Cabled crossings are built from Coxeter braids. For \(m,n\geq 0\), we have \[\begin{aligned} X_{m,n} &\simeq (X_{1,n}\boxtimes \mathbf{1}_{m-1}) \circ_1\cdots \circ_1 (\mathbf{1}_{m-1-i} \boxtimes X_{1,n}\boxtimes\mathbf{1}_{i}) \circ_1\cdots \circ_1 (\mathbf{1}_{m-1}\boxtimes X_{1,n})\\ & \stackrel{\text{h.e.}}{\simeq} (\mathbf{1}_{n-1}\boxtimes X_{m,1}) \circ_1\cdots \circ_1 (\mathbf{1}_{i} \boxtimes X_{m,1}\boxtimes\mathbf{1}_{n-1-i}) \circ_1\cdots \circ_1 (X_{m,1}\boxtimes \mathbf{1}_{n-1} ) \\ X'_{m,n} &\simeq (\mathbf{1}_{n-1}\boxtimes X'_{m,1}) \circ_1\cdots \circ_1 (\mathbf{1}_{i} \boxtimes X'_{m,1}\boxtimes\mathbf{1}_{n-1-i}) \circ_1\cdots \circ_1 (X'_{m,1}\boxtimes \mathbf{1}_{n-1} )\\ & \stackrel{\text{h.e.}}{\simeq} (X'_{1,n}\boxtimes \mathbf{1}_{m-1}) \circ_1\cdots \circ_1 (\mathbf{1}_{m-1-i} \boxtimes X'_{1,n}\boxtimes\mathbf{1}_{i}) \circ_1\cdots \circ_1 (\mathbf{1}_{m-1}\boxtimes X'_{1,n}) \end{aligned}\]
Proof.
The isomorphisms hold by associativity of \(\circ_1\). The homotopy equivalences come from applying braid relations, see Theorem 2.2.4. ◻
Original source: arXiv:2401.02956v2
Original source · 2401.02956v2