For \(m,n\geq 0\), we define the braiding complexes: \[\begin{aligned} X_{m,n}&:=F( (\sigma_{n}\cdots\sigma_{1})\cdots (\sigma_{i+n-1}\cdots\sigma_{i}) \cdots (\sigma_{m+n-1}\cdots\sigma_{m}) )\langle -m n \rangle\\ X'_{m,n}&:=F( (\sigma^{-1}_{n}\cdots\sigma^{-1}_{m+n-1}) \cdots (\sigma^{-1}_{i}\cdots\sigma^{-1}_{i+m-1}) \cdots (\sigma^{-1}_{1}\cdots\sigma^{-1}_{m}) )\langle m n \rangle \end{aligned}\] The braiding complexes \(X_{m,n}\) (resp. \(X'_{m,n}\)) will be called positive (resp. negative) cabled crossings complexes or just cabled crossings, since the underlying braids are cabled crossings.
The braids appearing in the special cabled crossings \(X_{m,1}\), \(X'_{m,1}\), \(X_{1,n}\), and \(X'_{1,n}\) will be called Coxeter braids, since they are braid versions of Coxeter words, see the illustrations in Figure [0016] (as for functions, we compose functors and read diagrams from right to left. The Artin generator \(\sigma_{i}\) acts on the \(i\)-th and \((i+1)\)-th strand from the bottom.).