Comparing $h$-genera, Bridge-1 genera and Heegaard genera of knots
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866909595633451008 |
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| author | Qiu, Ruifeng Wang, Chao Zou, Yanqing |
| author_facet | Qiu, Ruifeng Wang, Chao Zou, Yanqing |
| contents | Let $h(K)$, $g_H(K)$, $g_1(K)$, $t(K)$ be the $h$-genus, Heegaard genus, bridge-1 genus, tunnel number of a knot $K$ in the $3$-sphere $S^3$, respectively. It is known that $g_H(K)-1=t(K)\leq g_1(K)\leq h(K)\leq g_H(K)$. A natural question arises: when do these invariants become equal?
We provide the necessary and sufficient conditions for equality and use these to show that for each integer $n\geq 1$, the following three families of knots are infinite:
\begin{eqnarray}
A_{n}=\{K\mid t(K)=n<g_1(K)\},
B_{n}=\{K\mid g_1(K)=n<h(K)\},
C_{n}=\{K\mid h(K)=n<g_H(K)\}.
\end{eqnarray} This result resolves a conjecture in \cite{Mo2}, confirming that each of these families is infinite. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_19118 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Comparing $h$-genera, Bridge-1 genera and Heegaard genera of knots Qiu, Ruifeng Wang, Chao Zou, Yanqing Geometric Topology 57K10, 57K31 Let $h(K)$, $g_H(K)$, $g_1(K)$, $t(K)$ be the $h$-genus, Heegaard genus, bridge-1 genus, tunnel number of a knot $K$ in the $3$-sphere $S^3$, respectively. It is known that $g_H(K)-1=t(K)\leq g_1(K)\leq h(K)\leq g_H(K)$. A natural question arises: when do these invariants become equal? We provide the necessary and sufficient conditions for equality and use these to show that for each integer $n\geq 1$, the following three families of knots are infinite: \begin{eqnarray} A_{n}=\{K\mid t(K)=n<g_1(K)\}, B_{n}=\{K\mid g_1(K)=n<h(K)\}, C_{n}=\{K\mid h(K)=n<g_H(K)\}. \end{eqnarray} This result resolves a conjecture in \cite{Mo2}, confirming that each of these families is infinite. |
| title | Comparing $h$-genera, Bridge-1 genera and Heegaard genera of knots |
| topic | Geometric Topology 57K10, 57K31 |
| url | https://arxiv.org/abs/2504.19118 |