A diagrammatic approach to the three-page index

TL;DR

This paper studies the three-page index of links, providing a diagrammatic method to refine bounds on this invariant and characterizing links that attain the maximum bound, especially distinguishing the Hopf link.

Contribution

It introduces a diagrammatic construction for three-page presentations that refines bounds on the three-page index and characterizes links achieving the maximum bound.

Findings

For non-split, nontrivial links other than the Hopf link,  c(L)-1 bounds the three-page index.

The paper characterizes exactly when   c(L) holds, identifying the Hopf link as the unique case.

Provides a new diagrammatic approach to analyze the complexity of link representations.

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, \[ \alpha_3(L)\le 3c(L)-1, \] and hence characterize completely the links for which $\alpha_3(L)=3c(L)$.