The braid index of an oriented link

Assume the Axiom of Choice. For an oriented link $L\subset S^3$ (Oriented links in the three-sphere and ambient isotopy) put

$b(L):=\min \{n\in \mathbb{N}:\text{ there is an }n\text{-braid }\beta \text{ with }\widehat{\beta }\text{ equivalent to }L\},$

the braid index of $L$. Here an $n$-braid is a geometric braid based at $Q$ (The closure of a geometric braid) and $\widehat{\beta }$ is its oriented closure.

For the empty link, $b(\mathrm{\varnothing })=0$: the empty braid in $B_0$ has that closure, while a positive-strand braid has at least one permutation cycle and hence a nonempty closure.

The minimum is well defined. The defining set is nonempty: by Alexander's theorem (Alexander's theorem: every link is a closed braid, which assumes AC) every oriented link is equivalent to the closure of a braid on some number of strands. The set is a subset of $\mathbb{N}$ and $\mathbb{N}$ is well ordered (The natural numbers $\mathbb{N}$ (von Neumann)), so it has a least element. The value of the minimum is a property of $L$ alone: the closure construction is well defined on braid isotopy classes and, by the closure-invariance lemma (The closure depends only on the braid isotopy class), replacing a braid by a braid-isotopic one does not change the equivalence class of the closure, so the condition "$\widehat{\beta }$ equivalent to $L$" depends only on the braid class, not on the chosen representative. The existence statement is Alexander's theorem under AC, and the invariance statement assumes ${\mathrm{A}\mathrm{C}}_{\omega }$, which is discharged here from AC through the choice-implication bridge (AC implies DC implies countable choice); the definition therefore declares AC in accordance with the axiom-strength convention of this page.