Problem 2.20 — Let R be a commutative ring with 1, say, Z or Q.

Definitions and mathematical statement

Let $R$ be a commutative ring with identity, in particular $\mathbb Z$ or $\mathbb Q$. For a set $\mathcal S$, write $R\mathcal S$ for its free $R$-module. Let $\mathcal K$ and $\mathcal M_K$ be the knot and associated-object sets used in the source, and retain its decreasing filtrations labeled loop and $\Delta\Delta$.

For every relevant index $l$, describe the following $R$-modules: \[ \frac{F_l(R\mathcal M_K;\mathrm{loop})}{F_{l+1}(R\mathcal M_K;\mathrm{loop})}, \qquad \frac{F_l(R\mathcal K;\Delta\Delta)}{F_{l+1}(R\mathcal K;\Delta\Delta)}. \] Also describe the image of the source's filtered map \[ \Phi_l:F_l(R\mathcal K;\Delta\Delta) \longrightarrow F_l(R\mathcal M_K;\mathrm{loop}). \] The third request concerns the image at the filtration level as printed, not merely an image on the two associated graded quotients.

The source extract does not define $M_K$, the ambient knot category, either local move, the filtration generators/index convention, or the "above map." A map induced by an assignment $K\mapsto M_K$ could be extended $R$-linearly, but its actual construction and filtration compatibility must be recovered from the missing context. The spaces are modules for general $R$, not automatically vector spaces; rational ranks alone would not describe their integral structure. No surgery interpretation, move diagram, or generic finite-type filtration is substituted for the absent definitions.

Short English statement

Describe both associated graded move-filtration modules and the image of the source's filtered knot-to-object map, over a general unital commutative ring. The move, object, filtration, and map definitions are missing.

Sources

• Ohtsuki, Problems on invariants of knots and 3-manifolds (2002), Problem 2.20, PDF page 44, 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 $10400042$, source code catalogue entry, Problem 2.20. Dataset attribution: Ulam AI and the UnsolvedMath Contributors.