Crossing Numbers of Knots on Closed Surfaces
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866913151663996928 |
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| author | Ozawa, Makoto |
| author_facet | Ozawa, Makoto |
| contents | Let c(K;F) denote the surface crossing number of a knot K with respect to a closed connected surface F in S^3. We relate c(K;F) to the tunnel number t(K) and to the Heegaard deficiency delta(F)=g(M_1;F)+g(M_2;F)-g(F), where S^3=M_1 union_F M_2. The zero-crossing case gives a structural obstruction: if c(K;F)=0, then t(K) <= delta(F). Conversely, if t(K)>delta(F), then c(K;F) >= 2(t(K)-delta(F))+1. Thus the Heegaard deficiency of F measures the amount of tunnel complexity that can be absorbed by F without producing crossings. The proof combines a surface ascending-number estimate, a bridge-number estimate for surface diagrams, and an amalgamation argument for Heegaard splittings relative to F. We also construct connected-sum families showing that the lower bound has the correct linear order. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_21659 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Crossing Numbers of Knots on Closed Surfaces Ozawa, Makoto Geometric Topology Let c(K;F) denote the surface crossing number of a knot K with respect to a closed connected surface F in S^3. We relate c(K;F) to the tunnel number t(K) and to the Heegaard deficiency delta(F)=g(M_1;F)+g(M_2;F)-g(F), where S^3=M_1 union_F M_2. The zero-crossing case gives a structural obstruction: if c(K;F)=0, then t(K) <= delta(F). Conversely, if t(K)>delta(F), then c(K;F) >= 2(t(K)-delta(F))+1. Thus the Heegaard deficiency of F measures the amount of tunnel complexity that can be absorbed by F without producing crossings. The proof combines a surface ascending-number estimate, a bridge-number estimate for surface diagrams, and an amalgamation argument for Heegaard splittings relative to F. We also construct connected-sum families showing that the lower bound has the correct linear order. |
| title | Crossing Numbers of Knots on Closed Surfaces |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2602.21659 |