Four-Page Index and Linear Upper Bounds for Ribbonlength

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 [Formula: see text] arcs. Hence [Formula: see text] 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 [Formula: see text] This improves the previously known general linear upper bound for ribbonlength and provides a diagrammatic method for estimating ribbonlength.