Additivity of Heegaard genera under annulus sum
UDC 515.162 Let $M$ be a compact 3-manifold containing an essential annulus $A$ separating $M$ into two manifolds $M_1$ and $M_2.$ We prove that if, for each $i=1,2,$ $g(M_i) \geq 2$ and $M_i$ admits a Heegaard splitting with Heegaard distance not smaller than $2\bigl(g(M_1) + g(M_2)\bigr) - 1,$ t...
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| Datum: | 2026 |
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| Hauptverfasser: | , , |
| Format: | Artikel |
| Sprache: | Englisch |
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Institute of Mathematics, NAS of Ukraine
2026
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| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/9572 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1871646568182972416 |
|---|---|
| author | Guo, Qi-Long Dai, Qian-Hui Wang, Shu-Xin Guo, Qi-Long Dai, Qian-Hui Wang, Shu-Xin |
| author_facet | Guo, Qi-Long Dai, Qian-Hui Wang, Shu-Xin Guo, Qi-Long Dai, Qian-Hui Wang, Shu-Xin |
| author_institution_txt_mv | [
{
"author": "Qi-Long Guo",
"institution": "College of Science, China University of Petroleum-Beijing, Beijing, China"
},
{
"author": "Qian-Hui Dai",
"institution": "College of Science, China University of Petroleum-Beijing, Beijing, China"
},
{
"author": "Shu-Xin Wang",
"institution": "School of Mathematics, Liaoning Normal University, Dalian, Liaoning, China"
}
] |
| author_sort | Guo, Qi-Long |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2026-07-24T15:24:12Z |
| description | UDC 515.162
Let $M$ be a compact 3-manifold containing an essential annulus $A$ separating $M$ into two manifolds $M_1$ and $M_2.$ We prove that if, for each $i=1,2,$ $g(M_i) \geq 2$ and $M_i$ admits a Heegaard splitting with Heegaard distance not smaller than $2\bigl(g(M_1) + g(M_2)\bigr) - 1,$ then $g(M) = g(M_1) + g(M_2).$ As a corollary, we present a sufficient condition for the superadditivity of the tunnel number of knots under connected sum. |
| doi_str_mv | 10.3842/umzh.v78i7-8.9572 |
| first_indexed | 2026-07-25T01:01:01Z |
| format | Article |
| fulltext | |
| id | umjimathkievua-article-9572 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-07-25T01:01:01Z |
| publishDate | 2026 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | |
| spelling | umjimathkievua-article-95722026-07-24T15:24:12Z Additivity of Heegaard genera under annulus sum Additivity of Heegaard genera under annulus sum Guo, Qi-Long Dai, Qian-Hui Wang, Shu-Xin Guo, Qi-Long Dai, Qian-Hui Wang, Shu-Xin Heegaard genus, Heegaard distance, annulus sum, Topology UDC 515.162 Let $M$ be a compact 3-manifold containing an essential annulus $A$ separating $M$ into two manifolds $M_1$ and $M_2.$ We prove that if, for each $i=1,2,$ $g(M_i) \geq 2$ and $M_i$ admits a Heegaard splitting with Heegaard distance not smaller than $2\bigl(g(M_1) + g(M_2)\bigr) - 1,$ then $g(M) = g(M_1) + g(M_2).$ As a corollary, we present a sufficient condition for the superadditivity of the tunnel number of knots under connected sum. УДК 515.162 Адитивність родів Гіґаарда щодо кільцевої суми Нехай $M$ – компактний 3-многовид, що містить суттєве кільце $A,$ яке розділяє $M$ на два многовиди $M_1$ і $M_2.$ Доведено, що якщо для кожного $i=1,2$ виконується умова $g(M_i)\geq 2$ і многовид $M_i$ допускає розщеплення Гіґаарда з відстанню Гіґаарда не меншою, ніж $2\bigl(g(M_1)+g(M_2)\bigr)-1,$ то $g(M)=g(M_1)+g(M_2).$ Як наслідок отримано достатню умову суперадитивності тунельного числа вузлів у зв'язній сумі. Institute of Mathematics, NAS of Ukraine 2026-07-24 Article Article https://umj.imath.kiev.ua/index.php/umj/article/view/9572 10.3842/umzh.v78i7-8.9572 Ukrains’kyi Matematychnyi Zhurnal; Vol. 78 No. 7-8 (2026); 603–604 Український математичний журнал; Том 78 № 7-8 (2026); 603–604 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/9572/10685 Copyright (c) 2026 Qi-Long Guo, Qian-Hui Dai, Shu-Xin Wang |
| spellingShingle | Guo, Qi-Long Dai, Qian-Hui Wang, Shu-Xin Guo, Qi-Long Dai, Qian-Hui Wang, Shu-Xin Additivity of Heegaard genera under annulus sum |
| title | Additivity of Heegaard genera under annulus sum |
| title_alt | Additivity of Heegaard genera under annulus sum |
| title_full | Additivity of Heegaard genera under annulus sum |
| title_fullStr | Additivity of Heegaard genera under annulus sum |
| title_full_unstemmed | Additivity of Heegaard genera under annulus sum |
| title_short | Additivity of Heegaard genera under annulus sum |
| title_sort | additivity of heegaard genera under annulus sum |
| topic_facet | Heegaard genus Heegaard distance annulus sum, Topology |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/9572 |
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