Kirby Problem 1.81

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.

Here a knot is any topological embedding $k:S^1\hookrightarrow S^3$, including wild embeddings. An arc of embeddings is a continuous family $k_t:S^1\hookrightarrow S^3$, $t\in[0,1]$, in the compact-open topology. A knot is trivial in this question if such a family joins it to a standard round unknot. This is the weak isotopy convention of the source; it does not require an ambient isotopy of $S^3$. Let $B:S^1\hookrightarrow S^3$ denote the Bing sling, the wild knot specified by the construction and illustration in the cited problem and its Bing reference.

Statement recorded in the supplied source.

Is every topologically embedded circle in $S^3$ trivial under this arc-of-embeddings equivalence? In particular, the named conjecture asks whether the Bing sling is nontrivial: is there no arc of embeddings joining $B$ to a round unknot? [S2]

Short English statement

Using paths of embeddings, including wild knots, ask whether every knot can become the unknot and whether the Bing sling supplies a counterexample.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2740 (KP-1.81). 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 1.81 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.