Concordance Rigidity of Positive Braid Knots

Definitions and mathematical statement

For an integer $n\geq 1$, let $B_n$ be the braid group with standard generators $\sigma_1,\ldots,\sigma_{n-1}$. A positive braid is a braid represented by a word using only these generators with positive exponents. A positive braid knot is a knot in $S^3$ isotopic to the closure of such a braid, where the closure has one component. Allowing the empty braid includes the unknot; the supplied record states no nontriviality restriction.

Here is a precise smooth, unoriented formulation of the supplied question. Let $K_0,K_1\subset S^3$ be arbitrary smooth positive braid knots, with no restriction on their braid indices. They are smoothly concordant if there is a smooth proper embedding \[ F:S^1\times[0,1]\longrightarrow S^3\times[0,1] \] with \[ F(S^1\times\{i\})=K_i\times\{i\}\qquad(i=0,1), \] whose interior lies in $S^3\times(0,1)$. They are ambiently isotopic if there is a smooth ambient isotopy $H_t:S^3\to S^3$, $t\in[0,1]$, satisfying $H_0=\operatorname{id}_{S^3}$ and $H_1(K_0)=K_1$.

The question is whether, for every such pair, \[ K_0\text{ and }K_1\text{ are smoothly concordant} \quad\Longrightarrow\quad K_0\text{ and }K_1\text{ are ambiently isotopic}. \]

The supplied record does not specify the concordance category or orientation conventions. The displayed formulation explicitly uses smooth concordance and unoriented ambient isotopy; it is a candidate interpretation requiring review, not a verified identification of the source's intended conventions. A topological concordance version would instead require a locally flat properly embedded annulus. If knots are regarded as oriented, the concordance must respect the prescribed boundary orientations, with boundary $(-K_0)\sqcup K_1$, and the isotopy must preserve knot orientations. These choices remain unresolved in the supplied evidence.

Short English statement

Must any two concordant positive braid knots in $S^3$ be isotopic? The question concerns all positive braid knots, with unrestricted braid index. A precise smooth formulation is given above, but the source does not specify the smooth versus topological concordance category or orientation conventions. The supplied record labels the problem open; that status is not independently verified.

Sources

• Local UnsolvedMath registry, permanent ID $30004819$, title: Concordance Rigidity of Positive Braid Knots; exact source problem code: catalogue entry. The supplied record belongs to the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the recorded proposed year is $2021$.
• Supplied source citation: MATRIX-MFO Tandem Workshop: Invariants and Structures in Low-Dimensional Topology (2022), Oberwolfach Reports, pp. 2327--2364, DOI 10.4171/owr/2021/42. Supplied source URL and literature reference labeled "OWR report": https://doi.org/10.4171/owr/2021/42. The supplied background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied additional literature reference, labeled "OWR-2024-43": https://ems.press/content/serial-article-files/50050.
• Literature assessment checked at 2026-08-22: this is supplied dated curation, not independently verified. It labels the problem open and reports that recent work provides invariants and strong partial results for positive $3$-braids while explicitly leaving open whether concordant positive braid knots must be isotopic. The supplied background calls this a dated literature triage derived from the supplied OWR curation and requests independent verification of time-sensitive status and cited claims.
• The supplied statement status is corrected_verified; its accompanying verification says only that the sphere notation was dollar-delimited without changing the concordance question. This does not establish mathematical review of the formulation or verification of its current literature status.