Problem 7.21 — (S.

Definitions and mathematical statement

For an admissible positive integer $N$, write \[\mu_N=\{\zeta\in{\mathbb{C}}:\zeta^N=1\}.\] The state sum $H_N(T)$ in the source has a phase ambiguity given by multiplication by elements of $\mu_N$. For a nonzero complex representative $z$, this means that the available phase-independent data determine at most its orbit $z\mu_N$, unless further normalization is supplied. The problem attributed to S. Baseilhac and R. Benedetti first asks to understand this ambiguity: determine the actual root-of-unity factors arising from changes of the source's auxiliary data $T$ representing $(W,L,\rho)$, including how these factors depend on the choices and transformations. The excerpt does not define $T$, the state sum or the transformations, so no specific phase formula is supplied here.

The second, explicitly tentative request is to use this analysis to obtain an invariant of $(W,L,\rho)$ equipped with extra structure, refining $K_N(W,L,\rho)$. In precise structural terms, one would specify the extra datum $s$, its equivalence/transport rules and an invariant $\widetilde K_N(W,L,\rho,s)$ whose prescribed forgetting operation recovers $K_N(W,L,\rho)$, while retaining phase information discarded there. Neither the nature of $s$, the refined target nor the actual relation between $H_N$ and $K_N$ is stated. In particular this draft does not identify $K_N$ with a root-of-unity orbit or with a chosen power of $H_N$; those would require the missing definitions. The words "possibly derive" are retained as a proposed application, not an assertion that a canonical phase lift exists.

Short English statement

Determine the root-of-unity ambiguity in the state sum $H_N(T)$. Baseilhac and Benedetti also suggest using suitable extra structure to retain this phase and refine the invariant $K_N$.

Sources

• Local UnsolvedMath registry, permanent ID $10400136$, exact source code catalogue entry.
• S. Baseilhac and R. Benedetti, attribution in Ohtsuki, Problems on invariants of knots and 3-manifolds (2002), Problem 7.21, PDF page 114: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf. Registry access 2026-07-29, pdf-font-structure extraction.