Standard geometric pure braid generators A_ij

Fix $n\in \mathbb{N}$ and let $Q=(q_1,\dots ,q_n)$ be the base configuration and ${\sigma }_1,\dots ,{\sigma }_{n-1}$ the positive elementary half twists of The elementary geometric half twist, its support disc, and its opposite, with classes $[{\sigma }_i]$ in the geometric braid group $G_n$ at $Q$ and with $[{\sigma }_i{]}^{-1}=[{\sigma }_i^{-}]$. Write $\star$ for first-under-second stacking, so that $[\gamma \star \beta ]=[\gamma ][\beta ]$. For indices $1\le i<j\le n$ put

$W_{ij}:={\sigma }_{j-1}\star {\sigma }_{j-2}\star \cdots \star {\sigma }_{i+1}\star {\sigma }_i^2\star {\sigma }_{i+1}^{-1}\star \cdots \star {\sigma }_{j-2}^{-1}\star {\sigma }_{j-1}^{-1},$

the stacking of the displayed half twists and their opposites in the order written: by the convention of Stacking of geometric braids is a well-defined associative operation on isotopy classes the right factor of each stacking lies in the lower half of the height interval and runs first, so the rightmost factor ${\sigma }_{j-1}^{-1}$ is the bottom one and the leftmost factor ${\sigma }_{j-1}$ the top one. When $j=i+1$ both outer blocks are empty and $W_{i,i+1}={\sigma }_i^2$. The standard pure braid generators are the classes

$A_{ij}:=[W_{ij}]\in G_n(1\le i<j\le n)\mathrm{.}$

The same letters name the corresponding elements of the pure configuration braid group, in the following precise sense.

The generators are pure. The endpoint-permutation homomorphism ${\pi }_{\mathrm{g}\mathrm{e}\mathrm{o}}:G_n\to S_n$ (written $\pi$ in The Artin presentation surjects onto the geometric braid group) sends $[{\sigma }_r]$ to the transposition of $r$ and $r+1$, as computed for the half twists in The elementary geometric half twist, its support disc, and its opposite. Since ${\pi }_{\mathrm{g}\mathrm{e}\mathrm{o}}({\sigma }_i^2)=\mathrm{i}\mathrm{d}$ and ${\pi }_{\mathrm{g}\mathrm{e}\mathrm{o}}$ is a homomorphism, the outer word cancels its own inverse:

${\pi }_{\mathrm{g}\mathrm{e}\mathrm{o}}(A_{ij})={\pi }_{\mathrm{g}\mathrm{e}\mathrm{o}}({\sigma }_{j-1}\cdots {\sigma }_{i+1})\cdot \mathrm{i}\mathrm{d}\cdot {\pi }_{\mathrm{g}\mathrm{e}\mathrm{o}}({\sigma }_{j-1}\cdots {\sigma }_{i+1}{)}^{-1}=\mathrm{i}\mathrm{d}\mathrm{.}$

Hence $A_{ij}$ lies in the pure geometric braid subgroup $G_n^{\mathrm{p}\mathrm{u}\mathrm{r}\mathrm{e}}=\ker {\pi }_{\mathrm{g}\mathrm{e}\mathrm{o}}$. This uses only that ${\pi }_{\mathrm{g}\mathrm{e}\mathrm{o}}$ is a homomorphism and that two half twists of one pair return each of the two strands to its starting point; the intermediate permutation ${\pi }_{\mathrm{g}\mathrm{e}\mathrm{o}}({\sigma }_{j-1}\cdots {\sigma }_{i+1})$, which fixes label $i$ and permutes only the labels $i+1,\dots ,j$, is irrelevant for the computation.

Identification with the configuration group. By Pure geometric braids and ordered configuration loops the map

$\mathrm{\Psi }:G_n^{\mathrm{p}\mathrm{u}\mathrm{r}\mathrm{e}}⟶P{B}_n,\mathrm{\Psi }([\beta ])=({\iota }_{\ast }^F[z_{\beta }]{)}^{-1},$

is an isomorphism onto the pure configuration braid group, where $z_{\beta }$ is the coordinate path of the representative $\beta$ and ${\iota }_{\ast }^F$ is induced by the inclusion of the open disc. An element of $P{B}_n$ that equals $\mathrm{\Psi }(A_{ij})$ for some $1\le i<j\le n$ is again written $A_{ij}$. In particular, whenever a statement about $P{B}_n$ names $A_{ij}$, it means this image under the published isomorphism, and the inverse sign in $\mathrm{\Psi }$ is part of the definition. Ordering statements about the generators therefore refer to the geometric classes $[W_{ij}]$, or equivalently to their images in $P{B}_n$.

Convention and scope. The classes $[W_{ij}]$ are finite products of the geometric half twists and their inverses, and every step of the construction is explicit: no choice principle is used. The definition invokes The Artin presentation surjects onto the geometric braid group only to record that the same letters ${\sigma }_r$ may be read as the image of the corresponding letters of the Artin presentation; neither injectivity of that presentation nor completeness of its relations is asserted or used. Nor does the definition assert that the family $\{A_{ij}\}$ generates $P{B}_n$, nor that any particular list of relations between the $A_{ij}$ is complete. For $n\le 1$ the index set $\{1\le i<j\le n\}$ is empty and the family $\{A_{ij}\}$ is empty; $P{B}_n$ is trivial there.