Handle number is not always realized by a minimal genus Seifert surface
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866910689483816960 |
|---|---|
| author | Baker, Kenneth L. Manjarrez-Gutiérrez, Fabiola |
| author_facet | Baker, Kenneth L. Manjarrez-Gutiérrez, Fabiola |
| contents | We construct genus one knots whose handle number is only realized by Seifert surfaces of non-minimal genus. These are counterexamples to the conjecture that the Seifert genus of a knot is its Morse-Novikov genus. As the Morse-Novikov genus may be greater than the Seifert genus, we define the genus $g$ Morse-Novikov number $MN_g(L)$ as the minimum handle number among Seifert surfaces for $L$ of genus $g$. Since, as we further show, the Morse-Novikov genus and the minimal genus Morse-Novikov number are additive under connected sum of knots, it then follows that there exists examples for which the discrepancies between Seifert genus and Morse-Novikov genus and between the Morse-Novikov number and the minimal genus Morse-Novikov number can be made arbitrarily large. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_05177 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Handle number is not always realized by a minimal genus Seifert surface Baker, Kenneth L. Manjarrez-Gutiérrez, Fabiola Geometric Topology 57K10, 57K35, 57K99 We construct genus one knots whose handle number is only realized by Seifert surfaces of non-minimal genus. These are counterexamples to the conjecture that the Seifert genus of a knot is its Morse-Novikov genus. As the Morse-Novikov genus may be greater than the Seifert genus, we define the genus $g$ Morse-Novikov number $MN_g(L)$ as the minimum handle number among Seifert surfaces for $L$ of genus $g$. Since, as we further show, the Morse-Novikov genus and the minimal genus Morse-Novikov number are additive under connected sum of knots, it then follows that there exists examples for which the discrepancies between Seifert genus and Morse-Novikov genus and between the Morse-Novikov number and the minimal genus Morse-Novikov number can be made arbitrarily large. |
| title | Handle number is not always realized by a minimal genus Seifert surface |
| topic | Geometric Topology 57K10, 57K35, 57K99 |
| url | https://arxiv.org/abs/2411.05177 |