Four-page index and linear upper bounds for ribbonlength
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866917275410366464 |
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| author | Yoo, Hyungkee |
| author_facet | Yoo, Hyungkee |
| contents | 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 $2c(K)$ arcs. Hence $$ α_4(K) \le 2c(K), $$ 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 $$ \mathrm{Rib}(K) \le 2c(K). $$ This improves the previously known general linear upper bound for ribbonlength and provides a diagrammatic method for estimating ribbonlength. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_13973 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Four-page index and linear upper bounds for ribbonlength Yoo, Hyungkee Geometric Topology 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 $2c(K)$ arcs. Hence $$ α_4(K) \le 2c(K), $$ 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 $$ \mathrm{Rib}(K) \le 2c(K). $$ This improves the previously known general linear upper bound for ribbonlength and provides a diagrammatic method for estimating ribbonlength. |
| title | Four-page index and linear upper bounds for ribbonlength |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2602.13973 |