Kirby Problem 1.75

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $\mathcal U$ be the space of smooth unparametrized embedded circles in $S^3$ belonging to the unknot isotopy class, with the smooth topology; thus reparametrizations of $S^1$ are identified. Let $\mathcal G\subset\mathcal U$ be the subspace of great circles, the intersections of $S^3\subset\mathbb R^4$ with 2-dimensional linear subspaces. It is the Grassmannian $O(4)/(O(2)\times O(2))$. A knot energy is a reparametrization-invariant functional on smooth knots; a critical point has zero first variation. A deformation retraction is a continuous $H:\mathcal U\times[0,1]\to\mathcal U$ with $H(u,0)=u$, $H(u,1)\in\mathcal G$ and $H(g,s)=g$ for $g\in\mathcal G$.

Statement recorded in the supplied source.

Construct a knot energy whose critical points in $\mathcal U$ are all great circles. Can a gradient flow for this energy be defined so that it gives a deformation retraction of $\mathcal U$ onto $\mathcal G$? The desired construction must apply to every smooth unknot, with the flow staying in its embedding class and its limiting retraction continuous. [S2]

Short English statement

Find a knot energy with no nongreat-circle unknot critical points, together with a gradient flow that explicitly retracts the entire smooth unknot space onto the great circles.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2734 (KP-1.75). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 R. İnanç Baykur, Robion C. Kirby and Daniel Ruberman, eds., K3: A New Problem List in Low-Dimensional Topology, Mathematical Surveys and Monographs 295, American Mathematical Society (2026), Problem 1.75 and its remarks; Chapters 1 and 2 for the stated conventions. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.