Bounded ribbonlength for knot families and multi-twist Möbius bands

TL;DR

This work develops a unified framework for bounding folded ribbonlength and aspect ratios of folded ribbon knots via accordion and escape-accordion folds. By relating folded ribbon knots to smooth embedded paper bands and exploiting a theta-optimized accordion construction, the authors prove a universal upper bound of $3\sqrt{3}+\epsilon$ on the infimal aspect ratio of any multi-twist Möbius band, and derive uniform folded ribbonlength bounds for infinite families such as $(2,q)$-torus knots and twist knots, with Rib values approaching constants independent of crossing number. In particular, they establish Rib$([T(2,q)])\le 8\sqrt{3}+\epsilon$ and Rib$([T_n])$ bounded by $9\sqrt{3}+2+\epsilon$ (odd) or $8\sqrt{3}+2+\epsilon$ (even), yielding a concrete $\alpha=0$ in the ribbonlength–crossing-number inequality. The methods highlight a structural distinction between small- and large-crossing knots and connect geometric band embeddings to folded-ribbon realizations, with implications for the efficiency of knot representations in flat ribbon models and for understanding the limits of the ribbonlength crossing-number problem.

Abstract

Take a thin, rectangular strip of paper, add in an odd number of half-twists, then join the ends together. This gives a multi-twist paper Möbius band. We prove that any multi-twist paper Möbius band can be constructed so the aspect ratio of the rectangle is $3\sqrt{3}+ε$ for any $ε>0$. We could also take the thin, rectangular strip of paper and tie a knot in it, then join the ends and fold flat in the plane. This creates a folded ribbon knot. We apply the techniques used to prove the multi-twist paper Möbius band result to $(2,q)$ torus knots and twist knots. We prove that any $(2,q)$-torus knot can be constructed so that the folded ribbonlength $\leq 13.86$. We prove that any twist knot can be constructed so that the folded ribbonlength is $\leq 17.59$. Both of these results give the lower bound for the ribbonlength crossing number problem which relates the infimal folded ribbonlength of a knot type $[K]$ to its crossing number $\text{Cr}(K)$. That is, we have shown $α=0$ in the equation $c\cdot \text{Cr}(K)^α\leq \text{Rib}([K])$, where $c$ is a constant.

Figures (31)

• Figure 1: On the left, a Möbius band made of one half-twist, and in the center one made of three half-twists. On the right, a folded ribbon trefoil knot.
• Figure 2: We create a folded ribbon trefoil knot (right) by adding a ribbon with fold lines to a polygonal trefoil knot diagram (left).
• Figure 3: The distinction between an underfold (left image) and an overfold (right image) depends on the orientation of the link. Image re-used with permission from DKTZ.
• Figure 4: On the left and center, a side and top view of a pseudo-fold that replaces a fold line. On the right, a short piece of the knot diagram near a vertex is approximated by a short curved segment.
• Figure 5: On the left, a sequence of accordion folds with fold angle $\pi/2$. In the center, an escape accordion. On the right, the crease pattern for an escape accordion.

Theorems & Definitions (38)

• Remark 1
• Conjecture 2
• Lemma 3: Approximation
• Definition 4: Accordion fold
• Definition 5: $V$-unit, escape accordion
• Lemma 6
• proof
• Corollary 7
• proof
• Proposition 9