Polynomial bounds for rectilinear knot Ramsey numbers

Definitions and mathematical statement

For a tame link $L\subset\mathbb R^3$, let $\operatorname{cr}(L)$ denote its minimal crossing number and let $b(L)$ be its number of components. A rectilinear complete-graph embedding has distinct straight edges intersecting only at shared endpoints. It contains $L$ if some collection of pairwise vertex-disjoint cycles is ambient isotopic to $L$ or to its mirror image. Let $R_{\rm link}(L)$ be the least vertex count forcing such a collection in every rectilinear embedding. For a knot, which has one component, write $R_{\rm knot}(L)$.

The main question asks whether there are absolute constants $C,d>0$ such that every knot $K$ satisfies \[ R_{\rm knot}(K)\leq C(1+\operatorname{cr}(K))^d. \] The workshop also asks for the analogous polynomial phenomenon for links. An explicit nondegenerate version is whether \[ R_{\rm link}(L)\leq C(1+\operatorname{cr}(L)+b(L))^d \] holds for all links, with constants independent of their types and component counts. Including the component count is necessary for unrestricted links: arbitrarily many split unknotted components can have crossing number zero, while their disjoint cycles require arbitrarily many graph vertices. For links with a fixed number of components, the question reduces to polynomial dependence on crossing number alone. The reflection convention is the same as in the preceding workshop question. Both assertions quantify over all valid placements of the complete graph's vertices, not merely over generic random placements.

Short English statement

Can knots, and suitably measured links, be forced in straight-line complete graphs of only polynomially many vertices?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 20000937 (AIM-COMBINATORICS-0062), AIM Workshop Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 American Institute of Mathematics, Graph Ramsey theory, item 15. Original workshop question underlying AIM-COMBINATORICS-0062. https://www.overleaf.com/read/mnvcscjjysvg.