A diagrammatic approach to the three-page index
Fuente:
arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866909994520150016 |
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| author | Yoo, Hyungkee |
| author_facet | Yoo, Hyungkee |
| contents | 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 $α_3(L)=3c(L)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_12846 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A diagrammatic approach to the three-page index Yoo, Hyungkee Geometric Topology 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 $α_3(L)=3c(L)$. |
| title | A diagrammatic approach to the three-page index |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2601.12846 |