Finiteness of Fibered Strongly Quasipositive Knots in Concordance Classes

Definitions and mathematical statement

Interpret a knot as a smooth oriented embedding of $S^1$ in $S^3$, considered up to orientation-preserving ambient isotopy. This ambient manifold and orientation convention are standard interpretations that are not explicitly specified in the supplied record. Two knots $K_0$ and $K_1$ are smoothly concordant, written $K_0\sim K_1$, if they cobound a smoothly and properly embedded oriented annulus in $S^3\times[0,1]$, with boundary $(-K_0)\times\{0\}\sqcup K_1\times\{1\}$. Write $\mathcal C$ for the set of these concordance classes; this interprets the undefined symbol "C" in the original excerpt.

A knot $K$ is fibered if its exterior $S^3\setminus\operatorname{int}(\nu K)$, where $\nu K$ is a tubular neighborhood, fibers smoothly over $S^1$ with fiber a compact connected Seifert surface for $K$. Here a Seifert surface is a compact connected oriented embedded surface in $S^3$ whose oriented boundary is $K$, and its intersection with the exterior is understood when describing the fiber.

Let $B_n$ be the braid group on $n$ strands, with positive Artin generators $\sigma_1,\ldots,\sigma_{n-1}$. A knot is strongly quasipositive if it is the oriented closure of a braid that is a product of positive embedded band generators \[ a_{i,j}=(\sigma_i\sigma_{i+1}\cdots\sigma_{j-2})\,\sigma_{j-1}\,(\sigma_i\sigma_{i+1}\cdots\sigma_{j-2})^{-1},\qquad 1\leq i<j\leq n, \] for some $n\geq1$, with empty products allowed. In particular, $a_{i,i+1}=\sigma_i$.

The supplied question, under these conventions, is whether \[ \forall c\in\mathcal C,\qquad \#\left\{[K]_{\mathrm{iso}}: [K]_{\mathrm{conc}}=c,\ K\text{ is fibered and strongly quasipositive}\right\}<\infty. \] Thus the objects counted within each concordance class are distinct knot isotopy types.

The supplied exposition describes this question as weaker than Baker's conjecture, but does not supply that conjecture's full statement. It also discusses braid-positive knots, meaning closures of braids expressed using only positive Artin generators, and states that these knots are fibered and strongly quasipositive. That discussion supplies context for a narrower question; the target retained here is the fibered strongly quasipositive class specified in both the clean statement and the opening question of the original excerpt. The excerpt's subsequent reference to "the following question" is incomplete.

Short English statement

Does every smooth concordance class contain only finitely many distinct isotopy types of fibered strongly quasipositive knots? The formulation uses the standard convention of oriented knots in $S^3$, which the supplied record does not explicitly state. The supplied registry labels the question open; that status has not been independently verified.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30006058$, exact source problem code catalogue entry, title: Finiteness of Fibered Strongly Quasipositive Knots in Concordance Classes. The record belongs to the Oberwolfach Reports Open Problems collection and lists proposed year $2024$; no individual proposer is specified.
• Supplied source citation: MATRIX-MFO Tandem Workshop: Invariants in Low-Dimensional Topology: Combinatorics, Geometry, and Computation (2025), Oberwolfach Reports, pp. 2511--2550, DOI https://doi.org/10.4171/owr/2024/43. This is also the supplied literature reference labeled "Oberwolfach Report 43/2024." The citation's year $2025$ and the report designation $43/2024$ are retained as supplied.
• Supplied literature reference: Krishna--Morton, Twist positivity, L-space knots, and concordance, https://doi.org/10.1007/s00029-024-00996-6.
• Literature assessment checked on 2026-08-22: this is supplied dated curation, not an independently verified assessment. It labels the question open, reports that the source calls it a weaker version of Baker's conjecture that remains open, and reports finiteness results for narrower positive-knot classes rather than for the full fibered strongly quasipositive class. The supplied background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels its statement corrected_verified and reports that the official Oberwolfach report was checked, the question isolated from surrounding exposition, and notation repaired. These are supplied verification claims; no source visit or independent verification is asserted here. The excerpt cites an unidentified reference [10] for the assertion that braid-positive knots are fibered; its bibliographic details are not supplied.