A diagrammatic approach to the three-page index

Abstract

The three-page index ${\alpha }_3(L)$ is an invariant that measures the complexity of representing a link $L$ in a three-page book. It is known that ${\alpha }_3(L)$ admits a linear upper bound in terms of the crossing number, with equality realized by the Hopf link. In this paper, we investigate the equality case of this bound from a diagrammatic viewpoint. Starting from a reduced link diagram, we construct three-page presentations via binding circles arising as boundaries of suitable contractible subcomplexes of the induced cell decomposition of the $2$-sphere. This approach allows a refined control of the number of arcs in the resulting three-page presentation. As a consequence, we prove that for any non-split, nontrivial link $L$ other than the Hopf link, \[ α_3(L)\le 3c(L)-1, \] and hence characterize completely the links for which ${\alpha }_3(L)=3c(L)$.