Pure Braids

Pure Braid

A pure braid is a braid that each of its strands joins the top and bottom disks at the same points. Given braid group $B_n$, the set of pure braids is denoted as $K_n$.

$K_n$ is the kernel of $\alpha$

Notice that $K_n=\ker \alpha$ where $\alpha$ is the canonical homomorphism from $B_n$ to the symmetric group $S_n$. Because every pure braid is mapped to the identity permutation.

The Pure Braid Group

Equivalence classes of pure braids form a subgroup of the braid group i.e. $K_n\le B_n$.

Proof Stacking and inverse are both closed in $K_n$. If ${\beta }_1,{\beta }_2\in K_n$, then ${\beta }_1{\beta }_2\in K_n$, because $\alpha ({\beta }_1{\beta }_2)=\alpha ({\beta }_1)\alpha ({\beta }_2)$ is the identity permutation. Similarly, the inverse $\alpha ({\beta }_1^{-1})=\alpha ({\beta }_1{)}^{-1}$ is also the identity permutation. $\square$

Proposition

The pure braid group is generated by $\{{\sigma }_{ij}:1\le i<j\le n\}$ where ${\sigma }_{ij}:={\sigma }_1$