Cyclic Labelings and Full-Twist Braid Words

Definitions and mathematical statement

Let $B_n$ denote the braid group on $n$ strands, and let $a_{ij}$, for $1\leq i<j\leq n$, denote its Birman--Ko--Lee generators. The supplied passage does not specify the admissible values of $n$, the convention for these generators, or the convention for the full twist.

Conjecture as supplied. Cyclic labelings correspond exactly to reduced words representing the full twist in the squares $a_{ij}^{2}$ of the Birman--Ko--Lee generators.

This is an incomplete formulation: the supplied record does not define a cyclic labeling, its underlying combinatorial objects, the proposed correspondence, or the meaning of "reduced" for words in these squared generators. It also does not specify whether inverse letters are allowed or whether words or labelings are considered modulo equivalences. Consequently, a fully quantified mathematical statement cannot be recovered from this record alone.

Supplied local-move context. Write $(ij)$ for an edge with endpoints $i,j$ in the edge-list notation of the passage. The passage states that the following local replacements preserve the property of being a cyclic labeling, when the displayed edges have consecutive labels:

1. Disjoint edges commute: \[ (ij)(kl)\;\longleftrightarrow\;(kl)(ij), \] provided that it is not the case that $i<k<j<l$ cyclically.
2. Triangle edges can be rotated: \[ (ij)(jk)(ik)\;\longleftrightarrow\;(ik)(ij)(jk) \;\longleftrightarrow\;(jk)(ik)(ij), \] when $i<j<k$ cyclically.
3. A "pierced triangle" has the replacement \[ (ij)(jl)(jk)(ik)\;\longleftrightarrow\;(ik)(ij)(jl)(jk), \] when $i<j<k<l$ cyclically. The consecutive-label condition is explicit for the first two moves in the supplied passage but is not separately stated for this third move; its precise applicability remains unresolved.

These are recorded as replacements of edge lists, as in the source passage. The record does not establish that they generate every equivalence relevant to the conjectured global correspondence, and the moves do not supply the missing definition of a cyclic labeling.

Short English statement

The supplied conjecture asks whether cyclic labelings correspond exactly to reduced expressions for the full twist using squared Birman--Ko--Lee braid generators. The record supplies local moves but omits definitions needed to specify the objects, reducedness, and correspondence precisely. Its open status is supplied dated curation, not an independently verified conclusion.

Sources

• Local UnsolvedMath registry, permanent ID $30006528$, title Cyclic Labelings and Full-Twist Braid Words, exact source problem code catalogue entry. The supplied registry identifies the proposed year as 2026 and supplies no named proposer; the original passage uses the first person without identifying its speaker.
• Supplied report citation: Enumerative Combinatorics (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/2, pp. 81--156. https://doi.org/10.4171/owr/2026/2. The record attributes the conjecture to part (iii) and the local-move context to part (iv) of an unidentified contribution within this report; no precise passage page is supplied.
• Supplied literature source: Oberwolfach Report 2026/2, https://doi.org/10.4171/owr/2026/2. Literature assessment checked at 2026-08-22: the supplied curation labels the problem open and states that the report presents the correspondence as a conjecture, supplies local moves, and gives no proof of the global correspondence. This is supplied dated curation, not independently verified source access or a new literature assessment.
• The registry labels the statement corrected_verified, explaining that only formula delimiters, notation, grammar, and typography were repaired while preserving mathematical content and scope. This supplied editorial label does not establish mathematical correctness or resolve the missing definitions.