Average signature and 4-genus of 2-bridge knots
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866914848814661632 |
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| author | Cohen, Moshe Lowrance, Adam M. Madras, Neal Raanes, Steven |
| author_facet | Cohen, Moshe Lowrance, Adam M. Madras, Neal Raanes, Steven |
| contents | We show that the average or expected absolute value of the signatures of all 2-bridge knots with crossing number $c$ approaches $\sqrt{{2c}/π}$. Baader, Kjuchukova, Lewark, Misev, and Ray consider a model for 2-bridge knot diagrams indexed by diagrammatic crossing number $n$ and show that the average 4-genus is sublinear in $n$. We build upon this result in two ways to obtain an upper bound for the average 4-genus of a 2-bridge knot: our model is indexed by crossing number $c$ and gives a specific sublinear upper bound of $9.75c/\log c$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_17738 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Average signature and 4-genus of 2-bridge knots Cohen, Moshe Lowrance, Adam M. Madras, Neal Raanes, Steven Geometric Topology 57K10 (Primary) 05A05, 60J10 (Secondary) We show that the average or expected absolute value of the signatures of all 2-bridge knots with crossing number $c$ approaches $\sqrt{{2c}/π}$. Baader, Kjuchukova, Lewark, Misev, and Ray consider a model for 2-bridge knot diagrams indexed by diagrammatic crossing number $n$ and show that the average 4-genus is sublinear in $n$. We build upon this result in two ways to obtain an upper bound for the average 4-genus of a 2-bridge knot: our model is indexed by crossing number $c$ and gives a specific sublinear upper bound of $9.75c/\log c$. |
| title | Average signature and 4-genus of 2-bridge knots |
| topic | Geometric Topology 57K10 (Primary) 05A05, 60J10 (Secondary) |
| url | https://arxiv.org/abs/2406.17738 |