Alexander's theorem on closed braids

Remarks

The idea of the proof is to choose an axis and to make the link wind around it in one direction. Take a polygonal diagram of $L$ and a point $O$ of the plane off the diagram. Each edge that runs clockwise around $O$ can be replaced by two edges that pass on the other side of $O$, so that they run counterclockwise. After finitely many such moves every edge runs counterclockwise around $O$, and cutting along a ray from $O$ shows the link as a closed braid. For example, the trefoil knot is the closure of $\sigma_1^3$ in $B_2$, and the Hopf link is the closure of $\sigma_1^2$ in $B_2$. The braid $\beta$ is far from unique; the Markov theorem says exactly when two braids have the same closure.

Going deeper

For a student

The theorem turns questions about links into questions about the braid groups $B_n$, which are groups with simple presentations. Together with the Markov theorem, it shows that a link invariant is the same as a function on all braid groups that is unchanged by the two Markov moves. This is how Jones first found the Jones polynomial, from traces on representations of braid groups. The least $n$ for which $L$ is a closed $n$-braid is called the braid index of $L$.