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 , the set of pure braids is denoted as .

is the kernel of

Notice that where is the canonical homomorphism from to the symmetric group . 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. .

Proof Stacking and inverse are both closed in . If , then , because is the identity permutation. Similarly, the inverse is also the identity permutation.

Proposition

The pure braid group is generated by where