The Zariski combing words alpha_i and x_i in the Artin presentation

Fix $n\ge 2$ and let $B_n=⟨{\sigma }_1,\dots ,{\sigma }_{n-1}⟩$ be the abstract braid group of The braid group by Artin presentation, the group presented by the generators ${\sigma }_1,\dots ,{\sigma }_{n-1}$ and the relations

${\sigma }_i{\sigma }_{i+1}{\sigma }_i={\sigma }_{i+1}{\sigma }_i{\sigma }_{i+1}(1\le i\le n-2),$

${\sigma }_i{\sigma }_j={\sigma }_j{\sigma }_i(1\le i,j\le n-1,\mathrm{\mid }i-j\mathrm{\mid }>1)\mathrm{.}$

Reading convention for words. A word $w_1{w}_2\cdots w_m$ in this alphabet and its inverses is read first letter first: it denotes the product $[w_1][w_2]\cdots [w_m]$ in $B_n$, and under the published stacking convention $[\gamma ][\beta ]:=[\gamma \star \beta ]$ of Stacking of geometric braids is a well-defined associative operation on isotopy classes and The isotopy classes of geometric braids based at $Q$ form a group, and the endpoint permutation is a homomorphism, in which the first factor of a stacking is the upper one, its geometric image is the stacking $w_1\star w_2\star \cdots \star w_m$ whose first factor is the topmost layer. Here the geometric image is taken through the published surjection $\phi \text{\hspace{0.17em}⁣}:B_n\to G_n$, $\phi ({\sigma }_i)=[{\sigma }_i]$ of The Artin presentation surjects onto the geometric braid group. In particular the empty word is the identity of $B_n$. No injectivity or completeness of $\phi$ is asserted, and nothing below depends on how many factors the stacking has.

The words ${\alpha }_i$. For $1\le i\le n-1$ put

${\alpha }_i:={\sigma }_i{\sigma }_{i+1}\cdots {\sigma }_{n-1}\in B_n,\text{and}{\alpha }_n:=1,$

the last being the empty word. Each ${\alpha }_i$ is a word of length $n-i$ in the generators, and ${\alpha }_{n-1}={\sigma }_{n-1}$.

The words $x_i$. For $1\le i\le n-1$ put

$x_i:={\sigma }_{n-1}^{-1}\cdots {\sigma }_{i+1}^{-1}{\sigma }_i^2{\sigma }_{i+1}\cdots {\sigma }_{n-1}={\alpha }_{i+1}^{-1}{\sigma }_i^2{\alpha }_{i+1}\in B_n\mathrm{.}$

The two displayed words for $x_i$ are literally the same word written in two ways: expanding ${\alpha }_{i+1}^{-1}$ gives ${\sigma }_{n-1}^{-1}\cdots {\sigma }_{i+1}^{-1}$ and expanding ${\alpha }_{i+1}$ gives ${\sigma }_{i+1}\cdots {\sigma }_{n-1}$, so the middle factor ${\sigma }_i^2$ sits in the same position in both readings. In particular $x_{n-1}={\sigma }_{n-1}^2$, and $x_i$ is a word of length $2(n-i)$ in the generators and their inverses.

These words are the Zariski combing words of the Artin presentation. The definition is choice-free, it uses no relation of the presentation, and it asserts no property of ${\alpha }_i$ or $x_i$ inside any geometric braid model; all of that is established, when needed, by the items that cite this definition.