Problem 7.24 — (S.

Definitions and mathematical statement

Let $\mathcal L$ denote the class of links in $S^3$ to which the source invariants apply, with their required equivalence and auxiliary-data conventions. For each allowed integer parameter $N$, put \[ \omega_N=\exp(2\pi i/N),\qquad Q_N(L)=K_N(S^3,L),\qquad C_N(L)=J_N(L;\omega_N), \] where $J_N$ is the coloured Jones polynomial in the source's convention, normalized by $J_N(U)=1$ for the unknot $U$. The problem, attributed to S. Baseilhac and R. Benedetti, is to determine the precise relationship between the functions $Q_N$ and $C_N$ of links. In particular, the comparison must specify its range of $N$ and links and any normalization or auxiliary choices on which it depends. The source does not assert a particular equality between these functions. Its excerpt does not define $K_N$, the allowed parameters, or the orientation, framing and phase conventions, so no unique comparison identity can be extracted from it.

Short English statement

Determine exactly how the quantum hyperbolic invariant $K_N(S^3,L)$ is related, as a function of links, to the coloured Jones polynomial evaluated at $\exp(2\pi i/N)$ and normalized to be $1$ on the unknot.

Sources

• S. Baseilhac and R. Benedetti, attribution in Problem 7.24 of T. Ohtsuki, Problems on invariants of knots and 3-manifolds (2002), supplied PDF page 115. https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf. Recorded access: 2026-07-29; extraction: pdf-font-structure.
• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $10400139$, exact source code catalogue entry. The supplied research summary identifies the QHI/coloured Jones comparison and labels it OPEN-TRIAGE; it supplies no resolving reference.