Linear Upper Bounds on the Ribbonlength of Knots and Links

A knotted ribbon is one of physical aspect of a knot. A folded ribbon knot is a depiction of a knot obtained by folding a long and thin rectangular strip to become flat. The ribbonlength of a knot type can be defined as the minimum length required to tie the given knot type as a folded ribbon knot. The ribbonlength has been conjectured to grow linearly or sub-linearly with respect to a minimal crossing number. Several knot types provide evidence that this conjecture is true, but there is no proof for general cases. In this paper, we show that for any knot or link, the ribbonlength is bounded by a linear function of the crossing number. In more detail, $$ \text{Rib}(K) \leq 2.5 c(K)+1. $$ for a knot or link $K$. Our approach involves binary grid diagrams and bisected vertex leveling techniques.

💡 Research Summary

The paper addresses the long‑standing problem of bounding the ribbonlength of a knot or link in terms of its minimal crossing number. Ribbonlength, denoted Rib(K), is defined as the infimum of the lengths of a thin rectangular strip (of fixed width) required to realize a given knot type K as a folded ribbon knot—a planar diagram whose core consists of straight segments with prescribed over‑/under‑crossings. Earlier results gave polynomial upper bounds such as Rib(K) ≤ 2 c(K)² + 6 c(K) + 4 (Tian) and later refinements of order c(K)³⁄² for arbitrary knots (Denne). However, a conjecture attributed to Diao and Kusner predicts a linear relationship: there should exist constants c₁, c₂ with c₁·c(K) ≤ Rib(K) ≤ c₂·c(K). The upper‑linear part (β = 1) had only been proved for special families (e.g., 2‑bridge knots).

The authors prove the conjectured linear upper bound for all knots and links, establishing the explicit inequality

 Rib(K) ≤ 2.5 c(K) + 1.

The proof introduces two novel combinatorial tools:

1. Binary Grid Diagrams – A restricted form of the usual grid diagram where each horizontal segment crosses at most one vertical segment. This restriction forces the diagram to decompose into six block types (B₁, B₂, B₃ and their rotated versions ˚B₁, ˚B₂, ˚B₃). Each block contains exactly one horizontal segment together with a collection of vertical pieces.

2. Bisected Vertex Leveling – A planar isotopy of the underlying 4‑valent graph of a knot diagram that satisfies three conditions: every vertex lies between two consecutive horizontal lines, edges have no interior maxima or minima in the vertical direction, and each horizontal line cuts the graph into two connected components. For a minimal crossing diagram (which has no nugatory crossings), such a leveling always exists.

Using the bisected vertex leveling, the diagram can be sliced into portions T₀,…,T₄, each corresponding to a specific combination of block types. The authors then prove two lemmas:

• Lemma 4 shows that a pair of consecutive blocks (an upper B₁ or ˚B₁ block and a lower ˚B₃ block) can be swapped without changing the knot type, allowing reordering of blocks.
• Lemma 5 provides an algorithm to replace every B₂ and B₃ block by a combination of B₁ and ˚B₃ blocks, preserving the total number of blocks of each original type.

After applying Lemma 5, the diagram consists only of B₁, ˚B₁, and ˚B₃ blocks. Lemma 4 is then used to push all ˚B₃ blocks to the top of the diagram, yielding a new diagram D′ whose upper part contains only ˚B₃ blocks and whose lower part contains only B₁ and ˚B₁ blocks.

The next step translates each block into a paper plane, a three‑fold folded ribbon piece. In a paper plane, the horizontal segment becomes the central ribbon strip, while the vertical segments become “wings”—flexible extensions that can be lengthened arbitrarily to connect adjacent planes. The lower part of D′ is built inductively: starting from the bottommost block (always a ˚B₁), a “pile of k paper planes” is formed by inserting a new paper plane into the available gaps (insertable spaces) between existing wings. Each insertion doubles the number of wings, guaranteeing that the total ribbon length contributed by the lower part is at most twice the number of blocks it contains.

The upper part, consisting solely of ˚B₃ blocks, is treated similarly, yielding another collection of paper planes whose total length is bounded by twice the number of ˚B₃ blocks. Consequently,

 Rib(K) ≤ 2 (b₁ + b₂ + b₃ + ˚b₁),

where bᵢ and ˚b₁ count the blocks of each type in the original binary grid diagram. By analyzing how many blocks are needed to represent a diagram with c(K) crossings, the authors show that

 b₁ + b₂ + b₃ + ˚b₁ ≤ c(K) + ½ c(K) + ½,

which simplifies to the final linear bound Rib(K) ≤ 2.5 c(K) + 1.

Significance

• The result settles the upper‑linear part of the Diao‑Kusner conjecture for all knots and links, confirming that ribbonlength grows at most linearly with crossing number.
• The binary grid diagram and bisected vertex leveling provide a new combinatorial framework that could be applied to other knot complexity measures such as stick number or rope length.
• From a physical perspective, the construction gives explicit, algorithmic ways to fabricate folded‑ribbon realizations of arbitrary knots with a length guarantee, which may be useful in DNA nanotechnology, polymer physics, and the design of ribbon‑based mechanical devices.

Overall, the paper delivers a clear, constructive proof of a linear upper bound on ribbonlength, introduces powerful diagrammatic tools, and opens avenues for both theoretical and applied investigations in knot theory.