New Upper Bounds on the Ribbonlength of Alternating Links with Bipartite Dual Graphs

TL;DR

This paper derives a new upper bound for the folded ribbonlength of alternating links whose diagrams have bipartite dual graphs, showing $Rib(L)\\le \\sqrt{3}\\, c(L)$. It develops a framework based on rotated circular three-page presentations and equilateral-triangle foldings to translate diagrammatic data into folded ribbon realizations, and proves a key length lemma linking arc counts to ribbonlength. The results yield the exact value $Rib(Hopf)=2\\sqrt{3}$ and improved bounds for several small-crossing knots, with explicit bounds for trefoil, certain twist knots, and the $(2,2,2)$-pretzel knot, while showing the bound is preserved under connected sums. Overall, the work advances Kusner-type linear bounds by identifying a broader class of links for which tight ribbonlength bounds can be established and connects combinatorial diagram properties to geometric realizations.

Abstract

The ribbonlength of a link is a geometric invariant defined as the infimum of the ratio of the length to the width of a folded ribbon realization of the link. In this paper, we prove that if an alternating link admits an alternating diagram with a bipartite dual graph, then its ribbonlength satisfies $$ \mathrm{Rib}(L) \le \sqrt{3} \, c(L). $$ Using this result, we present improved upper bounds on the ribbonlength for several knots and links with small crossing numbers, and determines the exact ribbonlength of the Hopf link to be $2\sqrt{3}$.

Figures (13)

• Figure 1: Two folded ribbon knots of the trivial knot
• Figure 2: A folded ribbon link of the Hopf link
• Figure 3: A new folded ribbon Hopf link
• Figure 4: Construction process for the folded ribbon Hopf link
• Figure 5: An arc presentation and a three-page presentation of the trefoil knot

Theorems & Definitions (18)

• Definition
• Definition
• Theorem 1
• Theorem 2
• Definition
• Lemma 3
• Definition
• Lemma 4
• proof
• Lemma 5