0PAV Lemma 7.4.9. Let θ:I→J\theta:I\to J and θ′:I′→I\theta^{\prime}:I^{\prime}\to I be two braids such that θ∘θ′\theta\circ\theta^{\prime} is a braid. The element deg(θ)⋅deg(θ′)⋅deg(θ∘θ′)−1\deg(\theta)\cdot\deg(\theta^{\prime})\cdot\deg(\theta\circ\theta^{\prime})^{-1} of Γ(Z,Zexc+)\Gamma(Z,Z_{exc}^{+}) is in ⨁Ω12𝐙eΩ\bigoplus_{\Omega}\frac{1}{2}{\mathbf{Z}}e_{\Omega} and it is equal to i(θ∘θ′)−i(θ)−i(θ′)+12∑Ω,s′∈I′∩Zexc∩Ωθs′′=id,θs′≠idc′∈C(s′)+∖θs′(0+)(mc′−mι(c′))(⟦θ′⟧)eΩ+12∑Ω,s′∈I′∩Ωθs′′≠id,θθs′′(1)=idc∈C(θs′′(1))+∖ι(θs′′(1−))θs′′(1)∈Zexc(mc−mι(c))(⟦θ⟧)eΩ.i(\theta\circ\theta^{\prime})-i(\theta)-i(\theta^{\prime})\\ +\frac{1}{2}\sum_{\begin{subarray}{c}\Omega,\ s^{\prime}\in I^{\prime}\cap Z_{exc}\cap\Omega\\ \theta^{\prime}_{s^{\prime}}=\operatorname{id}\nolimits,\ \theta_{s^{\prime}}\neq\operatorname{id}\nolimits\\ c^{\prime}\in C(s^{\prime})^{+}\setminus\theta_{s^{\prime}}(0+)\end{subarray}}(m_{c^{\prime}}-m_{\iota(c^{\prime})})(\llbracket\theta^{\prime}\rrbracket)e_{\Omega}+\frac{1}{2}\sum_{\begin{subarray}{c}\Omega,\ s^{\prime}\in I^{\prime}\cap\Omega\\ \theta^{\prime}_{s^{\prime}}\neq\operatorname{id}\nolimits,\ \theta_{\theta^{\prime}_{s^{\prime}}(1)}=\operatorname{id}\nolimits\\ c\in C(\theta^{\prime}_{s^{\prime}}(1))^{+}\setminus\iota(\theta^{\prime}_{s^{\prime}}(1-))\\ \theta^{\prime}_{s^{\prime}}(1)\in Z_{exc}\end{subarray}}(m_{c}-m_{\iota(c)})(\llbracket\theta\rrbracket)e_{\Omega}. and is also equal to 12∑Ω,(s1′,s2′)∈(I′∩Ω)2(s1′,s2′)∉E∪E′(i(θs1∘θs1′′,θs2∘θs2′′)−i(θs1,θs2)−i(θs1′′,θs2′′))eΩ++∑Ω,(s1′,s2′)∈E∩Ω(i(θs1,θs2∘θ′s2′)−i(θs1,θs2)−mθs1(0+)+(θ′s2′))eΩ++∑Ω,(s1′,s2′)∈E′∩Ω(i(θ′s1′,θs2∘θ′s2′)−i(θ′s1′,θ′s2′)−mθs1′′(1−)−(θs2))eΩ\frac{1}{2}\sum_{\begin{subarray}{c}\Omega,(s^{\prime}_{1},s^{\prime}_{2})\in(I^{\prime}\cap\Omega)^{2}\\ (s^{\prime}_{1},s^{\prime}_{2}){\not\in}E\cup E^{\prime}\end{subarray}}\bigl(i(\theta_{s_{1}}\circ\theta^{\prime}_{s^{\prime}_{1}},\theta_{s_{2}}\circ\theta^{\prime}_{s^{\prime}_{2}})-i(\theta_{s_{1}},\theta_{s_{2}})-i(\theta^{\prime}_{s^{\prime}_{1}},\theta^{\prime}_{s^{\prime}_{2}})\bigr)e_{\Omega}+\\ +\sum_{\Omega,\ (s^{\prime}_{1},s^{\prime}_{2})\in E\cap\Omega}\bigl(i(\theta_{s_{1}},\theta_{s_{2}}\circ\theta^{\prime}_{s^{\prime}_{2}})-i(\theta_{s_{1}},\theta_{s_{2}})-m_{\theta_{s_{1}}(0+)}^{+}(\theta^{\prime}_{s^{\prime}_{2}})\bigr)e_{\Omega}+\\ +\sum_{\Omega,\ (s^{\prime}_{1},s^{\prime}_{2})\in E^{\prime}\cap\Omega}\bigl(i(\theta^{\prime}_{s^{\prime}_{1}},\theta_{s_{2}}\circ\theta^{\prime}_{s^{\prime}_{2}})-i(\theta^{\prime}_{s^{\prime}_{1}},\theta^{\prime}_{s^{\prime}_{2}})-m_{\theta^{\prime}_{s^{\prime}_{1}}(1-)}^{-}(\theta_{s_{2}})\bigr)e_{\Omega} where • given (s1′,s2′)∈I′2(s^{\prime}_{1},s^{\prime}_{2})\in I^{\prime 2}, we put si=θsi′′(1)s_{i}=\theta^{\prime}_{s^{\prime}_{i}}(1) • EE is the set of pairs (s1′,s2′)∈I′×I′(s^{\prime}_{1},s^{\prime}_{2})\in I^{\prime}\times I^{\prime} with s1′∈Zexcs^{\prime}_{1}\in\ Z_{exc}, θs1′′=id\theta^{\prime}_{s^{\prime}_{1}}=\operatorname{id}\nolimits, θs1′≠id\theta_{s^{\prime}_{1}}\neq\operatorname{id}\nolimits, θs2′′≠id\theta^{\prime}_{s^{\prime}_{2}}\neq\operatorname{id}\nolimits • E′E^{\prime} is the set of pairs (s1′,s2′)∈I′×I′(s^{\prime}_{1},s^{\prime}_{2})\in I^{\prime}\times I^{\prime} with s1∈Zexcs_{1}\in\ Z_{exc}, θs1′′≠id\theta^{\prime}_{s^{\prime}_{1}}\neq\operatorname{id}\nolimits, θs1=id\theta_{s_{1}}=\operatorname{id}\nolimits and θs2≠id\theta_{s_{2}}\neq\operatorname{id}\nolimits.
0PAW Proof. Given s′∈I′s^{\prime}\in I^{\prime} and s=θs′′(1)s=\theta^{\prime}_{s^{\prime}}(1), the class θs∘θs′′\theta_{s}\circ\theta^{\prime}_{s^{\prime}} is admissible, hence θs(0+)∪ι(θs(0+))=θs′′(1−)∪ι(θs′′(1−))\theta_{s}(0+)\cup\iota(\theta_{s}(0+))=\theta^{\prime}_{s^{\prime}}(1-)\cup\iota(\theta^{\prime}_{s^{\prime}}(1-)) unless s∈Zexcs\in Z_{exc} and one of θs\theta_{s} and θs′\theta_{s^{\prime}} is the identity, but not the other. Given c∈T(Z)c\in T(Z), we put vc=(mc−mι(c))(⟦θ⟧)ec=mc(⟦θ⟧)ec+mι(c)(⟦θ⟧)eι(c)=vι(c).v_{c}=(m_{c}-m_{\iota(c)})(\llbracket\theta\rrbracket)e_{c}=m_{c}(\llbracket\theta\rrbracket)e_{c}+m_{\iota(c)}(\llbracket\theta\rrbracket)e_{\iota(c)}=v_{\iota(c)}. Let a=∑s′∈I′∩Zexcθs′′=id,θs′≠idc′∈C(s′)∖((θs′(0+)∪ι(θs′(0+)))CLOSEmc′(⟦θ′⟧)ec′=∑s′∈I′∩Zexcθs′′=id,θs′≠idc′∈C(s′)+∖θs′(0+)(mc′−mι(c′))(⟦θ′⟧)ec′.a=\sum_{\begin{subarray}{c}s^{\prime}\in I^{\prime}\cap Z_{exc}\\ \theta^{\prime}_{s^{\prime}}=\operatorname{id}\nolimits,\ \theta_{s^{\prime}}\neq\operatorname{id}\nolimits\\ c^{\prime}\in C(s^{\prime})\setminus\bigl((\theta_{s^{\prime}}(0+)\cup\iota(\theta_{s^{\prime}}(0+))\bigr)\end{subarray}}m_{c^{\prime}}(\llbracket\theta^{\prime}\rrbracket)e_{c^{\prime}}=\sum_{\begin{subarray}{c}s^{\prime}\in I^{\prime}\cap Z_{exc}\\ \theta^{\prime}_{s^{\prime}}=\operatorname{id}\nolimits,\ \theta_{s^{\prime}}\neq\operatorname{id}\nolimits\\ c^{\prime}\in C(s^{\prime})^{+}\setminus\theta_{s^{\prime}}(0+)\end{subarray}}(m_{c^{\prime}}-m_{\iota(c^{\prime})})(\llbracket\theta^{\prime}\rrbracket)e_{c^{\prime}}. We have m(θ∘θ′)−m(θ)−m(θ′)=m(\theta\circ\theta^{\prime})-m(\theta)-m(\theta^{\prime})= =∑s′∈I′c′∈(θ∘θ′)s′(0+)∪ι((θ∘θ′)s′(0+))mc′(⟦θ⟧)ec′−∑s∈Ic∈θs(0+)∪ι(θs(0+))mc(⟦θ⟧)ec−a=\sum_{\begin{subarray}{c}s^{\prime}\in I^{\prime}\\ c^{\prime}\in(\theta\circ\theta^{\prime})_{s^{\prime}}(0+)\cup\iota((\theta\circ\theta^{\prime})_{s^{\prime}}(0+))\end{subarray}}m_{c^{\prime}}(\llbracket\theta\rrbracket)e_{c^{\prime}}-\sum_{\begin{subarray}{c}s\in I\\ c\in\theta_{s}(0+)\cup\iota(\theta_{s}(0+))\end{subarray}}m_{c}(\llbracket\theta\rrbracket)e_{c}-a =∑s′∈I′θs′′≠idvθs′′(0+)−∑s′∈I′θs′′≠idθθs′′(1)≠idvθs′′(1−)−12∑s′∈I′θs′′≠idθθs′′(1)=idc∈C(θs′′(1))vc−a=\sum_{\begin{subarray}{c}s^{\prime}\in I^{\prime}\\ \theta^{\prime}_{s^{\prime}}\neq\operatorname{id}\nolimits\end{subarray}}v_{\theta^{\prime}_{s^{\prime}}(0+)}-\sum_{\begin{subarray}{c}s^{\prime}\in I^{\prime}\\ \theta^{\prime}_{s^{\prime}}\neq\operatorname{id}\nolimits\\ \theta_{\theta^{\prime}_{s^{\prime}}(1)}\neq\operatorname{id}\nolimits\end{subarray}}v_{\theta^{\prime}_{s^{\prime}}(1-)}-\frac{1}{2}\sum_{\begin{subarray}{c}s^{\prime}\in I^{\prime}\\ \theta^{\prime}_{s^{\prime}}\neq\operatorname{id}\nolimits\\ \theta_{\theta^{\prime}_{s^{\prime}}(1)}=\operatorname{id}\nolimits\\ c\in C(\theta^{\prime}_{s^{\prime}}(1))\end{subarray}}v_{c}-a Using (7.3.1), we find ⟨⟦θ⟧,⟦θ′⟧⟩\displaystyle\langle\llbracket\theta\rrbracket,\llbracket\theta^{\prime}\rrbracket\rangle =−12∑s′∈I′c′∈θs′′(0+)∪ι(θs′′(0+))vc′+12∑s′∈I′c∈θs′′(1−)∪ι(θs′′(1−))vc\displaystyle=-\frac{1}{2}\sum_{\begin{subarray}{c}s^{\prime}\in I^{\prime}\\ c^{\prime}\in\theta^{\prime}_{s^{\prime}}(0+)\cup\iota(\theta^{\prime}_{s^{\prime}}(0+))\end{subarray}}v_{c^{\prime}}+\frac{1}{2}\sum_{\begin{subarray}{c}s^{\prime}\in I^{\prime}\\ c\in\theta^{\prime}_{s^{\prime}}(1-)\cup\iota(\theta^{\prime}_{s^{\prime}}(1-))\end{subarray}}v_{c} =−∑s′∈I′θs′′≠idvθs′′(0+)+∑s′∈I′θs′′≠idvθs′′(1−).\displaystyle=-\sum_{\begin{subarray}{c}s^{\prime}\in I^{\prime}\\ \theta^{\prime}_{s^{\prime}}\neq\operatorname{id}\nolimits\end{subarray}}v_{\theta^{\prime}_{s^{\prime}}(0+)}+\sum_{\begin{subarray}{c}s^{\prime}\in I^{\prime}\\ \theta^{\prime}_{s^{\prime}}\neq\operatorname{id}\nolimits\end{subarray}}v_{\theta^{\prime}_{s^{\prime}}(1-)}. We deduce that ⟨⟦θ⟧,⟦θ′⟧⟩+m(θ∘θ′)−m(θ)−m(θ′)=−∑s′∈I′θs′′≠idθθs′′(1)=idc∈C(θs′′(1))+∖ι(θs′′(1−))θs′′(1)∈Zexc(mc−mι(c))(⟦θ⟧)ec−a\langle\llbracket\theta\rrbracket,\llbracket\theta^{\prime}\rrbracket\rangle+m(\theta\circ\theta^{\prime})-m(\theta)-m(\theta^{\prime})=-\sum_{\begin{subarray}{c}s^{\prime}\in I^{\prime}\\ \theta^{\prime}_{s^{\prime}}\neq\operatorname{id}\nolimits\\ \theta_{\theta^{\prime}_{s^{\prime}}(1)}=\operatorname{id}\nolimits\\ c\in C(\theta^{\prime}_{s^{\prime}}(1))^{+}\setminus\iota(\theta^{\prime}_{s^{\prime}}(1-))\\ \theta^{\prime}_{s^{\prime}}(1)\in Z_{exc}\end{subarray}}(m_{c}-m_{\iota(c)})(\llbracket\theta\rrbracket)e_{c}-a and the first equality of the lemma follows. Consider s1′≠s2′s^{\prime}_{1}\neq s^{\prime}_{2} in I′I^{\prime}. If s1′∈Zexcs^{\prime}_{1}\in Z_{exc}, θs1′′=ids1′\theta^{\prime}_{s^{\prime}_{1}}=\operatorname{id}\nolimits_{s^{\prime}_{1}} and θs1≠ids1\theta_{s_{1}}\neq\operatorname{id}\nolimits_{s_{1}}, it follows from Lemma 7.3.21 that ∑s2′∈I′θs2′′≠idi(ids1′,θs2′′)=12∑s2′∈I′θs2′′≠ids2′c′∈C(s1′)+(mc′−mι(c′))(θs2′′)=12∑c′∈C(s1′)+(mc′−mι(c′))(⟦θ′⟧).\sum_{\begin{subarray}{c}s^{\prime}_{2}\in I^{\prime}\\ \theta^{\prime}_{s^{\prime}_{2}}\neq\operatorname{id}\nolimits\end{subarray}}i(\operatorname{id}\nolimits_{s^{\prime}_{1}},\theta^{\prime}_{s^{\prime}_{2}})=\frac{1}{2}\sum_{\begin{subarray}{c}s^{\prime}_{2}\in I^{\prime}\\ \theta^{\prime}_{s^{\prime}_{2}}\neq\operatorname{id}\nolimits_{s^{\prime}_{2}}\\ c^{\prime}\in C(s^{\prime}_{1})^{+}\end{subarray}}(m_{c^{\prime}}-m_{\iota(c^{\prime})})(\theta^{\prime}_{s^{\prime}_{2}})=\frac{1}{2}\sum_{c^{\prime}\in C(s^{\prime}_{1})^{+}}(m_{c^{\prime}}-m_{\iota(c^{\prime})})(\llbracket\theta^{\prime}\rrbracket). Similarly, if s1∈Zexcs_{1}\in Z_{exc}, θs1′′≠id\theta^{\prime}_{s^{\prime}_{1}}\neq\operatorname{id}\nolimits and θs1=id\theta_{s_{1}}=\operatorname{id}\nolimits, we have ∑s2′∈I′θs2≠idi(ids1,θs2)=12∑c∈C(s1)+(mc−mι(c))(⟦θ⟧).\sum_{\begin{subarray}{c}s^{\prime}_{2}\in I^{\prime}\\ \theta_{s_{2}}\neq\operatorname{id}\nolimits\end{subarray}}i(\operatorname{id}\nolimits_{s_{1}},\theta_{s_{2}})=\frac{1}{2}\sum_{c\in C(s_{1})^{+}}(m_{c}-m_{\iota(c)})(\llbracket\theta\rrbracket). The second equality of the lemma follows. ∎