Four-page index and linear upper bounds for ribbonlength

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 $$ \alpha_4(K) \le 2c(K), $$ 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.