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'∈ B_b(L), let hatbt and hatbt' denote their closures, and let [bt] denote the exponent sum of a braid word. Jones's braid-index uniqueness conjecture. If hatbt=hatbt'=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.

Jones's braid-index uniqueness conjecture

Jones’s conjecture says that two minimum-strand braid representations of the same link have the same exponent sum. It was presented as unresolved in 2001, particularly beyond the $3$ -braid case.

Known results

The $3$ -braid case was settled by Birman and Menasco; the $4$ -strand case was reported open in 2001.
Sharpness of the Morton–Franks–Williams inequality implies the conjecture.
Kawamuro (2006) proved it for many classes, including examples preserved under $(p,q)$ -cabling and connected sum, and found infinitely many cases where the inequality is non-sharp but the conjecture holds.

2015 reported affirmative solution

A 2015 paper states that the generalized Jones–Kawamuro inequality was solved affirmatively by Dynnikov–Prasolov and LaFountain–Menasco. Applied to two minimum-strand representatives, its right-hand side is $0$ , which would imply Jones’s conjecture; the scan found no independent verification or detailed proof assessment.

Current status (as of September 2026): The conjecture has a published report of an affirmative solution, but that resolution remains unverified; no independently confirmed unresolved case or counterexample is recorded.

Jones's braid-index uniqueness conjecture

Primary source

Progress summary

Solutions 0