Quadrisecants of wild knots

Definitions and mathematical statement

A knot $K\subset{\mathbb{R}}^3$ is the image of a topological embedding $S^1\hookrightarrow{\mathbb{R}}^3$. It is tame if it is ambient isotopic to a polygonal knot, and wild otherwise. Define its quadrisecant set by \[Q(K)=\{\ell\subset{\mathbb{R}}^3:\ell\text{ is an affine line and }|\ell\cap K|\geq4\}.\] Lines are counted without orientation, and four distinct intersection points suffice; a line meeting the knot in more points is also included and counts as one line.

The question attributed to Greg Kuperberg is \[\forall K\text{ wild},\qquad |Q(K)|=\infty.\] Equivalently, for every such $K$ and every integer $N\geq1$, there exist pairwise distinct lines $\ell_1,\ldots,\ell_N$, each meeting $K$ in at least four distinct points. Infinitely many point quadruples lying on one line would not suffice. No smoothness, generic-position, transversality, cyclic-order or alternating-quadrisecant condition is imposed.

The research summary discusses classical nontrivial tame-knot existence results attributed to Pannwitz and possible finite quadrisecant counts, but those are context for a different hypothesis and conclusion. Its wording about a generic isotopy-invariant "Q-number" has no definition or supporting reference here and is not adopted as a property of the raw set $Q(K)$. The summary's tentative nontrivial/knotted wording is not added as an extra restriction on the actual statement, which quantifies over every wild knot.

Short English statement

Must every wild embedded circle in three-space have infinitely many distinct affine lines each meeting it in at least four distinct points? The question does not require exactly four points or generic intersections.

Sources

• Local UnsolvedMath registry, permanent ID $11300004$, exact source code catalogue entry; proposed by Greg Kuperberg in the supplied record.
• Wikipedia, List of unsolved problems in mathematics, Topology, "Quadrisecants of wild knots," revision 1366636767: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Topology. Registry access 2026-07-29; MediaWiki-source extraction.
• The summary mentions Pannwitz and Kuperberg's 1994/1996 Quadrisecants of knots context but gives no full primary bibliographic entry. No authoritative primary citation is supplied in this record.