mathcal{A}} \in \tilde{\mathcal{A}}$. Moreover, by Lemma 6.4, $|\tilde{\mathcal{A}}| \le |S_u| + |S_v| \le 2\alpha{\text{sep}}(k) = 7 \cdot 2^{ck} + 4$. Additionally, by Lemmas 6.5 and 6.11, we have that $|V(P_S) \setminus V(\hat{P}S)| \le 2\alpha{\text{pat}}(k) = 200 \cdot 2^{ck}$. From this,
Thus, because the choice of $S$ was arbitrary, we conclude that $M \le 10^4 \cdot 4^{ck}$. $\square$
8.2 Pushing a Solution onto R
Let us now describe the process of pushing a solution onto $R$. To simplify this process, we define two “non-atomic” operations that encompass sequences of atomic operations in discrete homotopy. We remind that we only deal with walks that do not repeat edges.
Definition 8.11 (Non-Atomic Operations in Discrete Homotopy). Let $G$ be a triangulated plane graph with a weak linkage $W$, and $C$ be a cycle¹⁷ in $G$. Let $W \in W$.
Cycle Move. Applicable to $(W, C)$ if there exists a subpath $Q$ of $C$ such that (i) $Q$ is a subpath of $W$, (ii) $1 \le |E(Q)| \le |E(C)| - 1$, (iii) no edge in $E(C) \setminus E(Q)$ belongs to any walk in $W$, and (iii) no edge drawn in the strict interior of $C$ belongs to any walk in $W$. Then, the cycle move operation replaces $Q$ in $W$ by the unique subpath of $C$ between the endpoints of $Q$ that is edge-disjoint from $Q$.
Cycle Pull. Applicable to $(W, C)$ if (i) $C$ is a subwalk of $W$, and (ii) no edge drawn in the strict interior of $C$ belongs to any walk in $W$. Then, the cycle pull operation replaces $Q$ in $W$ by a single occurrence of the first vertex in $Q$.
We now prove that the operations above are compositions of atomic operations.
Lemma 8.3. Let $G$ be a triangulated plane graph with a weak linkage $W$, and $C$ be a cycle in $G$. Let $W \in W$ with a non-atomic operation applicable to $(W, C)$. Then, the result of the application is a weak linkage that is discretely homotopic to $W$.
Proof. We prove the claim by induction on the number of faces of $G$ in the interior of $C$. In the basis, where $C$ encloses only one face, then the cycle move and cycle pull operations are precisely the face move and face pull operations, respectively, and therefore the claim holds. Now, suppose that $C$ encloses $i \ge 2$ faces and that the claim is correct for cycles that enclose at most $i-1$ faces. We consider several cases as follows.
First, suppose that $C$ has a path $P$ fully drawn in its interior whose endpoints are two (distinct) vertices $u, v \in V(C)$, and whose internal vertices and all of its edges do not belong to $C$. (We remark that $P$ might consist of a single edge, and that edge might be parallel to some edge of $C$.) Now, notice that $P$ partitions the interior of $C$ into the interior of two cycles $C_1$ and $C_2$ that share only $P$ in common as follows: one cycle $C_1$ consists of one subpath of $C$
¹⁷A pair of parallel edges is considered to be a cycle.