Braid group ($B_n$)

For $n\ge 1$, the braid group $B_n$ consists geometrically of braids on $n$ strands, composed by stacking. Algebraically it is generated by ${\sigma }_1,\dots ,{\sigma }_{n-1}$ subject to the braid group relations: adjacent generators obey ${\sigma }_i{\sigma }_{i+1}{\sigma }_i={\sigma }_{i+1}{\sigma }_i{\sigma }_{i+1}$, and nonadjacent generators commute. There is no relation ${\sigma }_i^2=1$.

Recording only the strand endpoints gives a surjective group homomorphism $B_n\to S_n$, sending ${\sigma }_i$ to the transposition $(ii+1)$. Its group kernel is the pure braid group, whose strands return to their original endpoints. Thus endpoint permutations discard crossing information. For example, $B_2$ has one generator and no relations, so it is an infinite cyclic group, while its endpoint permutation group is $S_2$; the square of its generator has trivial endpoint permutation but is a nontrivial braid. This distinction does not conflict with Cayley's theorem, which realizes every group as some permutation group.

The pure braid group $P_n$ is the kernel of the endpoint-permutation homomorphism from the braid group $B_n$ to the symmetric group $S_n$:Each strand of a pure braid returns to its own original endpoint. It may still wind around other strands, so trivial endpoint permutation does not make the braid trivial.

For two strands, $B_2\cong \mathbb{Z}$ is generated by one crossing ${\sigma }_1$, and $P_2$ consists of its even powers. Thus $P_2\cong \mathbb{Z}$ but has index two in $B_2$. In general, the endpoint map is onto, so $[B_n:P_n]=n!$.

Geometrically, $P_n$ is the fundamental group of the configuration space of $n$ ordered distinct points in the plane. The ordering records which moving point must return to which starting point, matching the pure-strand condition.

A positive elementary braid acts on the free group of punctured-disk meridians by the displayed substitution, fixing other generators. The inverse sends $x_i$ to $x_{i+1}$ and $x_{i+1}$ to $x_{i+1}^{-1}{x}_i{x}_{i+1}$. Applying Seifert-van Kampen theorem at the crossings and closing the braid gives the knot-complement presentation $⟨x_i\mid x_i=\varphi (x_i)⟩$.

The generators ${\sigma }_i$ commute when $|i-j|>1$ and satisfy ${\sigma }_i{\sigma }_{i+1}{\sigma }_i={\sigma }_{i+1}{\sigma }_i{\sigma }_{i+1}$. Unlike a symmetric-group presentation, there is no ${\sigma }_i^2=1$ relation.