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

Ruifeng Qiu, Chao Wang, Yanqing Zou

TL;DR

The paper analyzes how the knot invariants $t(K)$, $g_1(K)$, $h(K)$, and $g_H(K)$ relate and when equalities occur. It provides necessary and sufficient conditions for equality via complementary disk pairs and spanning annuli on Heegaard surfaces, then uses these to prove that the three Morimoto-style families $A_n$, $B_n$, and $C_n$ are each infinite. The authors combine Heegaard distance, mapping class group techniques, and disk/annulus combinatorics to characterize equality cases and to construct infinite families with prescribed genus gaps. The results yield a clearer structural picture of how these genus invariants interact across classes of knots, including torus, cable, and high-distance hyperbolic examples.

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.

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

TL;DR

The paper analyzes how the knot invariants , , , and relate and when equalities occur. It provides necessary and sufficient conditions for equality via complementary disk pairs and spanning annuli on Heegaard surfaces, then uses these to prove that the three Morimoto-style families , , and are each infinite. The authors combine Heegaard distance, mapping class group techniques, and disk/annulus combinatorics to characterize equality cases and to construct infinite families with prescribed genus gaps. The results yield a clearer structural picture of how these genus invariants interact across classes of knots, including torus, cable, and high-distance hyperbolic examples.

Abstract

Let , , , be the -genus, Heegaard genus, bridge-1 genus, tunnel number of a knot in the -sphere , respectively. It is known that . 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 , 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.
Paper Structure (5 sections, 6 theorems, 7 equations, 1 figure)

This paper contains 5 sections, 6 theorems, 7 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The proof of Theorem \ref{['theorem1']}
  3. $A_{n}$ is infinite.
  4. $B_{n}$ is infinite.
  5. $C_{n}$ is infinite.

Key Result

Theorem 1.1

(I) Given $i\geq 0$ and $l>0$, $g_i(K)=g_{i+l}(K)+l$ holds if and only if there exists a $\Sigma_g$ realizing $g_i(K)$ with $l$ complementary pairs $(D_j,D_j')$, $1\leq j\leq l$, where the $l$ bouquets of circles $\partial D_1\cup\partial D_1',\ldots,\partial D_l\cup\partial D_l'$ are pairwise disjo

Figures (1)

  • Figure 1: The disk pair $(D,D')$ and the annulus $A$.

Theorems & Definitions (14)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Theorem 4.1
  • ...and 4 more