Concordance of an Explicit Pair of Positive Three-Braid Knots

Definitions and mathematical statement

Let $B_3=\langle \sigma_1,\sigma_2\mid \sigma_1\sigma_2\sigma_1=\sigma_2\sigma_1\sigma_2\rangle$ be the braid group on three strands, with $\sigma_i$ the positive Artin crossing of strands $i$ and $i+1$. A positive three-braid is a word in these generators using only positive powers. Write $\widehat{\beta}\subset S^3$ for the standard closure of a braid $\beta$, equipped with its braid orientation. Define the knots \[ K=\widehat{\sigma_1^3\sigma_2^3\sigma_1^6\sigma_2^6}, \qquad J=\widehat{\sigma_1^3\sigma_2^5\sigma_1^3\sigma_2^7}. \] Here smooth concordance means that there exists a smooth proper embedding \[ F:S^1\times[0,1]\longrightarrow S^3\times[0,1] \] whose image meets the boundary of $S^3\times[0,1]$ exactly in $K\times\{0\}$ and $J\times\{1\}$, with the restrictions to the two end circles parametrizing the prescribed knot orientations.

Question. Are $K$ and $J$ smoothly concordant?

Supplied context and qualifications. The source excerpt describes these knots as non-isotopic, attributing this to work of Birman--Menasco cited there as references [4, 5]. It reports that the concordance methods under discussion do not distinguish them and proposes computing further concordance invariants. These are supplied source claims, not conclusions established in this draft. The excerpt also states that a concordant but non-isotopic pair of positive three-braid knots would give a counterexample to the Slice--Ribbon Conjecture; that implication is retained as a source assertion without independent verification.

The accompanying source corollary concerns a knot $L$ that is the closure of a positive three-braid. It defines the minimum number of alternating positive blocks by \[ r(L)=\min\left\{r\in{\mathbb{Z}}_{\geq1}: L\text{ is isotopic to } \widehat{\sigma_1^{p_1}\sigma_2^{q_1}\cdots\sigma_1^{p_r}\sigma_2^{q_r}},\quad p_i,q_i\in{\mathbb{Z}}_{\geq1}\ (1\leq i\leq r)\right\}. \] As supplied, the corollary asserts \[ r(L)=g(L)+\upsilon(L)+1 \] and asserts that $r(L)=r(M)$ whenever $L$ and $M$ are concordant positive three-braid knots. The supplied excerpt does not define $g$ or $\upsilon$, specify the normalization of $\upsilon$, or discuss the existence of the displayed block representation in boundary cases. This auxiliary assertion is recorded for provenance and requires review against the full source; it is not an additional claim to be proved here. The explicit concordance question above does not depend on resolving that notation.

Short English statement

Determine whether the two explicitly specified positive three-braid closures cobound a smoothly embedded annulus in $S^3\times[0,1]$, using their braid orientations. The supplied record labels the question open and reports that the knots are non-isotopic but are not distinguished by the concordance methods discussed in the source. Those status and background claims require independent review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30006060$, source problem code catalogue entry, entitled "Concordance of an Explicit Pair of Positive Three-Braid Knots." The record gives proposed year 2024 and names no proposer.
• Supplied report citation: MATRIX-MFO Tandem Workshop: Invariants in Low-Dimensional Topology: Combinatorics, Geometry, and Computation (2025), Oberwolfach Reports, pp. 2511--2550, DOI 10.4171/owr/2024/43. Supplied source URL: https://doi.org/10.4171/owr/2024/43. The citation year 2025 differs from the report identifier 43/2024 and the proposed year 2024; this discrepancy is unresolved here.
• Supplied literature source: "Oberwolfach Report 43/2024," https://doi.org/10.4171/owr/2024/43. Literature assessment checked at 2026-08-22: the supplied curation labels the problem open, reports explicit non-isotopic positive three-braid pairs not distinguished by available concordance invariants, and says that no concordance counterexample or proof was located. This is supplied dated curation, not an independently verified literature assessment.
• The source excerpt attributes non-isotopy to Birman--Menasco, references [4, 5] in the report. Full bibliographic entries for those references are not supplied.
• The registry labels the statement corrected_verified and says that the official report was checked, the question isolated from surrounding exposition, and the notation repaired. These are supplied verification metadata; this draft does not claim an independent source visit or verification.