ScalingStacks

Remark 7.4.11. Let $\theta:I\to J$ and $\theta^{\prime}:I^{\prime}\to I$ be two braids such that $\theta\circ\theta^{\prime}$ is a braid. By Lemma 7.3.23, the terms $i(\theta_{s_{1}},\theta_{s_{2}})+i(\theta^{\prime}_{s^{\prime}_{1}},\theta^{\prime}_{s^{\prime}_{2}})-i(\theta_{s_{1}}\circ\theta^{\prime}_{s^{\prime}_{1}},\theta_{s_{2}}\circ\theta^{\prime}_{s^{\prime}_{2}})$, $i(\theta_{s_{1}},\theta_{s_{2}})+m_{\theta_{s_{1}}(0+)}^{+}(\theta^{\prime}_{s^{\prime}_{2}})-i(\theta_{s_{1}},\theta_{s_{2}}\circ\theta^{\prime}_{s^{\prime}_{2}})$ and $i(\theta^{\prime}_{s^{\prime}_{1}},\theta^{\prime}_{s^{\prime}_{2}})+m_{\theta^{\prime}_{s^{\prime}_{1}}(1-)}^{-}(\theta_{s_{2}})-i(\theta^{\prime}_{s^{\prime}_{1}},\theta_{s_{2}}\circ\theta^{\prime}_{s^{\prime}_{2}})$ in Lemma 7.4.9 are all non-negative.

We deduce that the following assertions are equivalent:

• •

  $\deg(\theta)\cdot\deg(\theta^{\prime})=\deg(\theta\circ\theta^{\prime})$

• •

  $\deg(\theta_{|E})\cdot\deg(\theta^{\prime}_{|E^{\prime}})=\deg(\theta_{|E}\circ\theta^{\prime}_{|E^{\prime}})$ for any two-element subset $E^{\prime}\subset I^{\prime}$, where $E=\chi(\theta^{\prime})(E^{\prime})$.

If given $s\in I^{\prime}$ with $\theta^{\prime}_{s}=\operatorname{id}\nolimits$ or $\theta_{\chi(\theta^{\prime})(s)}=\operatorname{id}\nolimits$, we have $s{\not\in}Z_{exc}$, then $\deg(\theta)\cdot\deg(\theta^{\prime})=\deg(\theta\circ\theta^{\prime})$ if and only if $i(\theta_{\theta^{\prime}_{s}(1)},\theta_{\theta^{\prime}_{s^{\prime}}(1)})+i(\theta^{\prime}_{s},\theta^{\prime}_{s^{\prime}})=i((\theta\circ\theta^{\prime})_{s},(\theta\circ\theta^{\prime})_{s^{\prime}})$ for all $s\neq s^{\prime}$ in $I^{\prime}$.