A diagrammatic approach to the three-page index
Hyungkee Yoo
TL;DR
The paper sharpens the known linear bound for the three-page index by proving that, for any non-split, nontrivial link $L$ other than the Hopf link, $α_3(L) ≤ 3c(L) - 1$, and that equality occurs only for split unions of Hopf links. It achieves this via a constructive, diagrammatic approach that builds circular three-page presentations from reduced diagrams using binding circles defined by extended spanning trees in the induced cell complex, thereby tightly controlling the number of arcs. A graph-theoretic reinterpretation through the dual diagram graph highlights the role of non-separating independent sets in improving bounds and motivates conjectures for further tightening, connecting link complexity to planar graph structure. Overall, the work provides a complete characterization of the equality case and opens new avenues for improving upper bounds with combinatorial graph techniques.
Abstract
The three-page index $α_3(L)$ is an invariant that measures the complexity of representing a link $L$ in a three-page book. It is known that $α_3(L)$ 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 $α_3(L)=3c(L)$.