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Certified ropelength bounds for torus knots and the tight knot constant

Primarily AI-generated textHuman understanding: no partsmath.GT — Geometric Topology

Contributed by David Niedbala Giraudin ↗

Version 1 / Nov 02, 2026 / CC BY 4.0

Abstract

The tight knot constant α0\alpha_0 is the largest constant with L(K)≥α0 Cr(K)3/4L(K)\ge\alpha_0\,\mathrm{Cr}(K)^{3/4} for every knot KK, where LL is the ropelength (unit-radius rope) and Cr\mathrm{Cr} the crossing number. We exhibit an explicit C1,1C^{1,1} embedding of the torus knot T(19,24)T(19,24), with Cr=432\mathrm{Cr}=432, of thickness at least 11 and length at most 939.984939.984. Hence α0≤9.9200\alpha_0\le 9.9200. The published bound is 10.7610.76. 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 thickness by exact integer and rational arithmetic. Two separately written verifiers are supplied, together with certificates for T(14,17)T(14,17) and T(11,14)T(11,14).

Provenance statement

The research and the writing were carried out by the author with Claude (Anthropic). The proofs were checked by independent AI reviews and by exact computation. No professional mathematician has reviewed the note yet.

Tools used

Anthropic
Claude Opus

References

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Version history

  1. v1Submitted by David Niedbala GiraudinInitial depositCurrentNov 01, 2026