Rational Formula for the Knot Invariant $\rho_1$

Definitions and mathematical statement

The supplied record identifies $\rho_1$ as a knot invariant and associates the proposed identity with the perturbed Alexander invariant. It does not define $\rho_1$ or the quantity $P^{(1)}$, specify their normalizations, or state their coefficient rings. The superscript $(1)$ is retained as supplied; its meaning is unspecified.

Conjectural identity, with unresolved definitions. For the knots intended by the source, the proposed relation is \[ \rho_1(t^2)=\frac{t^2}{(1-t^2)^2}P^{(1)}(t). \] Both quantities are understood to concern the same knot, with knot dependence suppressed as in the supplied statement. The intended class of knots, any orientation or framing conventions, and the precise quantification over knots are not supplied. Consequently, this is a transcription of the proposed relation rather than a self-contained conjecture with fully specified hypotheses.

The role and domain of $t$, and the algebraic setting in which the equality is asserted, are also unspecified. If the expression is interpreted pointwise over ${\mathbb{C}}$, its displayed right-hand side is undefined at $t=\pm1$ without an additional extension convention. The record does not specify such a convention or whether the intended equality is instead an identity of rational functions or formal series. No interpretation or boundary extension is imposed here.

Short English statement

The proposed formula relates the knot invariant $\rho_1$, evaluated at $t^2$, to the unspecified quantity $P^{(1)}(t)$. The supplied record labels it an open conjecture, but definitions, normalizations, the intended knot class, and the meaning of the variable must be recovered before a precise self-contained statement is possible.

Sources

• Local UnsolvedMath registry, supplied complete source record: permanent ID 30003560, source problem code catalogue entry, title "Rational Formula for the Knot Invariant $\rho_1$," in the Oberwolfach Reports Open Problems collection. The record gives proposed year 2017 and names no proposer.
• Supplied report citation: Low-dimensional Topology and Number Theory (2018), Oberwolfach Reports, pp. 2363--2426, DOI 10.4171/owr/2017/38. https://doi.org/10.4171/owr/2017/38. The background describes this report as documenting workshop discussions and open problems.
• Supplied literature reference: A perturbed Alexander invariant. https://doi.org/10.4171/QT/206.
• Supplied literature reference: Low dimensional Topology and Number Theory. https://doi.org/10.4171/owr/2017/38.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open, describes the displayed identity as conjectural, and states that the cited primary work still lacks a topological explanation of the invariant and does not establish this identity. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment. The record expressly requests independent verification of time-sensitive status and cited claims.
• The supplied statement status is corrected_verified, with an editorial explanation that mathematical content was preserved while notation, typography, or omitted definitions were repaired. This supplied label does not establish mathematical correctness or resolve the missing definitions in the present record.