Jones's braid-index uniqueness conjecture

Let $b(L)$ be the braid index of a link $L$, let $B_{b(L)}$ be the braid group on $b(L)$ strands, let $\bt,\bt'\in B_{b(L)}$, let $\hat{\bt}$ and $\hat{\bt'}$ denote their closures, and let $[\bt]$ denote the exponent sum of a braid word.

Jones's braid-index uniqueness conjecture. If

$$\hat{\bt}=\hat{\bt'}=L,$$

then

$$[\bt]=[\bt'].$$

The conjecture asserts that all braid representatives of a link having the minimal braid index have the same exponent sum. It is introduced as one of Jones's still unsolved conjectures and is relevant to whether the Morton–Williams–Franks inequalities for parallel cables can be sharp.