Problem 7.19 — (S.

Definitions and mathematical statement

Fix the source's three-manifold/link pair $(W,L)$ and an admissible parameter $N$. Let $\rho$ vary through the relevant flat-bundle/representation data for the group denoted $B$. A standard flat-bundle interpretation uses representations $\rho:\pi_1(W)\to B$ modulo the appropriate gauge/conjugacy equivalence; write $\mathcal X_B(W)$ for the intended character space.

Study \[ [\rho]\longmapsto K_N(W,L,\rho) \] as a function on this character space: determine its regularity, domains/loci of definition, and fibers, and other meaningful structural properties. For a well-defined target value $c$, a fiber is $\{[\rho]:K_N(W,L,\rho)=c\}$. The pair $(W,L)$ is held fixed; this is not an added large-$N$ limit question.

The group $B$, character-space construction, admissibility/genericity requirements, parameter range, and definition/value normalization of $K_N$ are not given in the record. In particular, if values are specified only up to a phase/root-of-unity ambiguity, the target or a lift must be fixed before ordinary function regularity and fibers are defined. A representation quotient is not silently assumed to be a well-behaved reductive GIT quotient. These missing conventions must be recovered from the preceding source construction.

Short English statement

For fixed $(W,L)$, describe how $K_N$ varies with the flat bundle on the intended $B$-character space, especially regularity and fibers. Fix the group, quotient, admissible domain, and value/phase convention.

Sources

• Ohtsuki, Problems on invariants of knots and 3-manifolds (2002), Problem 7.19, PDF page 114, attributed to S. Baseilhac and R. Benedetti, https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf; accessed 2026-07-29, PDF-font-structure extraction.
• Local registry data/unsolvedmath/problems.json, permanent UnsolvedMath ID $10400134$, source code catalogue entry, Problem 7.19. Dataset attribution: Ulam AI and the UnsolvedMath Contributors.