ScalingStacks

Let \(\underline{\beta}_1\) and \(\underline{\beta}_2\) be braid words representing the same braid \(\beta\), then there exist homotopy equivalences \[\psi_{\underline{\beta}_1,\,\underline{\beta}_2}\colon F(\underline{\beta}_1) \rightarrow F(\underline{\beta}_2)\] which form a transitive system, i.e. if \(\underline{\beta}_3\) is a third braid word representing the same braid, then \[\psi_{\underline{\beta}_2,\,\underline{\beta}_3}\circ_2\psi_{\underline{\beta}_1,\,\underline{\beta}_2} \sim\psi_{\underline{\beta}_1,\,\underline{\beta}_3}.\] If moreover \(\underline{\beta}_1'\), \(\underline{\beta}_2'\) are braid words representing another braid \(\beta'\) then we have \[ \psi_{\underline{\beta}_1\underline{\beta}_1',\underline{\beta}_2\underline{\beta}_2'} \sim\psi_{\underline{\beta}_1,\underline{\beta}_2}\circ_1\psi_{\underline{\beta}_1',\underline{\beta}_2'}.\]