The Alexander and Markov Theorems

Braid Closure

Braid Closure

For any braid $\beta \in B_n$, its closure $\hat{\beta }$ is the link that results from connecting the lower ends of the braid with the upper ends in the same order.

e.g. The closure of $\beta ={\sigma }_1{\sigma }_2^{-1}{\sigma }_1{\sigma }_3$:

Alexander's Theorem

Any knot or link can be represented as a closed braid. i.e. $\hat{}$ is surjective.

Proof

Remark

In a certain sense, braids are the quantum analog of permutations.

Markov Moves

Markov's Theorem