Four-page index and linear upper bounds for ribbonlength

TL;DR

This work links geometric ribbonlength to a new diagrammatic invariant, the four-page index $\alpha_4(K)$, by leveraging a Kauffman state determined from spanning trees to produce a binding circle and a four-page presentation. It proves the central bound $\mathrm{Rib}(K) \le \alpha_4(K)$ and establishes $\alpha_4(K) \le 2c(K)$, with strict inequality for non-alternating links, yielding the linear bound $\mathrm{Rib}(K) \le 2c(K)$ and providing a practical diagrammatic method to estimate ribbonlength. The paper also develops circular four-page presentations via binding circles and demonstrates sharpness of the coefficient, alongside concrete examples. Overall, the results give a sharper, diagrammatic linear bound for ribbonlength and connect open-book presentations to quantitative knot complexity.

Abstract

We introduce the four-page index of a knot or link as a presentation invariant arising from embeddings in a four-page open book decomposition. Using spanning trees of the checkerboard graph of a reduced non-split diagram, we construct a Kauffman state consisting of a single state circle. The associated Eulerian tour of the underlying 4-valent plane graph determines a binding circle intersecting each edge exactly once, producing a four-page presentation with at most $2c(K)$ arcs. Hence $$ α_4(K) \le 2c(K), $$ with strict inequality in the non-alternating case. We further prove that ribbonlength is bounded above by the four-page index, and therefore obtain the linear bound $$ \mathrm{Rib}(K) \le 2c(K). $$ This improves the previously known general linear upper bound for ribbonlength and provides a diagrammatic method for estimating ribbonlength.

Figures (6)

• Figure 1: A folded ribbon knot of the trefoil knot
• Figure 2: Four-page presentations of Hopf link, trefoil knot, and figure eight knot
• Figure 3: Local configurations near a binding point and the corresponding ribbon pieces
• Figure 4: A four-page presentation and its circular four-page presentation with an ordering of the pages
• Figure 5: A binding circle for circular four-page presentation

Theorems & Definitions (20)

• Definition 1
• Definition 2
• Theorem 1
• Theorem 2
• Corollary 3
• Definition 3
• Definition 4
• proof : Proof of Theorem \ref{['thm:rib_alpha4']}
• Definition 5
• Lemma 4