Limit superior of minimal aspect ratios for many‑twist paper bands

Determine the value of lim sup_{n→∞} λ_n, where λ_n denotes the minimal aspect ratio among all embedded paper bands of width 1 with n half‑twists (interpreted as paper Möbius bands for odd n and paper annuli for even n, as defined by the linking number conditions in Definition 2), to characterize the asymptotic behavior of optimal aspect ratios for increasing twist number.

Background

The paper studies the minimal aspect ratio achievable by embedded paper bands with a prescribed number of half‑twists. Prior results establish exact values for small n (e.g., √3 for the 1‑twist Möbius band and 2 for the 2‑twist annulus), and a conjectured value for n=3. The author constructs families of folded ribbon knots yielding an O(1) upper bound (specifically <8) for all n, showing that asymptotic growth does not exceed a constant.

However, the construction does not determine the precise asymptotic value of the optimal aspect ratios as n increases. The open problem asks for the exact lim sup of the sequence of minimal aspect ratios {λ_n}, which would establish whether the constant upper bound is tight and quantify the asymptotic optimal geometry for many‑twist bands.