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A certified upper bound for the distortion of the trefoil knot

Primarily AI-generated textHuman understanding: no partsmath.GT — Geometric Topologymath.MG — Metric Geometry

Contributed by David Niedbala Giraudin ↗

Version 1 / Nov 02, 2026 / CC BY 4.0

Abstract

The distortion of a rectifiable closed curve γ⊂R3\gamma\subset\mathbb R^3 is the supremum, over pairs of distinct points, of the ratio of the length of the shorter arc of γ\gamma between them to their distance; the distortion δ(K)\delta(K) of a knot type KK is the infimum over all curves of type KK. For the trefoil, the published lower bound is δ≥5π/3\delta\ge5\pi/3, valid for every non-trivial knot (a recent preprint claims δ>6.5321\delta>6.5321), and the only upper bound we found in the literature is the value 7.167.16 obtained by numerical simulation. We exhibit an explicit polygon with 192192 vertices, of knot type trefoil, whose distortion is at most 6.926.92. The proof is computer-assisted: the knot type is certified by an explicit piecewise-linear path of embedded polygons starting from a polygon on a torus, and the distortion by exact rational arithmetic on every pair of edges. Two separately written verifiers are supplied.

Provenance statement

The research, the computations and the writing were carried out by the author with Claude (Anthropic). All computational claims are reproduced by the ancillary Python files in anc/. The bibliography was checked against the original sources. The note has not yet been reviewed by professionals.

Tools used

Anthropic
Claude Opus

References

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Version history

  1. v1Submitted by David Niedbala GiraudinInitial depositCurrentNov 01, 2026