0PAS Lemma 7.4.7. Let θ:I→J\theta:I\to J be a braid in ZZ. Let EE be a subset of {s∈I∩Zo|θs=ids}\{s\in I\cap Z_{o}\ |\ \theta_{s}=\operatorname{id}\nolimits_{s}\} and let θ¯=(θs)s∈I−E\bar{\theta}=(\theta_{s})_{s\in I-E}. We have degE+(θ)=degE+(θ¯)\deg_{E^{+}}(\theta)=\deg_{E^{+}}(\bar{\theta}).
0PAT Proof. Note that ⟦θ⟧=⟦θ¯⟧\llbracket\theta\rrbracket=\llbracket\bar{\theta}\rrbracket. Let s∈Es\in E. We have ∑c∈C(s)mc(⟦θ⟧)ec=∑c∈C(s)+∑s′∈I,s′≠s(mc−mι(c))(⟦θs′⟧)ec→ec→12∑s′∈I,s′≠si(ids,θs′)\sum_{c\in C(s)}m_{c}(\llbracket\theta\rrbracket)e_{c}=\sum_{c\in C(s)^{+}}\sum_{s^{\prime}\in I,\ s^{\prime}\neq s}(m_{c}-m_{\iota(c)})(\llbracket\theta_{s^{\prime}}\rrbracket)e_{c}\xrightarrow{e_{c}\to 1}2\sum_{s^{\prime}\in I,\ s^{\prime}\neq s}i(\operatorname{id}\nolimits_{s},\theta_{s^{\prime}}) by Lemma 7.3.21. The lemma follows. ∎