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arXiv · 2608.07605

Random Knots via Stiefel manifolds

Alexander Kolpakov·Igor Rivin
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Abstract

A fixed simplex, randomly projected into three dimensions and joined in Hamiltonian order, produces a rich and unusually tractable model of random stick knots. We prove that Gaussian projections and Haar-random Stiefel projections have exactly the same knot-type law, despite having different metric shapes, and that at every stick budget the model gives positive probability to precisely the knot types realizable with that many sticks. Its linear-algebraic structure yields an exact marginal distance law, an exact mean planar-crossing count, and crossing concentration, while in the first nontrivial six-stick case the complete tetrahedral sign pattern gives an exact unknot-versus-handed-trefoil classifier for every generic sample. The result is a direct bridge from random projections and finite sign geometry to the topology of random knots.

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Alexander Kolpakov, Igor Rivin. 2026-08-06. Random Knots via Stiefel manifolds. https://arxiv.org/abs/2608.07605

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