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We fix the radius of the helix, and study the coiling in both directions: as the helix unravels to a straight line, and as it coils infinitely tight. Specifically, we study the arclength-rescaled Möbius energy density, which is a naturally tractable quantity under the Möbius energy's chord-arc comparison of inverse-square laws.
The asymptotics of the uncoiling helix, corresponding to an energy decay, can be proven with a short chain of estimates. However, the asymptotics of the helix as it coils infinitely tight, blowing up the energy, is a much more involved calculation. Our strategy for calculating the asymptotics, initially reminiscent of the work by Kim-Kusner, begins with a meromorphic extension of the integrand. However, proving the asymptotic equivalence requires a fundamentally distinct strategy because our integrand has infinitely many poles.
keywords: Möbius energy, helix, complex asymptotics, knot energies, physical knot theory, coiling, curves
From: Max Lipton [view email]
[v1]
Tue, 12 May 2026 23:15:15 UTC (555 KB)
[v2]
Fri, 29 May 2026 15:55:02 UTC (555 KB)
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