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arXiv · 2504.19118

Comparing $h$-genera, Bridge-1 genera and Heegaard genera of knots

Ruifeng Qiu·Chao Wang·Yanqing Zou
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Abstract

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.

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Ruifeng Qiu, Chao Wang, Yanqing Zou. 2025-04-27. Comparing $h$-genera, Bridge-1 genera and Heegaard genera of knots. https://arxiv.org/abs/2504.19118

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