The Artin braid group \(B_n\) on \(n\) strands is defined by the presentation \[ B_n = \langle \sigma_1, \sigma_2, \ldots, \sigma_{n-1} \mid\sigma_i \sigma_j = \sigma_j \sigma_i, ~|i-j| > 1;~~\sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1} \rangle. \] The Nielsen-Thurston theory of surface automorphisms, as well as the theory of orderable groups, produces actions of braid groups on the circle and real line leading to the translation/rotation number of braids. The twist number \(\tau_b\) for braids can be defined in the following way.
There exists a unique function \(\tau_b : B_n \to \mathbb{R}\) with the following properties:
(i) \(\sup_{\alpha,\beta\in B_n} |\tau_b(\alpha \beta) - \tau_b(\alpha) - \tau_b(\beta)| = 1\);
(ii) for all \(\beta \in B_n\) and \(k \in \mathbb{Z}\), we have \(\tau_b(\beta^k) = k \tau_b(\beta)\);
(iii) \(\tau_b(\beta) \geq 0\) whenever \(\beta\) can be written in Artin’s generators and their inverses with no occurrence of \(\sigma_1^{-1}\).
The level sets \(\tau_b^{-1}(x)\) are called layers. If \((r, s)\) is an interval in \(\mathbb{R}\), then \(\tau_b^{-1}(r, s) = \bigcup_{x \in (r,s)} \tau_b^{-1}(x)\).
In the first theorem the author proves that the layers \(\tau_b^{-1}(x)\) separate the Cayley graph \(\mathrm{Cay}_{\sigma}(B_n)\), \(n \geq 3\), if and only if \(x\) is an integer.
This theorem has the following refinement. If \(z\) is an integer, then:
(i) the layers \(\tau_b^{-1}(x)\) separate \(\mathrm{Cay}_{\sigma}(B_n)\) whenever \(n \geq 2\). Moreover, the distance between the sets \(\tau_b^{-1}(-\infty, z)\) and \(\tau_b^{-1}(z, \infty)\) in \(\mathrm{Cay}_{\sigma}(B_n)\) is at least \(n\);
(ii) the set \(\tau_b^{-1}(z, z+1)\), which is the union of the layers \(\tau_b^{-1}(x)\) with \(x \in (z, z+1)\), does not separate \(\mathrm{Cay}_{\sigma}(B_n)\) whenever \(n \geq 3\).
Further, the following statement shows that certain “noninteger” layers are extremely “thin” in a certain sense. If \(n \geq 2\), \(p\) and \(q\) are coprime integers, and \(q \in \{n-1, n\}\), then no edge of \(\mathrm{Cay}_{\sigma}(B_n)\) has both edges in the layer \(\tau_b^{-1}(p/q)\).