Quadrisecants of Wild Knots Conjecture

[franzhusch]

What is the conjecture

A wild knot is a knot (a closed curve homeomorphic to $S^1$ embedded in $\mathbb{R}^3$) that is not a smooth or piecewise-linear knot. Formally, it is a locally infinite knot that may have wildly oscillating behavior near certain points.

A quadrisecant of a knot $K \subset \mathbb{R}^3$ is a straight line that intersects $K$ at exactly four distinct points.

Conjecture: Every wild knot possesses infinitely many quadrisecants.

(This description may contain subtle errors especially on more complex problems; for exact details, refer to the sources.)

Sources:

• https://arxiv.org/abs/math/9712205, https://en.wikipedia.org/wiki/Wild_knot, https://mathworld.wolfram.com/WildKnot.html, https://en.wikipedia.org/wiki/Knot_theory, https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics#Topology

Prerequisites needed

Formalizability Rating: 4/5 (0 is best) (as of 2026-02-06)

Building blocks (1-3; from search results):

• Topological embeddings of circles in Euclidean space (manifold theory)
• Basic point-set topology and metric spaces

Missing pieces (exactly 2; unclear/absent from search results):

• Formalization of wild knots and their distinctive topological properties (local infiniteness, non-smoothness)
• Definition and theory of secant lines and their intersection multiplicities with topological curves

Rating justification: Knot theory is not substantially developed in Mathlib, and wild knots are a specialized concept in topology. Formalizing the statement would require building significant infrastructure for topological knot representations and defining quadrisecants precisely in the topological setting.

AMS categories

• ams-57

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• I plan on adding this conjecture to the repository
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