Kirby Problem 2.14

Definitions and mathematical statement

A Jordan curve is the image of a continuous embedding of the circle in the plane. A polygon is inscribed in it when all its vertices lie on the curve; its edges need not lie inside the bounded complementary region. All polygons below must be nondegenerate. The problem contains four related questions.

(a) Must every Jordan curve contain the four vertices of a square?

(b) Must every Jordan curve contain the six vertices of an affine image of a regular hexagon? Here the affine map is $x\mapsto Ax+b$ with $A\in\operatorname{GL}_2(\mathbb R)$, so it does not collapse the hexagon.

(c) Let $n\geq2$, let $z_1,\ldots,z_{2n}$ be distinct points on one circle in $\mathbb C$, and let $\gamma\subset\mathbb C$ be a smooth Jordan curve. Does there always exist a nonconstant complex polynomial $p$ of degree less than $n$ such that $p(z_j)\in\gamma$ for every $j$? The stated condition does not separately require the image points to be distinct.

(d) Let $h:S^2\to\mathbb R$ be continuous, and fix four vertices $x_1,\ldots,x_4\in S^2$ of a Euclidean square. Must there exist a rotation $R\in\operatorname{SO}(3)$ such that \[ h(Rx_1)=h(Rx_2)=h(Rx_3)=h(Rx_4)? \] The general questions, including their regularity and degree restrictions, are retained separately; special cases need not have the same status.

Short English statement

Ask whether arbitrary plane curves contain prescribed square or hexagonal configurations, and whether analogous polynomial and equal-value rotation constructions always exist.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2762 (KP-2.14), Kirby's Problems in Low-Dimensional Topology. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 K3: A New Problem List in Low-Dimensional Topology, Problem 2.14. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.